Numerical Analysis Homework #5
(b) 1980-01-01T00:00:00+00:00 image/svg+xml Matplotlib v3.7.0, https://matplotlib.org/ (c) 1980-01-01T00:00:00+00:00 image/svg+xml Matplotlib v3.7.0, https://matplotlib.org/ (d) 1980-01-01T00:00:00+00:00 image/svg+xml Matplotlib v3.7.0, https://matplotlib.org/ (a) 1980-01-01T00:00:00+00:00 image/svg+xml Matplotlib v3.7.0, https://matplotlib.org/ (b) 1980-01-01T00:00:00+00:00 image/svg+xml Matplotlib v3.7.0, https://matplotlib.org/ (c) 1980-01-01T00:00:00+00:00 image/svg+xml Matplotlib v3.7.0, https://matplotlib.org/ (d) 1980-01-01T00:00:00+00:00 image/svg+xml Matplotlib v3.7.0, https://matplotlib.org/ 1980-01-01T00:00:00+00:00 image/svg+xml Matplotlib v3.7.0, https://matplotlib.org/
(a) Eq 3.23 gives
u ( t ) = [ 2 ( u 0 − u 3 ) + ( ( u 1 − u 0 ) + ( u 3 − u 2 ) ) ] t 3 + [ 3 ( u 3 − u 0 ) − ( ( u 3 − u 2 ) + 2 ( u 1 − u 0 ) ) ] t 2 + ( u 1 − u 0 ) t + u 0 = [ u 0 + u 1 − ( u 2 + u 3 ) ] t 3 + [ u 2 + 2 u 3 − ( u 0 + 2 u 1 ) ] t 2 + ( u 1 − u 0 ) t + u 0
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
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% draw a box outlining some math
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% f \onall X = { f(x) : x ∈ X }
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% shorthands: various brackets
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% reverse \mapsto (FIXME: make better)
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% reverse-order composition
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% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
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\newcommand{\embeds}{ \hookrightarrow }
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\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
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\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\begin{align*} u(t) &= [ 2(u_0 - u_3) + ((u_1 - u_0) + (u_3 - u_2))]t^3 + [3(u_3 - u_0) - ((u_3 - u_2) + 2(u_1 - u_0))]t^2 + (u_1 - u_0) t + u_0
\\&= [ u_0 + u_1 - (u_2 + u_3)]t^3 + [u_2 + 2u_3 - (u_0 + 2u_1)]t^2 + (u_1 - u_0) t + u_0
\end{align*} u ( t ) = [ 2 ( u 0 − u 3 ) + (( u 1 − u 0 ) + ( u 3 − u 2 ))] t 3 + [ 3 ( u 3 − u 0 ) − (( u 3 − u 2 ) + 2 ( u 1 − u 0 ))] t 2 + ( u 1 − u 0 ) t + u 0 = [ u 0 + u 1 − ( u 2 + u 3 )] t 3 + [ u 2 + 2 u 3 − ( u 0 + 2 u 1 )] t 2 + ( u 1 − u 0 ) t + u 0
and likewise
v ( t ) = [ v 0 + v 1 − ( v 2 + v 3 ) ] t 3 + [ v 2 + 2 v 3 − ( v 0 + 2 v 1 ) ] t 2 + ( v 1 − v 0 ) t + v 0
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
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% draw a box outlining some math
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% f \onall X = { f(x) : x ∈ X }
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% shorthands: various brackets
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% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
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% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
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% hook arrows
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\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
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\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\begin{align*} v(t) &= [ v_0 + v_1 - (v_2 + v_3)]t^3 + [v_2 + 2v_3 - (v_0 + 2v_1)]t^2 + (v_1 - v_0) t + v_0
\end{align*} v ( t ) = [ v 0 + v 1 − ( v 2 + v 3 )] t 3 + [ v 2 + 2 v 3 − ( v 0 + 2 v 1 )] t 2 + ( v 1 − v 0 ) t + v 0 (b) B 3 f ( x ) = ∑ k = 0 3 ( 3 k ) f ( k 3 ) x k ( 1 − x ) 3 − k = [ ( 3 0 ) f ( 0 3 ) x 0 ( 1 − x ) 3 − 0 ] + [ ( 3 1 ) f ( 1 3 ) x 1 ( 1 − x ) 3 − 1 ] + [ ( 3 2 ) f ( 2 3 ) x 2 ( 1 − x ) 3 − 2 ] + [ ( 3 3 ) f ( 3 3 ) x 3 ( 1 − x ) 3 − 3 ] = [ f ( 0 ) ( 1 − x ) 3 ] + [ 3 f ( 1 3 ) x ( 1 − x ) 2 ] + [ 3 f ( 2 3 ) x 2 ( 1 − x ) ] + [ f ( 1 ) x 3 ] = [ u 0 ( 1 − x ) 3 ] + [ 3 u 1 x ( 1 − x ) 2 ] + [ 3 u 2 x 2 ( 1 − x ) ] + [ u 3 x 3 ] = [ 3 u 1 + u 3 − ( u 0 + 3 u 2 ) ] x 3 + [ 3 u 0 + 3 u 2 − 6 u 1 ] x 2 + [ 3 ( u 1 − u 0 ) ] x + u 0
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
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\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
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% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\begin{align*} B_3^f(x) &= \sum_{k=0}^3 \binom 3 k f\left(\frac k 3\right) x^k (1-x)^{3-k}
\\&= \begin{aligned}
& \left[ \binom 3 0 f \left(\frac 0 3\right) x^0 (1-x)^{3-0} \right]
+ \left[ \binom 3 1 f \left(\frac 1 3\right) x^1 (1-x)^{3-1} \right]
\\& + \left[ \binom 3 2 f \left(\frac 2 3\right) x^2 (1-x)^{3-2} \right]
+ \left[ \binom 3 3 f \left(\frac 3 3\right) x^3 (1-x)^{3-3} \right]
\end{aligned}
\\&=
\left[ f (0) (1-x)^3 \right]
+ \left[ 3 f \left(\frac 1 3\right) x (1-x)^2 \right]
+ \left[ 3 f \left(\frac 2 3\right) x^2 (1-x) \right]
+ \left[ f(1) x^3 \right]
\\&=
\left[ u_0 (1-x)^3 \right]
+ \left[ 3 u_1 x (1-x)^2 \right]
+ \left[ 3 u_2 x^2 (1-x) \right]
+ \left[ u_3 x^3 \right]
\\&= [3u_1 + u_3 - (u_0 + 3u_2)]x^3 + [3u_0 + 3u_2 - 6u_1]x^2 + [3(u_1 - u_0)]x + u_0
\end{align*} B 3 f ( x ) = k = 0 ∑ 3 ( k 3 ) f ( 3 k ) x k ( 1 − x ) 3 − k = [ ( 0 3 ) f ( 3 0 ) x 0 ( 1 − x ) 3 − 0 ] + [ ( 1 3 ) f ( 3 1 ) x 1 ( 1 − x ) 3 − 1 ] + [ ( 2 3 ) f ( 3 2 ) x 2 ( 1 − x ) 3 − 2 ] + [ ( 3 3 ) f ( 3 3 ) x 3 ( 1 − x ) 3 − 3 ] = [ f ( 0 ) ( 1 − x ) 3 ] + [ 3 f ( 3 1 ) x ( 1 − x ) 2 ] + [ 3 f ( 3 2 ) x 2 ( 1 − x ) ] + [ f ( 1 ) x 3 ] = [ u 0 ( 1 − x ) 3 ] + [ 3 u 1 x ( 1 − x ) 2 ] + [ 3 u 2 x 2 ( 1 − x ) ] + [ u 3 x 3 ] = [ 3 u 1 + u 3 − ( u 0 + 3 u 2 )] x 3 + [ 3 u 0 + 3 u 2 − 6 u 1 ] x 2 + [ 3 ( u 1 − u 0 )] x + u 0 f ′ ( 1.0 ) ≈ f ( 1.2 ) − f ( 1.0 ) 0.2 = 1.3125
%% general %%
% shorthands
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% draw a box outlining some math
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% f \onall X = { f(x) : x ∈ X }
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% shorthands: various brackets
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\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
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% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
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% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
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% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
f'(1.0) \approx \frac{f(1.2) - f(1.0)}{0.2} = 1.{3125} f ′ ( 1.0 ) ≈ 0.2 f ( 1.2 ) − f ( 1.0 ) = 1. 3125 f ′ ( 1.2 ) ≈ f ( 1.4 ) − f ( 1.2 ) 0.2 = 1.9855
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
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\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
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\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
f'(1.2) \approx \frac{f(1.4) - f(1.2)}{0.2} = 1.{9855} f ′ ( 1.2 ) ≈ 0.2 f ( 1.4 ) − f ( 1.2 ) = 1. 9855 f ′ ( 1.4 ) ≈ f ( 1.2 ) − f ( 1.4 ) − 0 . 2 = 1.9855
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
f'(1.4) \approx \frac{f(1.2) - f(1.4)}{{-0}.2} = 1.{9855} f ′ ( 1.4 ) ≈ − 0 .2 f ( 1.2 ) − f ( 1.4 ) = 1. 9855 ∣ 1.3125 − 1 ∣ ≈ 0.31 ∣ 1.9855 − 1.63757 ∣ ≈ 0.34 ∣ 1.9855 − 2.34212 ∣ ≈ 0.35
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\begin{align*} \mag { 1.{3125} - 1 } &\approx 0.{31}
\\ \mag { 1.{9855} - 1.{63757} } &\approx 0.{34}
\\ \mag { 1.{9855} - 2.{34212} } &\approx 0.{35}
\end{align*} ∣ 1. 3125 − 1 ∣ ∣ 1. 9855 − 1. 63757 ∣ ∣ 1. 9855 − 2. 34212 ∣ ≈ 0. 31 ≈ 0. 34 ≈ 0. 35 term is
∣ ( h / 2 ) f ′ ′ ( ξ ) ∣ = 0.1 ( 2 ln x + 3
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\mag{ (h/2) f''(\xi) } = 0.1 (2 \ln x + 3 ∣ ( h /2 ) f ′′ ( ξ ) ∣ = 0.1 ( 2 ln x + 3 ξ ∈ [ x 0 , x 0 + h ]
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\xi \in [x_0, x_0 + h] ξ ∈ [ x 0 , x 0 + h ] [ x 0 − h , x 0 ]
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
[x_0 - h, x_0] [ x 0 − h , x 0 ] 1.0 : ≈ 0.33 1.2 : ≈ 0.36 1.4 : ≈ 0.36
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\begin{align*} 1.0 &\colon \approx 0.{33}
\\ 1.2 &\colon \approx 0.{36}
\\ 1.4 &\colon \approx 0.{36}
\end{align*} 1.0 1.2 1.4 : ≈ 0. 33 : ≈ 0. 36 : ≈ 0. 36 term is smaller for the midpoint formula , so we use that when possible and use the endpoint formula otherwise. That produces the following approximations:
f ′ ( − 2 . 7 ) ≈ − 0 . 0366
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
f'({-2}.7) \approx {-0}.{0366} f ′ ( − 2 .7 ) ≈ − 0 . 0366 f ′ ( − 2 . 5 ) ≈ 0.0600
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
f'({-2}.5) \approx 0.{0600} f ′ ( − 2 .5 ) ≈ 0. 0600 f ′ ( − 2 . 3 ) ≈ 0.0871
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
f'({-2}.3) \approx 0.{0871} f ′ ( − 2 .3 ) ≈ 0. 0871 f ′ ( − 2 . 1 ) ≈ 0.0446
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
f'({-2}.1) \approx 0.{0446} f ′ ( − 2 .1 ) ≈ 0. 0446 − 2 . 7 : ≈ 1.3897 − 2 . 5 : ≈ 1.87739 − 2 . 3 : ≈ 2.72388 − 2 . 1 : ≈ 0.026278
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\begin{align*} {-2}.7 &\colon \approx 1.{3897}
\\ {-2}.5 &\colon \approx 1.{87739}
\\ {-2}.3 &\colon \approx 2.{72388}
\\ {-2}.1 &\colon \approx 0.{026278}
\end{align*} − 2 .7 − 2 .5 − 2 .3 − 2 .1 : ≈ 1. 3897 : ≈ 1. 87739 : ≈ 2. 72388 : ≈ 0. 026278 bounded by
h 2 3 f ( 3 ) ( ξ ) ≈ 0.106 ( 27 sin ( 3 ξ ) cos ( 3 ξ ) − e 2 ξ )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\frac{h^2}3 f^{(3)}(\xi) \approx 0.{106} ( {27} \sin(3 \xi) \cos(3 \xi) - e^{2 \xi} ) 3 h 2 f ( 3 ) ( ξ ) ≈ 0. 106 ( 27 sin ( 3 ξ ) cos ( 3 ξ ) − e 2 ξ ) ξ
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\xi ξ [ x 0 , x 0 + 2 h ]
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
[x_0, x_0 + 2h] [ x 0 , x 0 + 2 h ] − 2 . 7
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
{-2}.7 − 2 .7 ≈ 5.4
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\approx 5.4 ≈ 5.4 − 2 . 1
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
{-2}.1 − 2 .1 ≈ 0.4
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\approx 0.4 ≈ 0.4 bounded by
h 2 6 f ( 3 ) ( ξ )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\frac{h^2}6 f^{(3)}(\xi) 6 h 2 f ( 3 ) ( ξ ) ξ
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\xi ξ [ x 0 − h , x 0 + h ]
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
[x_0 - h, x_0 + h] [ x 0 − h , x 0 + h ] − 2 . 5
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
{-2}.5 − 2 .5 ≈ 2.7
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\approx 2.7 ≈ 2.7 − 2 . 3
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
{-2}.3 − 2 .3 ≈ 0.19
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\approx 0.{19} ≈ 0. 19 Order -4 Taylor Polynomial expansion of f
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
f f f ( x ) ≈ f ( x 0 ) + f ′ ( x 0 ) ( x − x 0 ) + f ( 2 ) ( x 0 ) 2 ( x − x 0 ) 2 + f ( 3 ) ( x 0 ) 6 ( x − x 0 ) 3 + f ( 4 ) ( x 0 ) 24 ( x − x 0 ) 4
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
f(x) \approx f(x_0) + f'(x_0)(x - x_0) + \frac{f^{(2)}(x_0)}2 (x-x_0)^2 + \frac{f^{(3)}(x_0)}6 (x-x_0)^3 + \frac{f^{(4)}(x_0)}{{24}}(x - x_0)^4 f ( x ) ≈ f ( x 0 ) + f ′ ( x 0 ) ( x − x 0 ) + 2 f ( 2 ) ( x 0 ) ( x − x 0 ) 2 + 6 f ( 3 ) ( x 0 ) ( x − x 0 ) 3 + 24 f ( 4 ) ( x 0 ) ( x − x 0 ) 4 x = x 0 + h
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
x = x_0 + h x = x 0 + h f ( x 0 + h ) ≈ f ( x 0 ) + f ′ ( x 0 ) h + f ( 2 ) ( x 0 ) 2 h 2 + f ( 3 ) ( x 0 ) 6 h 3 + f ( 4 ) ( x 0 ) 24 h 4 (1)
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
f(x_0 + h) \approx f(x_0) + f'(x_0)h + \frac{f^{(2)}(x_0)}2 h^2 + \frac{f^{(3)}(x_0)}6 h^3 + \frac{f^{(4)}(x_0)}{{24}}h^4 \tag 1 f ( x 0 + h ) ≈ f ( x 0 ) + f ′ ( x 0 ) h + 2 f ( 2 ) ( x 0 ) h 2 + 6 f ( 3 ) ( x 0 ) h 3 + 24 f ( 4 ) ( x 0 ) h 4 ( 1 ) x = x 0 − h
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
x = x_0 - h x = x 0 − h f ( x 0 − h ) ≈ f ( x 0 ) − f ′ ( x 0 ) h + f ( 2 ) ( x 0 ) 2 h 2 − f ( 3 ) ( x 0 ) 6 h 3 + f ( 4 ) ( x 0 ) 24 h 4 (2)
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
f(x_0 - h) \approx f(x_0) - f'(x_0)h + \frac{f^{(2)}(x_0)}2 h^2 - \frac{f^{(3)}(x_0)}6 h^3 + \frac{f^{(4)}(x_0)}{{24}}h^4 \tag 2 f ( x 0 − h ) ≈ f ( x 0 ) − f ′ ( x 0 ) h + 2 f ( 2 ) ( x 0 ) h 2 − 6 f ( 3 ) ( x 0 ) h 3 + 24 f ( 4 ) ( x 0 ) h 4 ( 2 ) ( 2 )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
(2) ( 2 ) ( 1 )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
(1) ( 1 ) f ( x 0 + h ) − f ( x 0 − h ) ≈ 2 f ′ ( x 0 ) h + 1 3 f ( 3 ) ( x 0 ) h 3 (A)
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
f(x_0 + h) - f(x_0 - h) \approx 2f'(x_0)h + \frac 1 3 f^{(3)}(x_0) h^3 \tag A f ( x 0 + h ) − f ( x 0 − h ) ≈ 2 f ′ ( x 0 ) h + 3 1 f ( 3 ) ( x 0 ) h 3 ( A ) evaluation at x = x 0 + 2 h
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
x = x_0 + 2h x = x 0 + 2 h f ( x 0 + 2 h ) ≈ f ( x 0 ) + 2 f ′ ( x 0 ) h + 2 f ( 2 ) ( x 0 ) h 2 + 4 3 f ( 3 ) ( x 0 ) h 3 + 2 3 f ( 4 ) ( x 0 ) h 4 (3)
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
f(x_0 + 2h) \approx f(x_0) + 2f'(x_0)h + 2f^{(2)}(x_0) h^2 + \frac 4 3 f^{(3)}(x_0) h^3 + \frac2 3 f^{(4)}(x_0)h^4 \tag 3 f ( x 0 + 2 h ) ≈ f ( x 0 ) + 2 f ′ ( x 0 ) h + 2 f ( 2 ) ( x 0 ) h 2 + 3 4 f ( 3 ) ( x 0 ) h 3 + 3 2 f ( 4 ) ( x 0 ) h 4 ( 3 ) x = x 0 − 2 h
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
x = x_0 - 2h x = x 0 − 2 h f ( x 0 − 2 h ) ≈ f ( x 0 ) − 2 f ′ ( x 0 ) h + 2 f ( 2 ) ( x 0 ) h 2 − 4 3 f ( 3 ) ( x 0 ) h 3 + 2 3 f ( 4 ) ( x 0 ) h 4 (4)
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
f(x_0 - 2h) \approx f(x_0) - 2f'(x_0)h + 2f^{(2)}(x_0) h^2 - \frac 4 3 f^{(3)}(x_0) h^3 + \frac2 3 f^{(4)}(x_0)h^4 \tag 4 f ( x 0 − 2 h ) ≈ f ( x 0 ) − 2 f ′ ( x 0 ) h + 2 f ( 2 ) ( x 0 ) h 2 − 3 4 f ( 3 ) ( x 0 ) h 3 + 3 2 f ( 4 ) ( x 0 ) h 4 ( 4 ) ( 4 )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
(4) ( 4 ) ( 3 )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
(3) ( 3 ) f ( x 0 + 2 h ) − f ( x 0 − 2 h ) ≈ 4 f ′ ( x 0 ) h + 8 3 f ( 3 ) ( x 0 ) h 3 (B)
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
f(x_0 + 2h) - f(x_0 - 2h) \approx 4f'(x_0)h + \frac 8 3 f^{(3)}(x_0)h^3 \tag B f ( x 0 + 2 h ) − f ( x 0 − 2 h ) ≈ 4 f ′ ( x 0 ) h + 3 8 f ( 3 ) ( x 0 ) h 3 ( B ) ( B )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
(B) ( B ) − 2
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
{-2} − 2 ( A )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
(A) ( A ) f ( x 0 + 2 h ) + 2 f ( x 0 − h ) − f ( x 0 − 2 h ) − 2 f ( x 0 + h ) ≈ 2 f ( 3 ) ( x 0 ) h 3
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
f(x_0 + 2h) + 2f(x_0 - h) - f(x_0 - 2h) - 2f(x_0 + h) \approx 2 f^{(3)}(x_0)h^3 f ( x 0 + 2 h ) + 2 f ( x 0 − h ) − f ( x 0 − 2 h ) − 2 f ( x 0 + h ) ≈ 2 f ( 3 ) ( x 0 ) h 3 re -writes as
f ( 3 ) ( x 0 ) ≈ 1 2 f ( x 0 + 2 h ) + f ( x 0 − h ) − 1 2 f ( x 0 − 2 h ) − f ( x 0 + h ) h 3
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
f^{(3)}(x_0) \approx \frac{ \frac 1 2 f(x_0 + 2h) + f(x_0 - h) - \frac 1 2 f(x_0 - 2h) - f(x_0 + h) }{h^3} f ( 3 ) ( x 0 ) ≈ h 3 2 1 f ( x 0 + 2 h ) + f ( x 0 − h ) − 2 1 f ( x 0 − 2 h ) − f ( x 0 + h ) f ′ ( 0 ) ≈ f ( h ) − f ( 0 ) h = h + e h − 1 h
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
f'(0) \approx \frac{f(h) - f(0)}{h} = \frac{h + e^h - 1}{h} f ′ ( 0 ) ≈ h f ( h ) − f ( 0 ) = h h + e h − 1 N 1 ( h )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
N_1(h) N 1 ( h ) N 2 ( h ) = 2 N 1 ( h / 2 ) − N 1 ( h )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
N_2(h) = 2 N_1(h/2) - N_1(h) N 2 ( h ) = 2 N 1 ( h /2 ) − N 1 ( h ) M = N 2 ( h ) − K 2 2 h 2 − 3 K 3 4 h 3 − ⋯
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
M = N_2(h) - \frac{K_2}2 h^2 - \frac{3 K_3}4 h^3 - \cdots M = N 2 ( h ) − 2 K 2 h 2 − 4 3 K 3 h 3 − ⋯ h / 2
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
h/2 h /2 M = N 2 ( h / 2 ) − K 2 8 h 2 − 3 K 3 32 h 3 − ⋯
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
M = N_2(h/2) - \frac{K_2}8 h^2 - \frac{3 K_3}{32} h^3 - \cdots M = N 2 ( h /2 ) − 8 K 2 h 2 − 32 3 K 3 h 3 − ⋯ 1
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
1 1 − 4
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
{-4} − 4 − 3 M = [ N 2 ( h ) − 4 N 2 ( h / 2 ) ] − 3 K 3 8 h 3 + O ( h 4 )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
{-3}M = [ N_2(h) - 4N_2(h/2) ] - \frac{3K_3}8 h^3 + O(h^4) − 3 M = [ N 2 ( h ) − 4 N 2 ( h /2 )] − 8 3 K 3 h 3 + O ( h 4 ) O ( h 3 )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
O(h^3) O ( h 3 ) N 3 ( h ) : = − 1 3 ( N 2 ( h ) − 4 N 2 ( h / 2 ) ) = − 1 3 ( [ 2 N 1 ( h / 2 ) − N 1 ( h ) ] − 4 [ 2 N 1 ( h / 4 ) − N 1 ( h / 2 ) ] ) = 1 3 ( N 1 ( h ) − 6 N 1 ( h / 2 ) + 8 N 1 ( h / 4 ) )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\begin{align*} N_3(h) &:= -\frac 1 3 (N_2(h) - 4 N_2(h/2))
\\&= -\frac 1 3 \left( [2N_1(h/2) - N_1(h)] - 4 [2N_1(h/4) - N_1(h/2)] \right)
\\&= \frac 1 3 \left( N_1(h) - 6N_1(h/2) + 8N_1(h/4) \right)
\end{align*} N 3 ( h ) := − 3 1 ( N 2 ( h ) − 4 N 2 ( h /2 )) = − 3 1 ( [ 2 N 1 ( h /2 ) − N 1 ( h )] − 4 [ 2 N 1 ( h /4 ) − N 1 ( h /2 )] ) = 3 1 ( N 1 ( h ) − 6 N 1 ( h /2 ) + 8 N 1 ( h /4 ) ) h = 0.4
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
h = 0.4 h = 0.4 N 1 ( h ) = 2.22956 N 1 ( h / 2 ) = 2.10701 N 1 ( h / 4 ) = 2.05171
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\begin{align*} N_1(h) &= 2.{22956}
\\ N_1(h/2) &= 2.{10701}
\\ N_1(h/4) &= 2.{05171}
\end{align*} N 1 ( h ) N 1 ( h /2 ) N 1 ( h /4 ) = 2. 22956 = 2. 10701 = 2. 05171 N 3 ( h ) = 2.00039
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
N_3(h) = 2.{00039} N 3 ( h ) = 2. 00039 term is in the form given by (4.14), so we can use the formula from that section. It tells us that
N X ( h ) = N X − 1 ( h 2 ) + N X − 1 ( h / 2 ) − N X − 1 ( h ) 4 X − 1 − 1
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
N_X(h) = N_{X{-1}} \tpar{ \frac h 2 } + \frac{N_{X{-1}}(h/2) - N_{X{-1}}(h)}{4^{X{-1}} - 1} N X ( h ) = N X − 1 ( 2 h ) + 4 X − 1 − 1 N X − 1 ( h /2 ) − N X − 1 ( h ) N 4 ( h ) = 4096 ⋅ N 1 ( h / 8 ) / 2835 − 64 ⋅ N 1 ( h / 4 ) / 135 + 4 ⋅ N 1 ( h / 2 ) / 135 − N 1 ( h ) / 2835
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
N_4(h) = {4096}\cdot{}N_1(h/8)/{2835} - {64}\cdot{}N_1(h/4)/{135} + 4\cdot{}N_1(h/2)/{135} - N_1(h)/{2835} N 4 ( h ) = 4096 ⋅ N 1 ( h /8 ) / 2835 − 64 ⋅ N 1 ( h /4 ) / 135 + 4 ⋅ N 1 ( h /2 ) / 135 − N 1 ( h ) / 2835 M ≈ 1.99999933403880
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
M \approx 1.{99999933403880} M ≈ 1. 99999933403880 f ′ ( x ) = f ( x + h ) − f ( x ) h − h 2 f ′ ′ ( x ) − h 2 6 f ′ ′ ′ ( x ) + O ( h 3 )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
f'(x) = \frac { f(x+h) - f(x) } h - \frac h 2 f''(x) - \frac {h^2} 6 f'''(x) + O(h^3) f ′ ( x ) = h f ( x + h ) − f ( x ) − 2 h f ′′ ( x ) − 6 h 2 f ′′′ ( x ) + O ( h 3 ) h
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
h h h / 2
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
h/2 h /2 f ′ ( x ) = 2 h ( f ( x + h 2 ) − f ( x ) ) − h 4 f ′ ′ ( x ) − h 2 24 f ′ ′ ′ ( x ) + O ( h 3 )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
f'(x) = \frac 2 h \tpar { f \tpar{ x+ \frac h 2 } - f(x) } - \frac h 4 f''(x) - \frac {h^2}{{24}} f'''(x) + O(h^3) f ′ ( x ) = h 2 ( f ( x + 2 h ) − f ( x ) ) − 4 h f ′′ ( x ) − 24 h 2 f ′′′ ( x ) + O ( h 3 ) 1
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
1 1 formula to − 2
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
{-2} − 2 − f ′ ( x ) = [ 1 h ( f ( x + h ) − f ( x ) ) − 4 h ( f ( x + h 2 ) − f ( x ) ) ] − h 2 12 f ′ ′ ′ ( x ) + O ( h 3 ) f ′ ( x ) = 1 h [ 4 f ( x + h 2 ) − f ( x + h ) − 3 f ( x ) ] + h 2 12 f ′ ′ ′ ( x ) + O ( h 3 )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\begin{align*} -f'(x) &= \tbrak{ \frac 1 h \tpar{ f(x+h) - f(x) } - \frac 4 h \tpar{ f\tpar{x+\frac h 2} - f(x) } } - \frac {h^2}{{12}} f'''(x) + O(h^3)
\\ f'(x) &= \frac 1 h \tbrak{ 4 f\tpar{x+\frac h 2} - f(x+h) - 3 f(x) } + \frac {h^2}{{12}} f'''(x) + O(h^3)
\end{align*} − f ′ ( x ) f ′ ( x ) = [ h 1 ( f ( x + h ) − f ( x ) ) − h 4 ( f ( x + 2 h ) − f ( x ) ) ] − 12 h 2 f ′′′ ( x ) + O ( h 3 ) = h 1 [ 4 f ( x + 2 h ) − f ( x + h ) − 3 f ( x ) ] + 12 h 2 f ′′′ ( x ) + O ( h 3 ) h / 2
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
h/2 h /2 f ′ ( x ) = 2 h [ 4 f ( x + h 4 ) − f ( x + h 2 ) − 3 f ( x ) ] + h 2 48 f ′ ′ ′ ( x ) + O ( h 3 )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\begin{align*} f'(x) &= \frac 2 h \tbrak{ 4 f\tpar{x+\frac h 4} - f\tpar{x+\frac h 2} - 3 f(x) } + \frac {h^2}{{48}} f'''(x) + O(h^3)
\end{align*} f ′ ( x ) = h 2 [ 4 f ( x + 4 h ) − f ( x + 2 h ) − 3 f ( x ) ] + 48 h 2 f ′′′ ( x ) + O ( h 3 ) 1
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
1 1 h
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
h h − 4
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
{-4} − 4 h / 2
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
h/2 h /2 − 3 f ′ ( x ) = 1 h [ 4 f ( x + h 2 ) − f ( x + h ) − 3 f ( x ) ] − 4 2 h [ 4 f ( x + h 4 ) − f ( x + h 2 ) − 3 f ( x ) ] + O ( h 3 ) = 1 h [ 4 f ( x + h 2 ) − f ( x + h ) − 3 f ( x ) − 32 f ( x + h 4 ) + 8 f ( x + h 2 ) + 24 f ( x ) ] + O ( h 3 ) = 1 h [ − f ( x + h ) + 12 f ( x + h 2 ) − 32 f ( x + h 4 ) + 21 f ( x ) ] + O ( h 3 ) f ′ ( x ) = 1 3 h [ f ( x + h ) − 12 f ( x + h 2 ) + 32 f ( x + h 4 ) − 21 f ( x ) ] + O ( h 3 )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\begin{align*} {-3} f'(x) &= \frac 1 h \tbrak{ 4 f\tpar{x+\frac h 2} - f(x+h) - 3 f(x) } - 4 \frac 2 h \tbrak{ 4 f\tpar{x+\frac h 4} - f\tpar{x+\frac h 2} - 3 f(x) } + O(h^3)
\\ &= \frac 1 h \tbrak{ 4 f\tpar{x+\frac h 2} - f(x+h) - 3 f(x) - {32} f\tpar{x+\frac h 4} + 8 f\tpar{x+\frac h 2} + {24} f(x) } + O(h^3)
\\ &= \frac 1 h \tbrak{ - f(x+h) + {12} f\tpar{x+\frac h 2} - {32} f\tpar{x+\frac h 4} + {21} f(x) } + O(h^3)
\\ f'(x) &= \frac 1 {3h} \tbrak{ f(x+h) - {12} f\tpar{x+\frac h 2} + {32} f\tpar{x+\frac h 4} - {21} f(x) } + O(h^3)
\end{align*} − 3 f ′ ( x ) f ′ ( x ) = h 1 [ 4 f ( x + 2 h ) − f ( x + h ) − 3 f ( x ) ] − 4 h 2 [ 4 f ( x + 4 h ) − f ( x + 2 h ) − 3 f ( x ) ] + O ( h 3 ) = h 1 [ 4 f ( x + 2 h ) − f ( x + h ) − 3 f ( x ) − 32 f ( x + 4 h ) + 8 f ( x + 2 h ) + 24 f ( x ) ] + O ( h 3 ) = h 1 [ − f ( x + h ) + 12 f ( x + 2 h ) − 32 f ( x + 4 h ) + 21 f ( x ) ] + O ( h 3 ) = 3 h 1 [ f ( x + h ) − 12 f ( x + 2 h ) + 32 f ( x + 4 h ) − 21 f ( x ) ] + O ( h 3 ) O ( h 3 )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
O(h^3) O ( h 3 ) M = N ( h ) + K 1 h + K 2 h 2 + K 3 h 3 + ⋯ (1)
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
M = N(h) + K_1 h + K_2 h^2 + K_3 h^3 + \cdots \tag 1 M = N ( h ) + K 1 h + K 2 h 2 + K 3 h 3 + ⋯ ( 1 ) h / 3
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
h/3 h /3 M = N ( h 3 ) + K 1 h 3 + K 2 h 2 9 + K 3 h 3 27 + ⋯ (2)
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
M = N\tpar{\frac h 3} + K_1 \frac h 3 + K_2 \frac{h^2}9 + K_3 \frac{h^3}{{27}} + \cdots \tag 2 M = N ( 3 h ) + K 1 3 h + K 2 9 h 2 + K 3 27 h 3 + ⋯ ( 2 ) 1 2 ( 3 × ( 2 ) − ( 1 ) )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\frac 1 2 ( 3 \times (2) - (1) ) 2 1 ( 3 × ( 2 ) − ( 1 )) M = [ 3 2 N ( h 3 ) − 1 2 N ( h ) ] + ( − 1 / 3 ) K 2 h 2 + ( − 4 / 9 ) K 3 h 3 + ⋯ (3)
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
M = \tbrak{ \frac 3 2 N\tpar{\frac h 3} - \frac 1 2 N(h) } + ({-1}/3)K_2h^2 + ({-4}/9)K_3h^3 + \cdots \tag 3 M = [ 2 3 N ( 3 h ) − 2 1 N ( h ) ] + ( − 1 /3 ) K 2 h 2 + ( − 4 /9 ) K 3 h 3 + ⋯ ( 3 ) h / 3
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
h/3 h /3 M = [ 3 2 N ( h 9 ) − 1 2 N ( h 3 ) ] + ( − 1 / 27 ) K 2 h 2 + ( − 4 / 247 ) K 3 h 3 + ⋯ (4)
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
M = \tbrak{ \frac 3 2 N\tpar{\frac h 9} - \frac 1 2 N\tpar{\frac h 3} } + ({-1}/{27})K_2h^2 + ({-4}/{247})K_3h^3 + \cdots \tag 4 M = [ 2 3 N ( 9 h ) − 2 1 N ( 3 h ) ] + ( − 1 / 27 ) K 2 h 2 + ( − 4 / 247 ) K 3 h 3 + ⋯ ( 4 ) 1 8 ( 9 × ( 4 ) − ( 3 ) )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
\frac 1 8 ( 9 \times (4) - (3) ) 8 1 ( 9 × ( 4 ) − ( 3 )) M = [ 27 2 N ( h 9 ) − 6 N ( h 3 ) + 1 2 N ( h ) ] + O ( h 3 )
%% general %%
% shorthands
\newcommand{\cl}[1]{ \mathcal{#1} }
\newcommand{\sc}[1]{ \mathscr{#1} }
\newcommand{\bb}[1]{ \mathbb{#1} }
\newcommand{\fk}[1]{ \mathfrak{#1} }
\renewcommand{\bf}[1]{ \mathbf{#1} }
\renewcommand{\sf}[1]{ \mathsf{#1} }
\renewcommand{\rm}[1]{ \mathrm{#1} }
\newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } }
\newcommand{\ceil}[1]{ { \lceil {#1} \rceil } }
\newcommand{\ol}[1]{ \overline{#1} }
\newcommand{\t}[1]{ \text{#1} }
\newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE)
\newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude
\newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag
\newcommand{\card}{ \t{cd} } % cardinality
\newcommand{\dcup}{ \sqcup } % disjoint untion
\newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples
\newcommand{\tl}{ \tilde }
\newcommand{\wt}{ \widetilde }
\newcommand{\To}{ \Rightarrow }
% draw a box outlining some math
\newcommand{\box}[1]{ \fbox{$ #1 $} }
% f \onall X = { f(x) : x ∈ X }
\newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } }
% shorthands: various brackets
\newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens"
\newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets"
\newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces"
% reverse \mapsto (FIXME: make better)
%\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} }
\newcommand{\mapsfrom}{ \mathrel{↤} }
% reverse-order composition
\newcommand{\then}{ \operatorname{\ ;\ } }
% Like f' represents "f after modification", \pre{f}
% represents "f before modification"
% TODO: remove this?
\newcommand{\pre}[1]{{ \small `{#1} }}
% hook arrows
\newcommand{\injects}{ \hookrightarrow }
\newcommand{\embeds}{ \hookrightarrow }
\newcommand{\surjects}{ \twoheadrightarrow }
\newcommand{\projects}{ \twoheadrightarrow }
\newcommand{\id}{ \,\mathrm d } % integration d
% derivatives: use {\ddn n x y} for (dy/dx)
\newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative
\newcommand{\dd}{ \ddn{} } % first derivative
\newcommand{\d}{ \dd{} } % first derivative (no numerator)
\newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator)
% derivatives: use {\D n x y} for (∂_x y)
\newcommand{\Dn}[2]{ \partial^{#1}_{#2} }
\newcommand{\D}{ \Dn{} } % no power
\newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral
%% category theory %%
% category names
\newcommand{\cat}[1]{{ \sf{#1} }}
% yoneda embedding
\newcommand{\yo}{よ}
% extra long right-arrows
\newcommand{\X}{-\!\!\!-\!\!\!}
\newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } }
\newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } }
\newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } }
\newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} }
\newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} }
\newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} }
\newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} }
\newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} }
% represents an anonymous parameter
% eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$
% TODO: remove this?
\newcommand{\apar}{ {-} }
%% computability %%
% turing machines
\newcommand{\halts}{ {\downarrow} }
\newcommand{\loops}{ {\uparrow} }
M = \tbrak{ \frac {{27}} 2 N\tpar{\frac h 9} - 6 N\tpar{\frac h 3} + \frac 1 2 N\tpar{h} } + O(h^3) M = [ 2 27 N ( 9 h ) − 6 N ( 3 h ) + 2 1 N ( h ) ] + O ( h 3 )