From 29e930b137cca382fa6af509461d0c79c51fd21b Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Tue, 28 May 2019 22:00:30 +1000 Subject: [PATCH] [spec] finalise reference notes before SAC --- spec/spec-collated.pdf | Bin 464037 -> 371859 bytes spec/spec-collated.tex | 134 ++++++++++++++++++++++++++++------------- 2 files changed, 92 insertions(+), 42 deletions(-) diff --git a/spec/spec-collated.pdf b/spec/spec-collated.pdf index 6e6dc208768da9be5b522be916a5644b69d5ccf9..9714796099c07445488f0159bfd3c9f571620a91 100644 GIT binary patch delta 113864 zcmV)RK(oK4sT`Bp7O*@Imoan&Du3-g*>W5=lJESAdTg#;HMkGYPQ*$YTQLrKXDDf6 zc0=QbX0t`LhTW8#q;2`@HxnqJfGSk=MJcS^eGvP~0uqV5GXat6x+2x{v!}1m{_vft zY-1Hu)$6yFRE@H{GL*w>_4=xMGyCsz!)MF+_Ppkt&nWY+uRfgD%*>XH)qnhEb$5SW zQ>GQ4J=t7sm%n(B7q{EhdbvIS@%8tA_)e%~7OrR`i2)Sc3>2Owgv_4(^@qQ``1Zg2 z)BpMRAD({u@)bUP_U-GlpUxg?i-q>HAiqO`G6vVUaRTFZuU;jwylyZ&&q{1vd#at5pAbas35adW$d72o+q zrIaI8%~ZoQ_fP+2v$%PrZD~KhxxqJLc6WR8 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+\begin{tikzpicture} + \begin{axis}[ytick={-1,1}, yticklabels={\(-1\), \(1\)}, xmin=-7,xmax=7,ymin=-3,ymax=3,enlargelimits=true, xtick={-6.2830, -3.1415, 3.1415, 6.2830},xticklabels={\(-2\pi\), \(-\pi\), \(\pi\), \(2\pi\)}] +% \addplot[blue, domain=-6.2830:6.2830,unbounded coords=jump,samples=80] {sec(deg(x))}; + \addplot[blue, restrict y to domain=-10:10, domain=-7:7,samples=100] {sec(deg(x))} node [pos=0.93, black, right] {\(\operatorname{sec} x\)}; + \addplot[red, dashed, domain=-7:7,samples=100] {cos(deg(x))}; + \draw [gray, dotted, thick] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{rel axis cs:0,1}); + \draw [gray, dotted, thick] ({axis cs:4.71239,0}|-{rel axis cs:0,0}) -- ({axis cs:4.71239,0}|-{rel axis cs:0,1}); + \draw [gray, dotted, thick] ({axis cs:-4.71239,0}|-{rel axis cs:0,0}) -- ({axis cs:-4.71239,0}|-{rel axis cs:0,1}); + \draw [gray, dotted, thick] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1}); +\end{axis} + \node [black] at (7,3.5) {\(\cos x\)}; +\end{tikzpicture} \[\operatorname{sec} \theta = \frac{1}{\cos \theta} \> \vert \> \cos \theta \ne 0\] @@ -628,7 +653,20 @@ \subsubsection*{Cotangent} - \begin{center}\includegraphics[width=0.7\columnwidth]{graphics/cot.png}\end{center} +\begin{tikzpicture} + \begin{axis}[xmin=-3,xmax=3,ymin=-1.5,ymax=1.5,enlargelimits=true, xtick={-3.1415, -1.5708, 1.5708, 3.1415},xticklabels={\(-\pi\), \(-\frac{\pi}{2}\), \(\frac{\pi}{2}\), \(\pi\)}] + \addplot[blue, smooth, domain=-3:-0.1,unbounded coords=jump,samples=105] {cot(deg(x))} node [pos=0.3, left] {\(\operatorname{cot} x\)}; +\addplot[blue, smooth, domain=0.1:3,unbounded coords=jump,samples=105] {cot(deg(x))}; +\addplot[red, smooth, dashed] gnuplot [domain=-1.5:1.5,unbounded coords=jump,samples=105] {tan(x)}; +\addplot[red, smooth, dashed] gnuplot [domain=-3.5:-1.8,unbounded coords=jump,samples=105] {tan(x)} node [pos=0.5, right] {\(\tan x\)}; +\addplot[red, smooth, dashed] gnuplot [domain=1.8:3.5,unbounded coords=jump,samples=105] {tan(x)}; + \draw [thick, red, dotted] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1}); + \draw [thick, blue, dotted] ({axis cs:-3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:-3.1415,0}|-{rel axis cs:0,1}); + \draw [thick, blue, dotted] ({axis cs:0,0}|-{rel axis cs:0,0}) -- ({axis cs:0,0}|-{rel axis cs:0,1}); + \draw [thick, blue, dotted] ({axis cs:3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:3.1415,0}|-{rel axis cs:0,1}); + \draw [thick, red, dotted] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{rel axis cs:0,1}); +\end{axis} +\end{tikzpicture} \[\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0\] @@ -689,23 +727,16 @@ \subsection*{Inverse circular functions} - \pgfplotsset{every axis/.append style={ - axis x line=middle, % put the x axis in the middle - axis y line=middle, % put the y axis in the middle - axis line style={<->}, % arrows on the axis - xlabel={$x$}, % default put x on x-axis - ylabel={$y$}, % default put y on y-axis - }} - -% arrows as stealth fighters -\tikzset{>=stealth} - -\begin{tikzpicture} - \begin{axis}[domain = -1:1, samples = 500] - \addplot[color = red] {rad(asin(x))} node [pos=0.25, below right] {\(\sin^{-1}x\)}; - \addplot[color = blue] {rad(acos(x))} node [pos=0.25, below left] {\(\cos^{-1}x\)}; - \end{axis} -\end{tikzpicture} + \begin{tikzpicture} + \begin{axis}[ymin=-2, ymax=4, xmin=-1.1, xmax=1.1, ytick={-1.5708, 1.5708, 3.14159},yticklabels={$-\frac{\pi}{2}$, $\frac{\pi}{2}$, $\pi$}] + \addplot[color=red, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {asin(x)} node [pos=0.25, below right] {\(\sin^{-1}x\)}; + \addplot[color=blue, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {acos(x)} node [pos=0.25, below left] {\(\cos^{-1}x\)}; + \addplot[mark=*, red] coordinates {(-1,-1.5708)} node[right, font=\footnotesize]{\((-1,-\frac{\pi}{2})\)} ; + \addplot[mark=*, red] coordinates {(1,1.5708)} node[left, font=\footnotesize]{\((1,\frac{\pi}{2})\)} ; + \addplot[mark=*, blue] coordinates {(1,0)}; + \addplot[mark=*, blue] coordinates {(-1,3.1415)} node[right, font=\footnotesize]{\((-1,\pi)\)} ; + \end{axis} + \end{tikzpicture}\\ Inverse functions: \(f(f^{-1}(x)) = x\) (restrict domain) @@ -718,7 +749,14 @@ \[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\] \hfill where \(\tan y = x, \> y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\) - + \begin{tikzpicture} + \begin{axis}[yticklabel style={yshift=1.0pt, anchor=north east},x=0.1cm, y=1cm, ymax=2, ymin=-2, xticklabels={}, ytick={-1.5708,1.5708},yticklabels={\(-\frac{\pi}{2}\),\(\frac{\pi}{2}\)}] + \addplot[color=orange, smooth] gnuplot [domain=-35:35, unbounded coords=jump,samples=350] {atan(x)} node [pos=0.5, above left] {\(\tan^{-1}x\)}; + \addplot[->, gray, dotted, thick, domain=-35:35] {1.5708}; + \addplot[->, gray, dotted, thick, domain=-35:35] {-1.5708}; + \end{axis} + \end{tikzpicture} +\columnbreak \section{Differential calculus} \subsection*{Limits} @@ -810,7 +848,7 @@ \therefore {\frac{dy}{dx}} &= \frac{1}{\frac{dx}{dy}} \end{align*} - \subsubsection*{Second derivative} + \subsection*{Second derivative} \begin{align*}f(x) \longrightarrow &f^\prime (x) \longrightarrow f^{\prime\prime}(x)\\ \implies y \longrightarrow &\frac{dy}{dx} \longrightarrow \frac{d^2 y}{dx^2}\end{align*} @@ -822,22 +860,37 @@ \(f^\prime(x)=0\)\\ \emph{Point of inflection} - max \(|\)gradient\(|\) (i.e. \(f^{\prime\prime} = 0\)) - %\begin{table*}[ht] - %\centering - % \begin{tabularx}{\textwidth}{XXXX} - %\hline - % \rowcolor{shade2} - % & \(\dfrac{d^2 y}{dx^2} > 0\) & \(\dfrac{d^2y}{dx^2}<0\) & \(\dfrac{d^2y}{dx^2}=0\) (inflection) \\ - %\hline - % \(\frac{dy}{dx}>0\) & \begin{tikzpicture} \draw[domain=1:2,smooth,variable=\x,blue] plot ({\x},{(1/10)*\x*\x*\x}) plot ({\x},{0.675*\x-0.677}); \end{tikzpicture} & cell 3\\ - %cell 1 & cell 2 & cell 3\\ - %\hline - %\end{tabularx} - %\end{table*} - - -\begin{itemize} - \item + + + \pgfplotsset{every axis/.append style={ + axis x line=none, % put the x axis in the middle + axis y line=none, % put the y axis in the middle + }} + \begin{table*}[ht] + \centering + \begin{tabularx}{\textwidth}{rXXX} + \hline + \rowcolor{shade2} + & \centering\(\dfrac{d^2 y}{dx^2} > 0\) & \centering \(\dfrac{d^2y}{dx^2}<0\) & \(\dfrac{d^2y}{dx^2}=0\) (inflection) \\ + \hline + \(\frac{dy}{dx}>0\) & + \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=-3, xmax=0.8, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(e^(x))}; \addplot[red] {x/2.5+0.75}; \end{axis}\end{tikzpicture} \\Rising (concave up)}& + \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=0.1, xmax=4, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(ln(x))}; \addplot[red] {x/1.5-0.56}; \end{axis}\end{tikzpicture} \\Rising (concave down)}& + \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=-1.5, xmax=1.5, scale=0.2, samples=100] \addplot[blue] {(sin((deg x)))}; \addplot[red] {x}; \end{axis}\end{tikzpicture} \\Rising inflection point}\\ + \hline + \(\dfrac{dy}{dx}<0\) & + \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=-.5, xmax=1, ymin=-.5, ymax=.5, scale=0.2, samples=100] \addplot[blue] {(1/(x+1)-1}; \addplot[red] {-x}; \end{axis}\end{tikzpicture} \\Falling (concave up)}& + \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=0, xmax=1.5, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(2-x*x)^(1/2)}; \addplot[red] {-x+2}; \end{axis}\end{tikzpicture} \\Falling (concave down)}& + \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=1.5, xmax=4.5, scale=0.2, samples=100] \addplot[blue] {(sin((deg x)))}; \addplot[red] {-x+3.1415}; \end{axis}\end{tikzpicture} \\Falling inflection point}\\ + \hline + \(\dfrac{dy}{dx}=0\)& + \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(x*x))}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \\Local minimum}& \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(-x*x))}; \addplot[red, very thick] {0}; \end{axis}\end{tikzpicture} \\Local maximum}& + \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(x*x*x))}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \(\>\) \begin{tikzpicture}\begin{axis}[xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(-x*x*x))}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \\Stationary inflection point}\\ + \hline + \end{tabularx} + \end{table*} + \begin{itemize} + \item if \(f^\prime (a) = 0\) and \(f^{\prime\prime}(a) > 0\), then point \((a, f(a))\) is a local min (curve is concave up) \item @@ -1066,7 +1119,7 @@ \[\int^b_a f(x) \> dx = F(b) - F(a)\] \hfill where \(F = \int f \> dx\) - + \subsection*{Differential equations} \noindent\textbf{Order} - highest power inside derivative\\ @@ -1085,10 +1138,7 @@ \(\frac{dx}{dy} = 1 \div \frac{dy}{dx} = \frac{1}{g(y)}\). Integrate both sides to solve equation. Only add \(c\) on one side. Express \(e^c\) as \(A\). - \begin{table*}[ht] - \centering - \includegraphics[width=0.7\textwidth]{graphics/second-derivatives.png} - \end{table*} + \subsubsection*{Mixing problems} -- 2.43.2