From: Andrew Lorimer Date: Tue, 28 May 2019 12:00:30 +0000 (+1000) Subject: [spec] finalise reference notes before SAC X-Git-Tag: yr12~115 X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/29e930b137cca382fa6af509461d0c79c51fd21b [spec] finalise reference notes before SAC --- diff --git a/spec/spec-collated.pdf b/spec/spec-collated.pdf index 6e6dc20..9714796 100644 Binary files a/spec/spec-collated.pdf and b/spec/spec-collated.pdf differ diff --git a/spec/spec-collated.tex b/spec/spec-collated.tex index e2ccd0e..39ab74e 100644 --- a/spec/spec-collated.tex +++ b/spec/spec-collated.tex @@ -6,6 +6,7 @@ \usepackage{amssymb} \usepackage{harpoon} \usepackage{tabularx} +\usepackage{makecell} \usepackage[dvipsnames, table]{xcolor} \usepackage{blindtext} \usepackage{graphicx} @@ -34,7 +35,19 @@ \definecolor{cas}{HTML}{e6f0fe} \linespread{1.5} \newcommand{\midarrow}{\tikz \draw[-triangle 90] (0,0) -- +(.1,0);} +\newcommand{\tg}{\mathop{\mathrm{tg}}} +\newcommand{\cotg}{\mathop{\mathrm{cotg}}} +\newcommand{\arctg}{\mathop{\mathrm{arctg}}} +\newcommand{\arccotg}{\mathop{\mathrm{arccotg}}} + + \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 + }} \begin{document} \begin{multicols}{2} @@ -607,7 +620,19 @@ \end{itemize} \subsubsection*{Secant} - \begin{center}\includegraphics[width=0.7\columnwidth]{graphics/sec.png}\end{center} + +\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}