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zAh(}7U1oi{E9apnuUmXvSRxn-6FP1C4X6NeKluMI=<8>X^TXpV-wY0w!N{OMLPBbK H>Y)Dt-_F~I literal 0 HcmV?d00001 diff --git a/methods/calculus.tex b/methods/calculus.tex index 6f57dc0..1896189 100644 --- a/methods/calculus.tex +++ b/methods/calculus.tex @@ -1,3 +1,7 @@ +\documentclass[methods-collated.tex]{subfiles} + +\begin{document} + \section{Calculus} \subsection*{Average rate of change} @@ -19,16 +23,12 @@ \subsection*{Limit theorems} \begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\) -\item - \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\) -\item - \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\) -\item - \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\) + \def\labelenumi{\arabic{enumi}.} + \tightlist + \item For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\) + \item \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\) + \item \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\) + \item \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\) \end{enumerate} A function is continuous if \(L^-=L^+=f(x)\) for all values of \(x\). @@ -39,13 +39,10 @@ A function is continuous if \(L^-=L^+=f(x)\) for all values of \(x\). Not differentiable at: \begin{itemize} -\tightlist -\item - discontinuous points -\item - sharp point/cusp -\item - vertical tangents (\(\infty\) gradient) + \tightlist + \item discontinuous points + \item sharp point/cusp + \item vertical tangents (\(\infty\) gradient) \end{itemize} \subsection*{Tangents \& gradients} @@ -64,25 +61,20 @@ Not differentiable at: For \(x_2\) and \(x_1\) where \(x_2 > x_1\): \begin{itemize} -\tightlist -\item - \textbf{strictly increasing}\\ where \(f(x_2) > f(x_1)\) or \(f^\prime(x)>0\) -\item - \textbf{strictly decreasing}\\ where \(f(x_2) < f(x_1)\) or \(f^\prime(x)<0\) -\item - Endpoints are included, even where gradient \(=0\) + \tightlist + \item \textbf{strictly increasing}\\ where \(f(x_2) > f(x_1)\) or \(f^\prime(x)>0\) + \item \textbf{strictly decreasing}\\ where \(f(x_2) < f(x_1)\) or \(f^\prime(x)<0\) + \item Endpoints are included, even where gradient \(=0\) \end{itemize} -\columnbreak - -\subsubsection*{Solving on CAS} - -\colorbox{cas}{\textbf{In main}}: type function. Interactive -\(\rightarrow\) Calculation \(\rightarrow\) Line \(\rightarrow\) (Normal -\textbar{} Tan line)\\ -\colorbox{cas}{\textbf{In graph}}: define function. Analysis -\(\rightarrow\) Sketch \(\rightarrow\) (Normal \textbar{} Tan line). -Type \(x\) value to solve for a point. Return to show equation for line. +\begin{cas} + \colorbox{cas}{\textbf{In main}}: type function. Interactive + \(\rightarrow\) Calculation \(\rightarrow\) Line \(\rightarrow\) (Normal + \textbar{} Tan line)\\ + \colorbox{cas}{\textbf{In graph}}: define function. Analysis + \(\rightarrow\) Sketch \(\rightarrow\) (Normal \textbar{} Tan line). + Type \(x\) value to solve for a point. Return to show equation for line. +\end{cas} \subsection*{Stationary points} @@ -91,81 +83,81 @@ Type \(x\) value to solve for a point. Return to show equation for line. \textbf{Point of inflection:} && f^{\prime\prime} &= 0 \end{align*} - \begin{tikzpicture} - \begin{axis}[xmin=-21, xmax=21, ymax=1400, ymin=-1000, ticks=none, axis lines=middle] - \addplot[color=red, smooth, thick] gnuplot [domain=-15:15,unbounded coords=jump,samples=500] {x^3-3*x^2-144*x+432} node [black, pos=1, right] {\(f(x)\)}; - \addplot[color=darkgray, dashed, smooth, thick] gnuplot [domain=-15:15,unbounded coords=jump,samples=500] {3*x^2-6*x-144} node [black, pos=1, right] {\(f^\prime(x)\)}; - \addplot[mark=*, blue] coordinates {(1,286)} node[above right, align=left, font=\footnotesize]{inflection \\ (falling)} ; - \addplot[mark=*, orange] coordinates {(-6,972)} node[above left, align=right, font=\footnotesize]{stationary \\ (local max)} ; - \addplot[mark=*, orange] coordinates {(8,-400)} node[below, align=left, font=\footnotesize]{stationary \\ (local min)} ; - \end{axis} - \end{tikzpicture}\\ - \begin{tikzpicture} - \begin{axis}[enlargelimits=true, xmax=3.5, ticks=none, axis lines=middle] - \addplot[color=blue, smooth, thick] gnuplot [domain=0.74:3,unbounded coords=jump,samples=500] {(x-2)^3+2} node [black, pos=0.9, left] {\(f(x)\)}; - \addplot[color=darkgray, dashed, smooth, thick] gnuplot [domain=1:3,unbounded coords=jump,samples=500] {3*(x-2)^2} node [black, pos=0.9, right] {\(f^\prime(x)\)}; - \addplot[mark=*, purple] coordinates {(2,2)} node[below right, align=left, font=\footnotesize]{stationary \\ inflection} ; - \end{axis} - \end{tikzpicture}\\ -\pagebreak +\begin{tikzpicture} + \begin{axis}[xmin=-21, xmax=21, ymax=1400, ymin=-1000, ticks=none, axis lines=middle] + \addplot[color=red, smooth, thick] gnuplot [domain=-15:15,unbounded coords=jump,samples=500] {x^3-3*x^2-144*x+432} node [black, pos=1, right] {\(f(x)\)}; + \addplot[color=darkgray, dashed, smooth, thick] gnuplot [domain=-15:15,unbounded coords=jump,samples=500] {3*x^2-6*x-144} node [black, pos=1, right] {\(f^\prime(x)\)}; + \addplot[mark=*, blue] coordinates {(1,286)} node[above right, align=left, font=\footnotesize]{inflection \\ (falling)} ; + \addplot[mark=*, orange] coordinates {(-6,972)} node[above left, align=right, font=\footnotesize]{stationary \\ (local max)} ; + \addplot[mark=*, orange] coordinates {(8,-400)} node[below, align=left, font=\footnotesize]{stationary \\ (local min)} ; + \end{axis} +\end{tikzpicture}\\ +\begin{tikzpicture} + \begin{axis}[enlargelimits=true, xmax=3.5, ticks=none, axis lines=middle] + \addplot[color=blue, smooth, thick] gnuplot [domain=0.74:3,unbounded coords=jump,samples=500] {(x-2)^3+2} node [black, pos=0.9, left] {\(f(x)\)}; + \addplot[color=darkgray, dashed, smooth, thick] gnuplot [domain=1:3,unbounded coords=jump,samples=500] {3*(x-2)^2} node [black, pos=0.9, right] {\(f^\prime(x)\)}; + \addplot[mark=*, purple] coordinates {(2,2)} node[below right, align=left, font=\footnotesize]{stationary \\ inflection} ; + \end{axis} +\end{tikzpicture} + \subsection*{Derivatives} -\definecolor{shade1}{HTML}{ffffff} -\definecolor{shade2}{HTML}{F0F9E4} -\rowcolors{1}{shade1}{shade2} - \renewcommand{\arraystretch}{1.4} - \begin{tabularx}{\columnwidth}{rX} - \hline - \hspace{6em}\(f(x)\) & \(f^\prime(x)\)\\ - \hline - \(\sin x\) & \(\cos x\)\\ - \(\sin ax\) & \(a\cos ax\)\\ - \(\cos x\) & \(-\sin x\)\\ - \(\cos ax\) & \(-a \sin ax\)\\ - \(\tan f(x)\) & \(f^2(x) \sec^2f(x)\)\\ - \(e^x\) & \(e^x\)\\ - \(e^{ax}\) & \(ae^{ax}\)\\ - \(ax^{nx}\) & \(an \cdot e^{nx}\)\\ - \(\log_e x\) & \(\dfrac{1}{x}\)\\ - \(\log_e {ax}\) & \(\dfrac{1}{x}\)\\ - \(\log_e f(x)\) & \(\dfrac{f^\prime (x)}{f(x)}\)\\ - \(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\\ - \(\sin^{-1} x\) & \(\dfrac{1}{\sqrt{1-x^2}}\)\\ - \(\cos^{-1} x\) & \(\dfrac{-1}{\sqrt{1-x^2}}\)\\ - \(\tan^{-1} x\) & \(\dfrac{1}{1 + x^2}\)\\ - \(\frac{d}{dy}f(y)\) & \(\dfrac{1}{\frac{dx}{dy}}\) \hfill(reciprocal)\\ - \(uv\) & \(u \frac{dv}{dx}+v\frac{du}{dx}\) \hfill(product rule)\\ - \(\dfrac{u}{v}\) & \(\dfrac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\) \hfill(quotient rule)\\ - \(f(g(x))\) & \(f^\prime(g(x))\cdot g^\prime(x)\)\\ - \hline - \end{tabularx} - \vspace*{\fill} - \columnbreak +\rowcolors{1}{white}{orange} +\renewcommand{\arraystretch}{1.4} + +\begin{tabularx}{\columnwidth}{rX} + \hline + \hspace{6em}\(f(x)\) & \(f^\prime(x)\)\\ + \hline + \(\sin x\) & \(\cos x\)\\ + \(\sin ax\) & \(a\cos ax\)\\ + \(\cos x\) & \(-\sin x\)\\ + \(\cos ax\) & \(-a \sin ax\)\\ + \(\tan f(x)\) & \(f^2(x) \sec^2f(x)\)\\ + \(e^x\) & \(e^x\)\\ + \(e^{ax}\) & \(ae^{ax}\)\\ + \(ax^{nx}\) & \(an \cdot e^{nx}\)\\ + \(\log_e x\) & \(\dfrac{1}{x}\)\\ + \(\log_e {ax}\) & \(\dfrac{1}{x}\)\\ + \(\log_e f(x)\) & \(\dfrac{f^\prime (x)}{f(x)}\)\\ + \(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\\ + \(\sin^{-1} x\) & \(\dfrac{1}{\sqrt{1-x^2}}\)\\ + \(\cos^{-1} x\) & \(\dfrac{-1}{\sqrt{1-x^2}}\)\\ + \(\tan^{-1} x\) & \(\dfrac{1}{1 + x^2}\)\\ + \(\frac{d}{dy}f(y)\) & \(\dfrac{1}{\frac{dx}{dy}}\) \hfill(reciprocal)\\ + \(uv\) & \(u \frac{dv}{dx}+v\frac{du}{dx}\) \hfill(product rule)\\ + \(\dfrac{u}{v}\) & \(\dfrac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\) \hfill(quotient rule)\\ + \(f(g(x))\) & \(f^\prime(g(x))\cdot g^\prime(x)\)\\ + \hline +\end{tabularx} + \subsection*{Antiderivatives} -\rowcolors{1}{shade1}{cas} - \renewcommand{\arraystretch}{1.4} - \begin{tabularx}{\columnwidth}{rX} - \hline - \(f(x)\) & \(\int f(x) \cdot dx\) \\ - \hline - \(k\) (constant) & \(kx + c\)\\ - \(x^n\) & \(\dfrac{1}{n+1} x^{n+1}\) \\ - \(a x^{-n}\) &\(a \cdot \log_e |x| + c\)\\ - \(\dfrac{1}{ax+b}\) &\(\dfrac{1}{a} \log_e (ax+b) + c\)\\ - \((ax+b)^n\) & \(\dfrac{1}{a(n+1)}(ax+b)^{n-1} + c\>|\>n\ne 1\)\\ - \((ax+b)^{-1}\) & \(\dfrac{1}{a}\log_e |ax+b|+c\)\\ - \(e^{kx}\) & \(\dfrac{1}{k} e^{kx} + c\)\\ - \(e^k\) & \(e^kx + c\)\\ - \(\sin kx\) & \(\dfrac{-1}{k} \cos (kx) + c\)\\ - \(\cos kx\) & \(\dfrac{1}{k} \sin (kx) + c\)\\ - \(\sec^2 kx\) & \(\dfrac{1}{k} \tan(kx) + c\)\\ - \(\dfrac{1}{\sqrt{a^2-x^2}}\) & \(\sin^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\ - \(\dfrac{-1}{\sqrt{a^2-x^2}}\) & \(\cos^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\ - \(\frac{a}{a^2-x^2}\) & \(\tan^{-1} \frac{x}{a} + c\)\\ - \(\frac{f^\prime (x)}{f(x)}\) & \(\log_e f(x) + c\)\\ - \(\int f(u) \cdot \frac{du}{dx} \cdot dx\) & \(\int f(u) \cdot du\) \hfill(substitution)\\ - \(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\ - \hline - \end{tabularx} -\vspace*{\fill} - \columnbreak + +\rowcolors{1}{white}{lblue} +\renewcommand{\arraystretch}{1.4} + +\begin{tabularx}{\columnwidth}{rX} + \hline + \(f(x)\) & \(\int f(x) \cdot dx\) \\ + \hline + \(k\) (constant) & \(kx + c\)\\ + \(x^n\) & \(\dfrac{1}{n+1} x^{n+1}\) \\ + \(a x^{-n}\) &\(a \cdot \log_e |x| + c\)\\ + \(\dfrac{1}{ax+b}\) &\(\dfrac{1}{a} \log_e (ax+b) + c\)\\ + \((ax+b)^n\) & \(\dfrac{1}{a(n+1)}(ax+b)^{n-1} + c\>|\>n\ne 1\)\\ + \((ax+b)^{-1}\) & \(\dfrac{1}{a}\log_e |ax+b|+c\)\\ + \(e^{kx}\) & \(\dfrac{1}{k} e^{kx} + c\)\\ + \(e^k\) & \(e^kx + c\)\\ + \(\sin kx\) & \(\dfrac{-1}{k} \cos (kx) + c\)\\ + \(\cos kx\) & \(\dfrac{1}{k} \sin (kx) + c\)\\ + \(\sec^2 kx\) & \(\dfrac{1}{k} \tan(kx) + c\)\\ + \(\dfrac{1}{\sqrt{a^2-x^2}}\) & \(\sin^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\ + \(\dfrac{-1}{\sqrt{a^2-x^2}}\) & \(\cos^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\ + \(\frac{a}{a^2-x^2}\) & \(\tan^{-1} \frac{x}{a} + c\)\\ + \(\frac{f^\prime (x)}{f(x)}\) & \(\log_e f(x) + c\)\\ + \(\int f(u) \cdot \frac{du}{dx} \cdot dx\) & \(\int f(u) \cdot du\) \hfill(substitution)\\ + \(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\ + \hline +\end{tabularx} + +\end{document} diff --git a/methods/methods-collated.pdf b/methods/methods-collated.pdf index 12df5ce4067b0bbb5084e0c5eca856038f475d11..6c67429189e49d12f22f03884dd47ac8e2957943 100644 GIT binary patch delta 156827 zcmV(%K;pl~of(9J9FQXeFgP`nVTCAv?Okh+8^@LXu3yocPa|x|)cc`q7RUy1;vf$= 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@@ -32,17 +33,30 @@ \usetikzlibrary{% angles, + arrows, + arrows.meta, calc, datavisualization.formats.functions, decorations, decorations.markings, + decorations.text, decorations.pathreplacing, decorations.text, scopes } + \newcommand{\midarrow}{\tikz \draw[-triangle 90] (0,0) -- +(.1,0);} + \usepgflibrary{arrows.meta} -\pgfplotsset{compat=1.6} +\pgfplotsset{compat=1.16} +\pgfplotsset{every axis/.append style={ + axis x line=middle, % centre axes + axis y line=middle, + axis line style={->}, % arrows on axes + xlabel={$x$}, % axes labels + ylabel={$y$} +}} + \psset{dimen=monkey,fillstyle=solid,opacity=.5} \def\object{% \psframe[linestyle=none,fillcolor=blue](-2,-1)(2,1) @@ -51,29 +65,23 @@ \psline{->}(0,-2)% \uput[-90]{*0}(0,-2){$\vec{w}$}} } -\newcommand{\tg}{\mathop{\mathrm{tg}}} -\newcommand{\cotg}{\mathop{\mathrm{cotg}}} -\newcommand{\arctg}{\mathop{\mathrm{arctg}}} -\newcommand{\arccotg}{\mathop{\mathrm{arccotg}}} -\pgfplotsset{every axis/.append style={ - axis x line=middle, % centre axes - axis y line=middle, - axis line style={->}, % arrows on axes - xlabel={$x$}, % axes labels - ylabel={$y$} -}} \pagestyle{fancy} \fancyhead[LO,LE]{Year 12 Methods} \fancyhead[CO,CE]{Andrew Lorimer} \fancypagestyle{plain}{\fancyhead[LO,LE]{} \fancyhead[CO,CE]{}} % rm title & author for first page +\newcommand{\tg}{\mathop{\mathrm{tg}}} +\newcommand{\cotg}{\mathop{\mathrm{cotg}}} +\newcommand{\arctg}{\mathop{\mathrm{arctg}}} +\newcommand{\arccotg}{\mathop{\mathrm{arccotg}}} + \providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} +\newcommand*\leftlap[3][\,]{#1\hphantom{#2}\mathllap{#3}} +\newcommand*\rightlap[2]{\mathrlap{#2}\hphantom{#1}} \linespread{1.5} \setlength{\parindent}{0cm} \setlength\fboxsep{0pt} \setlength\fboxrule{.2pt} % for the \fboxes -\newcommand*\leftlap[3][\,]{#1\hphantom{#2}\mathllap{#3}} -\newcommand*\rightlap[2]{\mathrlap{#2}\hphantom{#1}} \newcolumntype{L}[1]{>{\hsize=#1\hsize\raggedright\arraybackslash}X} 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