+\documentclass[methods-collated.tex]{subfiles}
+
+\begin{document}
+
\section{Calculus}
\subsection*{Average rate of change}
\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 \(x\).
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}
(\(m_{{tan}} \cdot m_{\operatorname{norm}} = -1\))\\
\textbf{Secant} \(={{f(x+h)-f(x)} \over h}\)
-\colorbox{cas}{On CAS:} \\ Action \(\rightarrow\) Calculation
-\(\rightarrow\) Line \(\rightarrow\) \texttt{tanLine} or \texttt{normal}
+\begin{cas}
+ \textbf{In main}: Interactive \(\rightarrow\) Calculation \(\rightarrow\) Line \\
+ \-\hspace{1em} \texttt{tanLine(f(x), x, p)} \\
+ \-\hspace{1em} \texttt{normal(f(x), x, p)} \\
+ where \texttt{p} is the \(x\)-value of the coordinate
+
+ \textbf{In graph}: define function, then 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*{Strictly increasing/decreasing}
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.
-
\subsection*{Stationary points}
\begin{align*}
\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
-\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}
- \columnbreak
-\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}
-
+\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}
+
+\include{../spec/calculus-rules}
+
+\end{document}