\section{Calculus}

\subsection*{Average rate of change}

\[m \operatorname{of} x \in [a,b] = \dfrac{f(b)-f(a)}{b - a} = \frac{dy}{dx}\]

\colorbox{cas}{On CAS:} Action \(\rightarrow\) Calculation
\(\rightarrow\) \texttt{diff}

\subsection*{Average value}

\[ f_{\text{avg}} = \dfrac{1}{b-a} \int^b_a f(x) \> dx \]

\subsection*{Instantaneous rate of change}

\textbf{Secant} - line passing through two points on a curve\\
\textbf{Chord} - line segment joining two points on a curve

\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
  \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\)
\end{enumerate}

A function is continuous if \(L^-=L^+=f(x)\) for all values of \(x\).

\subsection*{First principles derivative}

\[f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}\]

Not differentiable at:
\begin{itemize}
\tightlist
\item
  discontinuous points
\item
  sharp point/cusp
\item
  vertical tangents (\(\infty\) gradient)
\end{itemize}

\subsection*{Tangents \& gradients}

\textbf{Tangent line} - defined by \(y=mx+c\) where
\(m={dy \over dx}\)\\
\textbf{Normal line} - \(\perp\) tangent
(\(m_{{tan}} \cdot m_{\operatorname{norm}} = -1\))\\
\textbf{Secant} \(={{f(x+h)-f(x)} \over h}\)

\colorbox{cas}{On CAS:} \\ Action \(\rightarrow\) Calculation
\(\rightarrow\) Line \(\rightarrow\) \texttt{tanLine} or \texttt{normal}

\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 \(=0\)
\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{Stationary point:} && f^\prime(x) &= 0 \\
  \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}

