methods / calculus.texon commit [chem] start organic reactions summary document (5447e6c)
   1\documentclass[methods-collated.tex]{subfiles}
   2
   3\begin{document}
   4
   5\section{Calculus}
   6
   7\subsection*{Average rate of change}
   8
   9\[m \operatorname{of} x \in [a,b] = \dfrac{f(b)-f(a)}{b - a} = \frac{dy}{dx}\]
  10
  11\colorbox{cas}{On CAS:} Action \(\rightarrow\) Calculation
  12\(\rightarrow\) \texttt{diff}
  13
  14\subsection*{Average value}
  15
  16\[ f_{\text{avg}} = \dfrac{1}{b-a} \int^b_a f(x) \> dx \]
  17
  18\subsection*{Instantaneous rate of change}
  19
  20\textbf{Secant} - line passing through two points on a curve\\
  21\textbf{Chord} - line segment joining two points on a curve
  22
  23\subsection*{Limit theorems}
  24
  25\begin{enumerate}
  26    \def\labelenumi{\arabic{enumi}.}
  27    \tightlist
  28  \item For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\)
  29  \item \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\)
  30  \item \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\)
  31  \item \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\)
  32\end{enumerate}
  33
  34A function is continuous if \(L^-=L^+=f(x)\) for all values of \(x\).
  35
  36\subsection*{First principles derivative}
  37
  38\[f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}\]
  39
  40Not differentiable at:
  41\begin{itemize}
  42    \tightlist
  43  \item discontinuous points
  44  \item sharp point/cusp
  45  \item vertical tangents (\(\infty\) gradient)
  46\end{itemize}
  47
  48\subsection*{Tangents \& gradients}
  49
  50\textbf{Tangent line} - defined by \(y=mx+c\) where
  51\(m={dy \over dx}\)\\
  52\textbf{Normal line} - \(\perp\) tangent
  53(\(m_{{tan}} \cdot m_{\operatorname{norm}} = -1\))\\
  54\textbf{Secant} \(={{f(x+h)-f(x)} \over h}\)
  55
  56\colorbox{cas}{On CAS:} \\ Action \(\rightarrow\) Calculation
  57\(\rightarrow\) Line \(\rightarrow\) \texttt{tanLine} or \texttt{normal}
  58
  59\subsection*{Strictly increasing/decreasing}
  60
  61For \(x_2\) and \(x_1\) where \(x_2 > x_1\):
  62
  63\begin{itemize}
  64    \tightlist
  65  \item \textbf{strictly increasing}\\ where \(f(x_2) > f(x_1)\) or \(f^\prime(x)>0\)
  66  \item \textbf{strictly decreasing}\\ where \(f(x_2) < f(x_1)\) or \(f^\prime(x)<0\)
  67  \item Endpoints are included, even where gradient \(=0\)
  68\end{itemize}
  69
  70\begin{cas}
  71  \colorbox{cas}{\textbf{In main}}: type function. Interactive
  72  \(\rightarrow\) Calculation \(\rightarrow\) Line \(\rightarrow\) (Normal
  73  \textbar{} Tan line)\\
  74  \colorbox{cas}{\textbf{In graph}}: define function. Analysis
  75  \(\rightarrow\) Sketch \(\rightarrow\) (Normal \textbar{} Tan line).
  76  Type \(x\) value to solve for a point. Return to show equation for line.
  77\end{cas}
  78
  79\subsection*{Stationary points}
  80
  81\begin{align*}
  82  \textbf{Stationary point:} && f^\prime(x) &= 0 \\
  83  \textbf{Point of inflection:} && f^{\prime\prime} &= 0
  84\end{align*}
  85
  86\begin{tikzpicture}
  87  \begin{axis}[xmin=-21, xmax=21, ymax=1400, ymin=-1000, ticks=none, axis lines=middle]
  88    \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] {\(f(x)\)};
  89    \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] {\(f^\prime(x)\)};
  90    \addplot[mark=*, blue] coordinates {(1,286)} node[above right, align=left, font=\footnotesize]{inflection \\ (falling)} ;
  91    \addplot[mark=*, orange] coordinates {(-6,972)} node[above left, align=right, font=\footnotesize]{stationary \\ (local max)} ;
  92    \addplot[mark=*, orange] coordinates {(8,-400)} node[below, align=left, font=\footnotesize]{stationary \\ (local min)} ;
  93  \end{axis}
  94\end{tikzpicture}\\
  95\begin{tikzpicture}
  96  \begin{axis}[enlargelimits=true, xmax=3.5, ticks=none, axis lines=middle]
  97    \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] {\(f(x)\)};
  98    \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] {\(f^\prime(x)\)};
  99    \addplot[mark=*, purple] coordinates {(2,2)} node[below right, align=left, font=\footnotesize]{stationary \\ inflection} ;
 100  \end{axis}
 101\end{tikzpicture}
 102
 103\subsection*{Derivatives}
 104
 105\rowcolors{1}{white}{peach}
 106\renewcommand{\arraystretch}{1.4}
 107
 108\begin{tabularx}{\columnwidth}{rX}
 109  \hline
 110  \hspace{6em}\(f(x)\) & \(f^\prime(x)\)\\
 111  \hline
 112  \(\sin x\) & \(\cos x\)\\
 113  \(\sin ax\) & \(a\cos ax\)\\
 114  \(\cos x\) & \(-\sin x\)\\
 115  \(\cos ax\) & \(-a \sin ax\)\\
 116  \(\tan f(x)\) & \(f^2(x) \sec^2f(x)\)\\
 117  \(e^x\) & \(e^x\)\\
 118  \(e^{ax}\) & \(ae^{ax}\)\\
 119  \(ax^{nx}\) & \(an \cdot e^{nx}\)\\
 120  \(\log_e x\) & \(\dfrac{1}{x}\)\\
 121  \(\log_e {ax}\) & \(\dfrac{1}{x}\)\\
 122  \(\log_e f(x)\) & \(\dfrac{f^\prime (x)}{f(x)}\)\\
 123  \(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\\
 124  \(\sin^{-1} x\) & \(\dfrac{1}{\sqrt{1-x^2}}\)\\
 125  \(\cos^{-1} x\) & \(\dfrac{-1}{\sqrt{1-x^2}}\)\\
 126  \(\tan^{-1} x\) & \(\dfrac{1}{1 + x^2}\)\\
 127  \(\frac{d}{dy}f(y)\) & \(\dfrac{1}{\frac{dx}{dy}}\) \hfill(reciprocal)\\
 128  \(uv\) & \(u \frac{dv}{dx}+v\frac{du}{dx}\) \hfill(product rule)\\
 129  \(\dfrac{u}{v}\) & \(\dfrac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\) \hfill(quotient rule)\\
 130  \(f(g(x))\) & \(f^\prime(g(x))\cdot g^\prime(x)\)\\
 131  \hline
 132\end{tabularx}
 133
 134\subsection*{Antiderivatives}
 135
 136\rowcolors{1}{white}{lblue}
 137\renewcommand{\arraystretch}{1.4}
 138
 139\begin{tabularx}{\columnwidth}{rX}
 140  \hline
 141  \(f(x)\) & \(\int f(x) \cdot dx\) \\
 142  \hline
 143  \(k\) (constant) & \(kx + c\)\\
 144  \(x^n\) & \(\dfrac{1}{n+1} x^{n+1}\) \\
 145  \(a x^{-n}\) &\(a \cdot \log_e |x| + c\)\\
 146  \(\dfrac{1}{ax+b}\) &\(\dfrac{1}{a} \log_e (ax+b) + c\)\\
 147  \((ax+b)^n\) & \(\dfrac{1}{a(n+1)}(ax+b)^{n-1} + c\>|\>n\ne 1\)\\
 148  \((ax+b)^{-1}\) & \(\dfrac{1}{a}\log_e |ax+b|+c\)\\
 149  \(e^{kx}\) & \(\dfrac{1}{k} e^{kx} + c\)\\
 150  \(e^k\) & \(e^kx + c\)\\
 151  \(\sin kx\) & \(\dfrac{-1}{k} \cos (kx) + c\)\\
 152  \(\cos kx\) & \(\dfrac{1}{k} \sin (kx) + c\)\\
 153  \(\sec^2 kx\) & \(\dfrac{1}{k} \tan(kx) + c\)\\
 154  \(\dfrac{1}{\sqrt{a^2-x^2}}\) & \(\sin^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
 155  \(\dfrac{-1}{\sqrt{a^2-x^2}}\) & \(\cos^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
 156  \(\frac{a}{a^2-x^2}\) & \(\tan^{-1} \frac{x}{a} + c\)\\
 157  \(\frac{f^\prime (x)}{f(x)}\) & \(\log_e f(x) + c\)\\
 158  \(\int f(u) \cdot \frac{du}{dx} \cdot dx\) & \(\int f(u) \cdot du\) \hfill(substitution)\\
 159  \(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\
 160  \hline
 161\end{tabularx}
 162
 163\end{document}