spec / calculus-ref.texon commit Merge branch 'master' of ssh://charles/tank/andrew/school/notes (06ac421)
   1\PassOptionsToPackage{unicode=true}{hyperref} % options for packages loaded elsewhere
   2\PassOptionsToPackage{hyphens}{url}
   3%
   4\documentclass[twocolumn]{article}
   5\usepackage{array}
   6\usepackage{lmodern}
   7\usepackage{amssymb,amsmath}
   8\usepackage{ifxetex,ifluatex}
   9\usepackage{multicol}
  10\usepackage{fixltx2e} % provides \textsubscript
  11\ifnum 0\ifxetex 1\fi\ifluatex 1\fi=0 % if pdftex
  12  \usepackage[T1]{fontenc}
  13  \usepackage[utf8]{inputenc}
  14  \usepackage{textcomp} % provides euro and other symbols
  15\else % if luatex or xelatex
  16  \usepackage{unicode-math}
  17  \defaultfontfeatures{Ligatures=TeX,Scale=MatchLowercase}
  18\fi
  19% use upquote if available, for straight quotes in verbatim environments
  20\IfFileExists{upquote.sty}{\usepackage{upquote}}{}
  21% use microtype if available
  22\IfFileExists{microtype.sty}{%
  23\usepackage[]{microtype}
  24\UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts
  25}{}
  26\IfFileExists{parskip.sty}{%
  27\usepackage{parskip}
  28}{% else
  29\setlength{\parindent}{0pt}
  30\setlength{\parskip}{6pt plus 2pt minus 1pt}
  31}
  32\usepackage{hyperref}
  33\hypersetup{
  34            pdfauthor={Andrew Lorimer},
  35            pdfborder={0 0 0},
  36            breaklinks=true}
  37\urlstyle{same}  % don't use monospace font for urls
  38\usepackage[margin=2cm]{geometry}
  39\usepackage{supertabular, booktabs}
  40% Fix footnotes in tables (requires footnote package)
  41\IfFileExists{footnote.sty}{\usepackage{footnote}\makesavenoteenv{supertabular}}{}
  42\usepackage{graphicx,grffile}
  43\makeatletter
  44\def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth\else\Gin@nat@width\fi}
  45\def\maxheight{\ifdim\Gin@nat@height>\textheight\textheight\else\Gin@nat@height\fi}
  46\makeatother
  47% Scale images if necessary, so that they will not overflow the page
  48% margins by default, and it is still possible to overwrite the defaults
  49% using explicit options in \includegraphics[width, height, ...]{}
  50\setkeys{Gin}{width=\maxwidth,height=\maxheight,keepaspectratio}
  51\setlength{\emergencystretch}{3em}  % prevent overfull lines
  52\providecommand{\tightlist}{%
  53  \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
  54\setcounter{secnumdepth}{0}
  55% Redefines (sub)paragraphs to behave more like sections
  56\ifx\paragraph\undefined\else
  57\let\oldparagraph\paragraph
  58\renewcommand{\paragraph}[1]{\oldparagraph{#1}\mbox{}}
  59\fi
  60\ifx\subparagraph\undefined\else
  61\let\oldsubparagraph\subparagraph
  62\renewcommand{\subparagraph}[1]{\oldsubparagraph{#1}\mbox{}}
  63\fi
  64
  65% set default figure placement to htbp
  66\makeatletter
  67\def\fps@figure{htbp}
  68\makeatother
  69
  70\usepackage{supertabular}
  71
  72\author{Andrew Lorimer}
  73\date{}
  74
  75\pagenumbering{gobble}
  76
  77\begin{document}
  78% \begin{multicols}{2}
  79\renewcommand{\arraystretch}{1.5}
  80
  81\hypertarget{spec---calculus}{%
  82\section{Spec - Calculus}\label{spec---calculus}}
  83
  84\hypertarget{gradients}{%
  85\subsection{Gradients}\label{gradients}}
  86
  87\[m \operatorname{of} x \in [a,b] = {{f(b)-f(a)}\over {b - a}} = {dy \over dx}\]
  88
  89\hypertarget{limit-theorems}{%
  90\subsection{Limit theorems}\label{limit-theorems}}
  91
  92\begin{enumerate}
  93\def\labelenumi{\arabic{enumi}.}
  94\tightlist
  95\item
  96  For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\)
  97\item
  98  \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\)
  99\item
 100  \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\)
 101\item
 102  \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\)
 103\end{enumerate}
 104
 105\hypertarget{first-principles-derivative}{%
 106\subsection{First principles
 107derivative}\label{first-principles-derivative}}
 108
 109\[f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}\]
 110
 111\hypertarget{tangents-gradients}{%
 112\subsection{Tangents \& gradients}\label{tangents-gradients}}
 113
 114\textbf{Tangent line} - defined by \(y=mx+c\) where
 115\(m={dy \over dx}\)\\
 116\textbf{Normal line} - \(\perp\) tangent
 117(\(m_{\operatorname{tan}} \cdot m_{\operatorname{norm}} = -1\))\\
 118\textbf{Secant} \(={{f(x+h)-f(x)} \over h}\)
 119
 120\hypertarget{derivatives}{%
 121\subsection{Derivatives}\label{derivatives}}
 122
 123\tablehead{\hline \(f(x)\) & \(f^\prime(x)\) \\ \hline \\}
 124\begin{supertabular}[]{@{}ll@{}}
 125
 126\(kx^n\) & \(knx^{n-1}\)\tabularnewline
 127\(g(x) \pm h(x)\) & \(g^\prime (x) \pm h^\prime (x)\)\tabularnewline
 128\(c\) & \(0\)\tabularnewline
 129\({u \over v}\) &
 130\({{(v{du \over dx} - u{dv \over dx}}) \div v^2}\)\tabularnewline
 131\(uv\) & \(u{dv \over dx} + v{du \over dx}\)\tabularnewline
 132\(f \circ g\) & \({dy \over du} \cdot {du \over dx}\)\tabularnewline
 133\(\sin ax\) & \(a\cos ax\)\tabularnewline
 134\(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\tabularnewline
 135\(\cos ax\) & \(-a \sin ax\)\tabularnewline
 136\(\cos(f(x))\) & \(f^\prime(x)(-\sin(f(x)))\) \\
 137\(e^{ax}\) & \(ae^{ax}\)\tabularnewline
 138\(\log_e {ax}\) & \(1 \over x\)\tabularnewline
 139\(\log_e f(x)\) & \(f^\prime (x) \over f(x)\)\tabularnewline
 140
 141\end{supertabular}
 142
 143\hypertarget{product-rule-for-yuv}{%
 144\subsection{\texorpdfstring{Product rule for
 145\(y=uv\)}{Product rule for y=uv}}\label{product-rule-for-yuv}}
 146
 147\[{dy \over dx} = u{dv \over dx} + v{du \over dx}\]
 148
 149\subsection{Chain rule for $(f\circ g)$}
 150
 151$${dy \over dx} = {dy \over du} \cdot {du \over dx}$$
 152
 153% Function notation:
 154
 155$${\displaystyle (f\circ g)'=(f'\circ g)\cdot g'}=f'(g(x)) \cdot g'(x)$$
 156
 157% $$(f\circ g)^\prime(x)=f^\prime(g(x))g^\prime(x),\quad \text{where}\hspace{0.3em} (f\circ g)(x)=f(g(x))$$}
 158
 159\hypertarget{logarithms}{%
 160\subsection{Logarithms}\label{logarithms}}
 161
 162\[\log_b (x) = n \quad \operatorname{where} \hspace{0.5em} b^n=x\]
 163
 164\subsubsection{Logarithmic identities}
 165$\log_b (xy)=\log_b x + \log_b y$ \\
 166$\log_b x^n = n \log_b x$ \\
 167$\log_b y^{x^n} = x^n \log_b y$
 168
 169\hypertarget{integration}{%
 170\subsection{Integration}\label{integration}}
 171
 172\[\int f(x) dx = F(x) + c\]
 173
 174% \begin{itemize}
 175% \tightlist
 176% \item
 177  area enclosed by curves
 178% \end{itemize}
 179
 180\tablehead{\hline \(f(x)\) & \(\int f(x) \cdot dx\) \\ \hline \\}
 181\begin{supertabular}[]{@{}ll@{}}
 182
 183
 184\(k\) (constant) & \(kx + c\) \\
 185\(x^n\) & \({1 \over {n+1}}x^{n+1} + c\) \\
 186\(a x^{-n}\) & \(a \cdot \log_e x + c\) \\
 187\(e^{kx}\) & \({1 \over k} e^{kx} + c\) \\
 188\(e^k\) & \(e^kx + c\) \\
 189\(\sin kx\) & \(-{1 \over k} \cos (kx) + c\) \\
 190\(\cos kx\) & \({1 \over k} \sin (kx) + c\) \\
 191\({f^\prime (x)} \over {f(x)}\) & \(\log_e f(x) + c\) \\
 192\(g^\prime(x)\cdot f^\prime(g(x)\) & \(f(g(x))\) (chain rule) \\
 193\(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\) \\
 194\({1 \over {ax+b}}\) & \({1 \over a} \log_e (ax+b) + c\) \\
 195\((ax+b)^n\) & \({1 \over {a(n+1)}}(ax+b)^{n-1} + c\) \\
 196
 197\end{supertabular}
 198
 199\hypertarget{definite-integrals}{%
 200\subsection{Definite integrals}\label{definite-integrals}}
 201
 202\[\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)_{}\]
 203
 204\hypertarget{kinematics}{%
 205\subsection{Kinematics}\label{kinematics}}
 206
 207\textbf{position \(x\)} - distance from origin or fixed point\\
 208\textbf{displacement \(s\)} - change in $x$ from starting point \\
 209\textbf{velocity \(v\)} - change in position with respect to time\\
 210\textbf{acceleration \(a\)} - change in velocity\\
 211\textbf{speed} - magnitude of velocity
 212
 213\large{
 214\tablehead{}
 215\begin{supertabular}[]{@{}ll@{}}
 216% \toprule
 217& no\tabularnewline
 218\(v=u+at\) & \(s\)\tabularnewline
 219\(s=ut + {1 \over 2} at^2\) & \(v\)\tabularnewline
 220\(v^2 = u^2 + 2as\) & \(t\)\tabularnewline
 221\(s= {1 \over 2}(u+v)t\) & \(a\)\tabularnewline
 222
 223
 224
 225\end{supertabular}
 226}
 227% \end{multicols}
 228\end{document}