From 4cafa00c92e478d7e1af90d89416a8852569b808 Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Tue, 11 Sep 2018 19:17:09 +1000 Subject: [PATCH] latex conversion of calculus ref --- spec/calculus-ref.tex | 228 ++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 228 insertions(+) create mode 100644 spec/calculus-ref.tex diff --git a/spec/calculus-ref.tex b/spec/calculus-ref.tex new file mode 100644 index 0000000..59d0c0c --- /dev/null +++ b/spec/calculus-ref.tex @@ -0,0 +1,228 @@ +\PassOptionsToPackage{unicode=true}{hyperref} % options for packages loaded elsewhere +\PassOptionsToPackage{hyphens}{url} +% +\documentclass[twocolumn]{article} +\usepackage{array} +\usepackage{lmodern} +\usepackage{amssymb,amsmath} +\usepackage{ifxetex,ifluatex} +\usepackage{multicol} +\usepackage{fixltx2e} % provides \textsubscript +\ifnum 0\ifxetex 1\fi\ifluatex 1\fi=0 % if pdftex + \usepackage[T1]{fontenc} + \usepackage[utf8]{inputenc} + \usepackage{textcomp} % provides euro and other symbols +\else % if luatex or xelatex + \usepackage{unicode-math} + \defaultfontfeatures{Ligatures=TeX,Scale=MatchLowercase} +\fi +% use upquote if available, for straight quotes in verbatim environments +\IfFileExists{upquote.sty}{\usepackage{upquote}}{} +% use microtype if available +\IfFileExists{microtype.sty}{% +\usepackage[]{microtype} +\UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts +}{} +\IfFileExists{parskip.sty}{% +\usepackage{parskip} +}{% else +\setlength{\parindent}{0pt} +\setlength{\parskip}{6pt plus 2pt minus 1pt} +} +\usepackage{hyperref} +\hypersetup{ + pdfauthor={Andrew Lorimer}, + pdfborder={0 0 0}, + breaklinks=true} +\urlstyle{same} % don't use monospace font for urls +\usepackage[margin=2cm]{geometry} +\usepackage{supertabular, booktabs} +% Fix footnotes in tables (requires footnote package) +\IfFileExists{footnote.sty}{\usepackage{footnote}\makesavenoteenv{supertabular}}{} +\usepackage{graphicx,grffile} +\makeatletter +\def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth\else\Gin@nat@width\fi} +\def\maxheight{\ifdim\Gin@nat@height>\textheight\textheight\else\Gin@nat@height\fi} +\makeatother +% Scale images if necessary, so that they will not overflow the page +% margins by default, and it is still possible to overwrite the defaults +% using explicit options in \includegraphics[width, height, ...]{} +\setkeys{Gin}{width=\maxwidth,height=\maxheight,keepaspectratio} +\setlength{\emergencystretch}{3em} % prevent overfull lines +\providecommand{\tightlist}{% + \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} +\setcounter{secnumdepth}{0} +% Redefines (sub)paragraphs to behave more like sections +\ifx\paragraph\undefined\else +\let\oldparagraph\paragraph +\renewcommand{\paragraph}[1]{\oldparagraph{#1}\mbox{}} +\fi +\ifx\subparagraph\undefined\else +\let\oldsubparagraph\subparagraph +\renewcommand{\subparagraph}[1]{\oldsubparagraph{#1}\mbox{}} +\fi + +% set default figure placement to htbp +\makeatletter +\def\fps@figure{htbp} +\makeatother + +\usepackage{supertabular} + +\author{Andrew Lorimer} +\date{} + +\pagenumbering{gobble} + +\begin{document} +% \begin{multicols}{2} +\renewcommand{\arraystretch}{1.5} + +\hypertarget{spec---calculus}{% +\section{Spec - Calculus}\label{spec---calculus}} + +\hypertarget{gradients}{% +\subsection{Gradients}\label{gradients}} + +\[m \operatorname{of} x \in [a,b] = {{f(b)-f(a)}\over {b - a}} = {dy \over dx}\] + +\hypertarget{limit-theorems}{% +\subsection{Limit theorems}\label{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} + +\hypertarget{first-principles-derivative}{% +\subsection{First principles +derivative}\label{first-principles-derivative}} + +\[f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}\] + +\hypertarget{tangents-gradients}{% +\subsection{Tangents \& gradients}\label{tangents-gradients}} + +\textbf{Tangent line} - defined by \(y=mx+c\) where +\(m={dy \over dx}\)\\ +\textbf{Normal line} - \(\perp\) tangent +(\(m_{\operatorname{tan}} \cdot m_{\operatorname{norm}} = -1\))\\ +\textbf{Secant} \(={{f(x+h)-f(x)} \over h}\) + +\hypertarget{derivatives}{% +\subsection{Derivatives}\label{derivatives}} + +\tablehead{\hline \(f(x)\) & \(f^\prime(x)\) \\ \hline \\} +\begin{supertabular}[]{@{}ll@{}} + +\(kx^n\) & \(knx^{n-1}\)\tabularnewline +\(g(x) \pm h(x)\) & \(g^\prime (x) \pm h^\prime (x)\)\tabularnewline +\(c\) & \(0\)\tabularnewline +\({u \over v}\) & +\({{(v{du \over dx} - u{dv \over dx}}) \div v^2}\)\tabularnewline +\(uv\) & \(u{dv \over dx} + v{du \over dx}\)\tabularnewline +\(f \circ g\) & \({dy \over du} \cdot {du \over dx}\)\tabularnewline +\(\sin ax\) & \(a\cos ax\)\tabularnewline +\(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\tabularnewline +\(\cos ax\) & \(-a \sin ax\)\tabularnewline +\(\cos(f(x))\) & \(f^\prime(x)(-\sin(f(x)))\) \\ +\(e^{ax}\) & \(ae^{ax}\)\tabularnewline +\(\log_e {ax}\) & \(1 \over x\)\tabularnewline +\(\log_e f(x)\) & \(f^\prime (x) \over f(x)\)\tabularnewline + +\end{supertabular} + +\hypertarget{product-rule-for-yuv}{% +\subsection{\texorpdfstring{Product rule for +\(y=uv\)}{Product rule for y=uv}}\label{product-rule-for-yuv}} + +\[{dy \over dx} = u{dv \over dx} + v{du \over dx}\] + +\subsection{Chain rule for $(f\circ g)$} + +$${dy \over dx} = {dy \over du} \cdot {du \over dx}$$ + +% Function notation: + +$${\displaystyle (f\circ g)'=(f'\circ g)\cdot g'}=f'(g(x)) \cdot g'(x)$$ + +% $$(f\circ g)^\prime(x)=f^\prime(g(x))g^\prime(x),\quad \text{where}\hspace{0.3em} (f\circ g)(x)=f(g(x))$$} + +\hypertarget{logarithms}{% +\subsection{Logarithms}\label{logarithms}} + +\[\log_b (x) = n \quad \operatorname{where} \hspace{0.5em} b^n=x\] + +\subsubsection{Logarithmic identities} +$\log_b (xy)=\log_b x + \log_b y$ \\ +$\log_b x^n = n \log_b x$ \\ +$\log_b y^{x^n} = x^n \log_b y$ + +\hypertarget{integration}{% +\subsection{Integration}\label{integration}} + +\[\int f(x) dx = F(x) + c\] + +% \begin{itemize} +% \tightlist +% \item + area enclosed by curves +% \end{itemize} + +\tablehead{\hline \(f(x)\) & \(\int f(x) \cdot dx\) \\ \hline \\} +\begin{supertabular}[]{@{}ll@{}} + + +\(k\) (constant) & \(kx + c\) \\ +\(x^n\) & \({1 \over {n+1}}x^{n+1} + c\) \\ +\(a x^{-n}\) & \(a \cdot \log_e x + c\) \\ +\(e^{kx}\) & \({1 \over k} e^{kx} + c\) \\ +\(e^k\) & \(e^kx + c\) \\ +\(\sin kx\) & \(-{1 \over k} \cos (kx) + c\) \\ +\(\cos kx\) & \({1 \over k} \sin (kx) + c\) \\ +\({f^\prime (x)} \over {f(x)}\) & \(\log_e f(x) + c\) \\ +\(g^\prime(x)\cdot f^\prime(g(x)\) & \(f(g(x))\) (chain rule) \\ +\(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\) \\ +\({1 \over {ax+b}}\) & \({1 \over a} \log_e (ax+b) + c\) \\ +\((ax+b)^n\) & \({1 \over {a(n+1)}}(ax+b)^{n-1} + c\) \\ + +\end{supertabular} + +\hypertarget{definite-integrals}{% +\subsection{Definite integrals}\label{definite-integrals}} + +\[\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)_{}\] + +\hypertarget{kinematics}{% +\subsection{Kinematics}\label{kinematics}} + +\textbf{position \(x\)} - distance from origin or fixed point\\ +\textbf{displacement \(s\)} - change in $x$ from starting point \\ +\textbf{velocity \(v\)} - change in position with respect to time\\ +\textbf{acceleration \(a\)} - change in velocity\\ +\textbf{speed} - magnitude of velocity + +\large{ +\tablehead{} +\begin{supertabular}[]{@{}ll@{}} +% \toprule +& no\tabularnewline +\(v=u+at\) & \(s\)\tabularnewline +\(s=ut + {1 \over 2} at^2\) & \(v\)\tabularnewline +\(v^2 = u^2 + 2as\) & \(t\)\tabularnewline +\(s= {1 \over 2}(u+v)t\) & \(a\)\tabularnewline + + + +\end{supertabular} +} +% \end{multicols} +\end{document} -- 2.47.1