From f2f603c4d052742995463d39f42b9de55843cafc Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Sun, 8 Sep 2019 22:10:33 +1000 Subject: [PATCH] [methods] add statistics notes to main document --- methods/calculus.tex | 4 +- methods/circ-functions.tex | 34 +- methods/methods-collated.pdf | Bin 431179 -> 445070 bytes methods/methods-collated.poly.gnuplot | 2 +- methods/methods-collated.poly.table | 1082 +++---------------------- methods/methods-collated.tex | 679 ++++++++++++---- 6 files changed, 629 insertions(+), 1172 deletions(-) diff --git a/methods/calculus.tex b/methods/calculus.tex index 4478faa..6f57dc0 100644 --- a/methods/calculus.tex +++ b/methods/calculus.tex @@ -139,6 +139,7 @@ Type \(x\) value to solve for a point. Return to show equation for line. \(f(g(x))\) & \(f^\prime(g(x))\cdot g^\prime(x)\)\\ \hline \end{tabularx} + \vspace*{\fill} \columnbreak \subsection*{Antiderivatives} \rowcolors{1}{shade1}{cas} @@ -166,4 +167,5 @@ Type \(x\) value to solve for a point. Return to show equation for line. \(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\ \hline \end{tabularx} - +\vspace*{\fill} + \columnbreak diff --git a/methods/circ-functions.tex b/methods/circ-functions.tex index 7580791..2b615cd 100644 --- a/methods/circ-functions.tex +++ b/methods/circ-functions.tex @@ -5,20 +5,26 @@ \[1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi}\] \subsection*{Exact values} - - - \begin{tikzpicture}[scale=0.75] - \draw [orange, thick] (0,0) -- (3,3) node [black, pos=0.5, above left] {\(\sqrt{2}\)}; - \draw [orange, thick] (0,0) -- (3,0) node [black, below, pos=0.5] {\(1\)} node[black, above, pos=0.3] {\(\frac{\pi}{4}\)}; - \draw [orange, thick] (3,0) -- (3,3) node [black, right, pos=0.5] {1} node[black, left, pos=0.7] {\(\frac{\pi}{4}\)}; - \draw [black] (0,0) coordinate (A) (3,0) coordinate (B) (3,3) coordinate (C) pic [draw,black,angle radius=2mm] {right angle = A--B--C}; - \end{tikzpicture} - \begin{tikzpicture}[scale=0.75] - \draw [orange, thick] (0,3) -- (5.19,0) node [black, pos=0.5, above right] {2}; - \draw [orange, thick] (0,0) -- (5.19,0) node [black, below, pos=0.5] {\(\sqrt{3}\)} node[black, above, pos=0.7] {\(\frac{\pi}{6}\)}; - \draw [orange, thick] (0,0) -- (0,3) node [black, left, pos=0.5] {1} node [black, pos=0.8, right] {\(\frac{\pi}{3}\)}; - \draw [black] (5.19,0) coordinate (A) (0,0) coordinate (B) (0,3) coordinate (C) pic [draw,black,angle radius=2mm] {right angle = A--B--C}; - \end{tikzpicture} +\adjustbox{trim=0.7cm 0cm}{ + \begin{tikzpicture} + \matrix{ + \begin{scope}[scale=0.8] + \draw [orange, thick] (0,0) -- (3,3) node [black, pos=0.5, above left] {\(\sqrt{2}\)}; + \draw [orange, thick] (0,0) -- (3,0) node [black, below, pos=0.5] {\(1\)} node[black, above, pos=0.3] {\(\dfrac{\pi}{4}\)}; + \draw [orange, thick] (3,0) -- (3,3) node [black, right, pos=0.5] {1} node[black, left, pos=0.7] {\(\dfrac{\pi}{4}\)}; + \draw [black] (0,0) coordinate (A) (3,0) coordinate (B) (3,3) coordinate (C) pic [draw,black,angle radius=2mm] {right angle = A--B--C}; + \end{scope} + & + \begin{scope}[scale=0.8] + \draw [orange, thick] (0,3) -- (5.19,0) node [black, pos=0.5, above right] {2}; + \draw [orange, thick] (0,0) -- (5.19,0) node [black, below, pos=0.5] {\(\sqrt{3}\)} node[black, above, pos=0.7] {\(\dfrac{\pi}{6}\)}; + \draw [orange, thick] (0,0) -- (0,3) node [black, left, pos=0.5] {1} node [black, pos=0.8, right] {\(\dfrac{\pi}{3}\)}; + \draw [black] (5.19,0) coordinate (A) (0,0) coordinate (B) (0,3) coordinate (C) pic [draw,black,angle radius=2mm] {right angle = A--B--C}; + \end{scope} + \\ + }; + \end{tikzpicture} +} \subsection*{Compound angle formulas} diff --git a/methods/methods-collated.pdf b/methods/methods-collated.pdf index acdd213f7b5a629d240207140dbf1d93941fdadf..9c6cbbb907fc6cdf86b9a9fdd742b794ba8a90a8 100644 GIT binary patch delta 373167 zcmV(`K-0g=s2Yxy8;~WJF&YCYf9+jqj~vI5{mx&}>rXRScRM5Rq}L004a=6{12Sb8 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zr76>UuT|p%>phv**9nIQ_VPM}w4~D{QF(&lU%4Ujtb?26|2V*nav=Z8*Ao zwc8Fl#Ph7GSwHnUxEcQG-Z_nTVB$`MXou!NK*rlbwi(NEwz<+TPsZ+v50dON>Z| zPnb!BNt~07jfsg}RFq4MSy-HlosE-~U4oTOoQac7On~VBcR~D~NY2dO!qt+9m7AF; zy#Wc53P3)?E2pA_H+=JTwdoQ+(jP^lNb}p07Q3fxn85c>kC*+oucO1m96XdnQpu9WoE%~cmFg~*QQMl#!LBMP8(f~Ok8P4p6>@EtJbj`^nIyO=F zJ~g*6ohfNieoT@PQS$r^88wA?!q=7SQw|?-@SpUIOBrdHYd7U#l@fSOt@#@SZh270 z+buT@;i*I!W$`NMg!u$yZUyU62viCbWl4r4Suz>peU?f}4XGTq1PN#KsdzR1K6WmE zJD?fZk8o+V5wIpuwE+I)CsU1m`OmKwe7#h@7NhNKPMg!nRO<{wZn<~OEjw!rTo7%G>MxD2-=Bo<-THwvW3v=^5O`WnU2yn>Y`DNa6**6xX zj>FMY>qf}I30y$n{kiBZw3!66j06SB%lxDh2a?h#_T&Hd%zw5VjICi^T#cMvJ)O0s9{#_kW!L diff --git a/methods/methods-collated.poly.gnuplot b/methods/methods-collated.poly.gnuplot index 1c600c1..c816031 100644 --- a/methods/methods-collated.poly.gnuplot +++ b/methods/methods-collated.poly.gnuplot @@ -1,2 +1,2 @@ set table "methods-collated.poly.table"; set format "%.5f" -set format "%.7e";; set samples 1000; set dummy x; plot [x=-2:2] sgn(x)*(abs(x)**(1./3)) ; +set format "%.7e";; set samples 100; set dummy x; plot [x=-2:2] sgn(x)*(abs(x)**(1./3)) ; diff --git a/methods/methods-collated.poly.table b/methods/methods-collated.poly.table index e981d3b..02e0dc8 100644 --- a/methods/methods-collated.poly.table +++ b/methods/methods-collated.poly.table @@ -1,1005 +1,105 @@ -# Curve 0 of 1, 1000 points +# Curve 0 of 1, 100 points # Curve title: "sgn(x)*(abs(x)**(1./3))" # x y type -2.0000000e+00 -1.2599210e+00 i --1.9959960e+00 -1.2590797e+00 i --1.9919920e+00 -1.2582372e+00 i --1.9879880e+00 -1.2573936e+00 i --1.9839840e+00 -1.2565489e+00 i --1.9799800e+00 -1.2557030e+00 i --1.9759760e+00 -1.2548560e+00 i --1.9719720e+00 -1.2540078e+00 i --1.9679680e+00 -1.2531585e+00 i --1.9639640e+00 -1.2523080e+00 i --1.9599600e+00 -1.2514564e+00 i --1.9559560e+00 -1.2506036e+00 i --1.9519520e+00 -1.2497497e+00 i --1.9479479e+00 -1.2488946e+00 i --1.9439439e+00 -1.2480383e+00 i --1.9399399e+00 -1.2471808e+00 i --1.9359359e+00 -1.2463222e+00 i --1.9319319e+00 -1.2454624e+00 i --1.9279279e+00 -1.2446013e+00 i --1.9239239e+00 -1.2437391e+00 i --1.9199199e+00 -1.2428757e+00 i --1.9159159e+00 -1.2420111e+00 i --1.9119119e+00 -1.2411453e+00 i --1.9079079e+00 -1.2402783e+00 i --1.9039039e+00 -1.2394100e+00 i --1.8998999e+00 -1.2385406e+00 i --1.8958959e+00 -1.2376699e+00 i --1.8918919e+00 -1.2367980e+00 i --1.8878879e+00 -1.2359249e+00 i --1.8838839e+00 -1.2350505e+00 i --1.8798799e+00 -1.2341749e+00 i --1.8758759e+00 -1.2332980e+00 i --1.8718719e+00 -1.2324199e+00 i --1.8678679e+00 -1.2315406e+00 i --1.8638639e+00 -1.2306599e+00 i --1.8598599e+00 -1.2297781e+00 i --1.8558559e+00 -1.2288949e+00 i --1.8518519e+00 -1.2280105e+00 i --1.8478478e+00 -1.2271248e+00 i --1.8438438e+00 -1.2262378e+00 i --1.8398398e+00 -1.2253496e+00 i --1.8358358e+00 -1.2244600e+00 i --1.8318318e+00 -1.2235692e+00 i --1.8278278e+00 -1.2226771e+00 i --1.8238238e+00 -1.2217836e+00 i --1.8198198e+00 -1.2208889e+00 i --1.8158158e+00 -1.2199928e+00 i --1.8118118e+00 -1.2190954e+00 i --1.8078078e+00 -1.2181967e+00 i --1.8038038e+00 -1.2172967e+00 i --1.7997998e+00 -1.2163953e+00 i --1.7957958e+00 -1.2154926e+00 i --1.7917918e+00 -1.2145885e+00 i --1.7877878e+00 -1.2136831e+00 i --1.7837838e+00 -1.2127764e+00 i --1.7797798e+00 -1.2118683e+00 i --1.7757758e+00 -1.2109588e+00 i --1.7717718e+00 -1.2100480e+00 i --1.7677678e+00 -1.2091358e+00 i --1.7637638e+00 -1.2082222e+00 i --1.7597598e+00 -1.2073072e+00 i --1.7557558e+00 -1.2063908e+00 i --1.7517518e+00 -1.2054731e+00 i --1.7477477e+00 -1.2045539e+00 i --1.7437437e+00 -1.2036334e+00 i --1.7397397e+00 -1.2027114e+00 i --1.7357357e+00 -1.2017880e+00 i --1.7317317e+00 -1.2008632e+00 i --1.7277277e+00 -1.1999370e+00 i --1.7237237e+00 -1.1990093e+00 i --1.7197197e+00 -1.1980802e+00 i --1.7157157e+00 -1.1971497e+00 i --1.7117117e+00 -1.1962177e+00 i --1.7077077e+00 -1.1952842e+00 i --1.7037037e+00 -1.1943493e+00 i --1.6996997e+00 -1.1934129e+00 i --1.6956957e+00 -1.1924751e+00 i --1.6916917e+00 -1.1915357e+00 i --1.6876877e+00 -1.1905949e+00 i --1.6836837e+00 -1.1896526e+00 i --1.6796797e+00 -1.1887088e+00 i --1.6756757e+00 -1.1877635e+00 i --1.6716717e+00 -1.1868167e+00 i --1.6676677e+00 -1.1858684e+00 i --1.6636637e+00 -1.1849186e+00 i --1.6596597e+00 -1.1839672e+00 i --1.6556557e+00 -1.1830143e+00 i --1.6516517e+00 -1.1820599e+00 i --1.6476476e+00 -1.1811039e+00 i --1.6436436e+00 -1.1801464e+00 i --1.6396396e+00 -1.1791873e+00 i --1.6356356e+00 -1.1782267e+00 i --1.6316316e+00 -1.1772645e+00 i --1.6276276e+00 -1.1763007e+00 i --1.6236236e+00 -1.1753353e+00 i --1.6196196e+00 -1.1743684e+00 i --1.6156156e+00 -1.1733998e+00 i --1.6116116e+00 -1.1724297e+00 i --1.6076076e+00 -1.1714579e+00 i --1.6036036e+00 -1.1704845e+00 i --1.5995996e+00 -1.1695095e+00 i --1.5955956e+00 -1.1685329e+00 i --1.5915916e+00 -1.1675546e+00 i --1.5875876e+00 -1.1665747e+00 i --1.5835836e+00 -1.1655932e+00 i --1.5795796e+00 -1.1646100e+00 i --1.5755756e+00 -1.1636251e+00 i --1.5715716e+00 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+\usepackage[dvipsnames, table]{xcolor} +\usepackage{adjustbox} \usepackage{amsmath} \usepackage{amssymb} +\usepackage{blindtext} +\usepackage{enumitem} +\usepackage{fancyhdr} +\usepackage[a4paper,margin=2cm]{geometry} +\usepackage{graphicx} \usepackage{harpoon} +\usepackage{listings} +\usepackage{longtable} +\usepackage{makecell} +\usepackage{mathtools} +\usepackage{multicol} +\usepackage{multirow} +\usepackage{newclude} +\usepackage{pgfplots} +\usepackage{pst-plot} +\usepackage{standalone} \usepackage{tabularx} \usepackage{tabu} -\usepackage{makecell} -\usepackage[dvipsnames, table]{xcolor} -\usepackage{blindtext} -\usepackage{graphicx} -\usepackage{wrapfig} -\usepackage{tikz} +\usepackage{tcolorbox} \usepackage{tikz-3dplot} -\usepackage{pgfplots} -\pgfplotsset{compat=1.8} -\usepackage{mathtools} -\usetikzlibrary{calc} -\usetikzlibrary{angles} -\usetikzlibrary{datavisualization.formats.functions} -\usetikzlibrary{decorations.markings} +\usepackage{tikz} +\usepackage{tkz-fct} +\usepackage[obeyspaces]{url} +\usepackage{wrapfig} + + +\usetikzlibrary{% + angles, + calc, + datavisualization.formats.functions, + decorations, + decorations.markings, + decorations.pathreplacing, + decorations.text, + scopes +} +\newcommand{\midarrow}{\tikz \draw[-triangle 90] (0,0) -- +(.1,0);} \usepgflibrary{arrows.meta} -\usepackage{longtable} -\usepackage{fancyhdr} +\pgfplotsset{compat=1.8} +\psset{dimen=monkey,fillstyle=solid,opacity=.5} +\def\object{% + \psframe[linestyle=none,fillcolor=blue](-2,-1)(2,1) + \psaxes[linecolor=gray,labels=none,ticks=none]{->}(0,0)(-3,-3)(3,2)[$x$,0][$y$,90] + \rput{*0}{% + \psline{->}(0,-2)% + \uput[-90]{*0}(0,-2){$\vec{w}$}} +} +\newcommand{\tg}{\mathop{\mathrm{tg}}} +\newcommand{\cotg}{\mathop{\mathrm{cotg}}} +\newcommand{\arctg}{\mathop{\mathrm{arctg}}} +\newcommand{\arccotg}{\mathop{\mathrm{arccotg}}} +\pgfplotsset{every axis/.append style={ + axis x line=middle, % centre axes + axis y line=middle, + axis line style={->}, % arrows on axes + xlabel={$x$}, % axes labels + ylabel={$y$} +}} + \pagestyle{fancy} \fancyhead[LO,LE]{Year 12 Methods} \fancyhead[CO,CE]{Andrew Lorimer} \fancypagestyle{plain}{\fancyhead[LO,LE]{} \fancyhead[CO,CE]{}} % rm title & author for first page + \providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} +\linespread{1.5} \setlength{\parindent}{0cm} -\usepackage{mathtools} -\usepackage{xcolor} % used only to show the phantomed stuff \setlength\fboxsep{0pt} \setlength\fboxrule{.2pt} % for the \fboxes \newcommand*\leftlap[3][\,]{#1\hphantom{#2}\mathllap{#3}} \newcommand*\rightlap[2]{\mathrlap{#2}\hphantom{#1}} + \newcolumntype{L}[1]{>{\hsize=#1\hsize\raggedright\arraybackslash}X} \newcolumntype{R}[1]{>{\hsize=#1\hsize\raggedleft\arraybackslash}X} +\newcolumntype{Y}{>{\centering\arraybackslash}X} + \definecolor{cas}{HTML}{e6f0fe} +\definecolor{important}{HTML}{fc9871} +\definecolor{dark-gray}{gray}{0.2} \definecolor{shade1}{HTML}{ffffff} \definecolor{shade2}{HTML}{e6f2ff} \definecolor{shade3}{HTML}{cce2ff} -\linespread{1.5} -\newcommand{\midarrow}{\tikz \draw[-triangle 90] (0,0) -- +(.1,0);} -\newcommand{\tg}{\mathop{\mathrm{tg}}} -\newcommand{\cotg}{\mathop{\mathrm{cotg}}} -\newcommand{\arctg}{\mathop{\mathrm{arctg}}} -\newcommand{\arccotg}{\mathop{\mathrm{arccotg}}} -\pgfplotsset{every axis/.append style={ - axis x line=middle, % centre axes - axis y line=middle, - axis line style={->}, % arrows on axes - xlabel={$x$}, % axes labels - ylabel={$y$}, -}} + +\newtcolorbox{cas}{colframe=cas!75!black, title=On CAS, left*=3mm} +\newtcolorbox{warning}{colback=white!90!black, leftrule=3mm, colframe=important, coltext=important, fontupper=\sffamily\bfseries} + \begin{document} @@ -64,158 +96,199 @@ \date{} \maketitle -\begin{multicols}{2} - - \section{Functions} - - \begin{itemize} - \tightlist - \item vertical line test - \item each \(x\) value produces only one \(y\) value - \end{itemize} - \subsection*{One to one functions} +\section{Functions} - \begin{itemize} \tightlist - \item - \(f(x)\) is \emph{one to one} if \(f(a) \ne f(b)\) if - \(a, b \in \operatorname{dom}(f)\) and \(a \ne b\)\\ - \(\implies\) unique \(y\) for each \(x\) (\(\sin x\) is not 1:1, - \(x^3\) is) - \item - horizontal line test - \item - if not one to one, it is many to one - \end{itemize} +\begin{itemize} \tightlist + \item vertical line test + \item each \(x\) value produces only one \(y\) value +\end{itemize} - \subsection*{Odd and even functions} +\subsection*{One to one functions} + +\begin{itemize} \tightlist + \item + \(f(x)\) is \emph{one to one} if \(f(a) \ne f(b)\) if + \(a, b \in \operatorname{dom}(f)\) and \(a \ne b\)\\ + \(\implies\) unique \(y\) for each \(x\) (\(\sin x\) is not 1:1, + \(x^3\) is) + \item + horizontal line test + \item + if not one to one, it is many to one +\end{itemize} - \begin{align*} - \text{Even:}&& f(x) &= f(-x) \\ - \text{Odd:} && -f(x) &= f(-x) - \end{align*} +\subsection*{Odd and even functions} + +\begin{align*} + \text{Even:}&& f(x) &= f(-x) \\ + \text{Odd:} && -f(x) &= f(-x) +\end{align*} + +Even \(\implies\) symmetrical across \(y\)-axis \\ +\(x^{\pm {p \over q}}\) is odd if \(q\) is odd\\ +For \(x^n\), parity of \(n \equiv\) parity of function + +\begin{tabularx}{\columnwidth}{XX} + \textbf{Even:} & \textbf{Odd:} \\ + \begin{tikzpicture}\begin{axis}[ticks=none, yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[blue, mark=none] {(x^2)}; \end{axis}\end{tikzpicture} & + \begin{tikzpicture}\begin{axis}[ticks=none, yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[blue, mark=none] {(x^3)}; \end{axis}\end{tikzpicture} +\end{tabularx} + +\subsection*{Inverse functions} + +\begin{itemize} \tightlist + \item Inverse of \(f(x)\) is denoted \(f^{-1}(x)\) + \item \(f\) must be one to one + \item If \(f(g(x)) = x\), then \(g\) is the inverse of \(f\) + \item Represents reflection across \(y=x\) + \item \(\implies f^{-1}(x)=f(x)\) intersections lie on \(y=x\) + \item \(\operatorname{ran} \> f = \operatorname{dom} \> f^{-1} \\ + \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}\) + \item ``Inverse'' \(\ne\) ``inverse \emph{function}'' (functions must pass vertical line test)\\ +\end{itemize} - Even \(\implies\) symmetrical across \(y\)-axis \\ - \(x^{\pm {p \over q}}\) is odd if \(q\) is odd\\ - For \(x^n\), parity of \(n \equiv\) parity of function +\subsubsection*{Finding \(f^{-1}\)} - \begin{tabularx}{\columnwidth}{XX} - \textbf{Even:} & \textbf{Odd:} \\ - \begin{tikzpicture}\begin{axis}[ticks=none, yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[blue, mark=none] {(x^2)}; \end{axis}\end{tikzpicture} & - \begin{tikzpicture}\begin{axis}[ticks=none, yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[blue, mark=none] {(x^3)}; \end{axis}\end{tikzpicture} - \end{tabularx} +\begin{enumerate} \tightlist + \item Let \(y=f(x)\) + \item Swap \(x\) and \(y\) (``take inverse'' + \item Solve for \(y\) \\ + Sqrt: state \(\pm\) solutions then restrict + \item State rule as \(f^{-1}(x)=\dots\) + \item For inverse \emph{function}, state in function notation +\end{enumerate} - \subsection*{Inverse functions} +\subsection*{Simultaneous equations (linear)} - \begin{itemize} \tightlist - \item Inverse of \(f(x)\) is denoted \(f^{-1}(x)\) - \item \(f\) must be one to one - \item If \(f(g(x)) = x\), then \(g\) is the inverse of \(f\) - \item Represents reflection across \(y=x\) - \item \(\implies f^{-1}(x)=f(x)\) intersections lie on \(y=x\) - \item \(\operatorname{ran} \> f = \operatorname{dom} \> f^{-1} \\ - \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}\) - \item ``Inverse'' \(\ne\) ``inverse \emph{function}'' (functions must pass vertical line test)\\ - \end{itemize} +\begin{itemize} \tightlist + \item \textbf{Unique solution} - lines intersect at point + \item \textbf{Infinitely many solutions} - lines are equal + \item \textbf{No solution} - lines are parallel +\end{itemize} - \subsubsection*{Finding \(f^{-1}\)} +\subsubsection*{Solving \(\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>\) for \(\{0,1,\infty\}\) solutions} + where all coefficients are known except for one, and \(a, b\) are known \begin{enumerate} \tightlist - \item Let \(y=f(x)\) - \item Swap \(x\) and \(y\) (``take inverse'' - \item Solve for \(y\) \\ - Sqrt: state \(\pm\) solutions then restrict - \item State rule as \(f^{-1}(x)=\dots\) - \item For inverse \emph{function}, state in function notation + \item Write as matrices: \(\begin{bmatrix}p & q \\ r & s \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} a \\ b \end{bmatrix}\) + \item Find determinant of first matrix: \(\Delta = ps-qr\) + \item Let \(\Delta = 0\) for number of solutions \(\ne 1\)\\ + or let \(\Delta \ne 0\) for one unique solution. + \item Solve determinant equation to find variable \\ + \textbf{For infinite/no solutions:} + \item Substitute variable into both original equations + \item Rearrange equations so that LHS of each is the same + \item \(\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x\) (\(\infty\) solns)\\ + \(\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x\) (0 solns) \end{enumerate} - - \subsection*{Simultaneous equations (linear)} - - \begin{itemize} \tightlist - \item \textbf{Unique solution} - lines intersect at point - \item \textbf{Infinitely many solutions} - lines are equal - \item \textbf{No solution} - lines are parallel - \end{itemize} - - \subsubsection*{Solving \(\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>\) for \(\{0,1,\infty\}\) solutions} - where all coefficients are known except for one, and \(a, b\) are known - - \begin{enumerate} \tightlist - \item Write as matrices: \(\begin{bmatrix}p & q \\ r & s \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} a \\ b \end{bmatrix}\) - \item Find determinant of first matrix: \(\Delta = ps-qr\) - \item Let \(\Delta = 0\) for number of solutions \(\ne 1\)\\ - or let \(\Delta \ne 0\) for one unique solution. - \item Solve determinant equation to find variable \\ - \textbf{For infinite/no solutions:} - \item Substitute variable into both original equations - \item Rearrange equations so that LHS of each is the same - \item \(\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x\) (\(\infty\) solns)\\ - \(\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x\) (0 solns) - \end{enumerate} - - \colorbox{cas}{On CAS:} Matrix \(\rightarrow\) \texttt{det} - - \subsubsection*{Solving \(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)} - - \begin{itemize} \tightlist - \item Use elimination - \item Generate two new equations with only two variables - \item Rearrange \& solve - \item Substitute one variable into another equation to find another variable - \end{itemize} -\subsection*{Piecewise functions} + \colorbox{cas}{On CAS:} Matrix \(\rightarrow\) \texttt{det} -\[\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\] + \subsubsection*{Solving \(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)} -\textbf{Open circle:} point included\\ -\textbf{Closed circle:} point not included + \begin{itemize} \tightlist + \item Use elimination + \item Generate two new equations with only two variables + \item Rearrange \& solve + \item Substitute one variable into another equation to find another variable + \end{itemize} -\subsection*{Operations on functions} + \subsection*{Piecewise functions} -For \(f \pm g\) and \(f \times g\): -\quad \(\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)\) + \[\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\] -Addition of linear piecewise graphs: add \(y\)-values at key points + \textbf{Open circle:} point included\\ + \textbf{Closed circle:} point not included -Product functions: + \subsection*{Operations on functions} -\begin{itemize} -\tightlist -\item - product will equal 0 if \(f=0\) or \(g=0\) -\item - \(f^\prime(x)=0 \veebar g^\prime(x)=0 \not\Rightarrow (f \times g)^\prime(x)=0\) -\end{itemize} + For \(f \pm g\) and \(f \times g\): + \quad \(\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)\) -\subsection*{Composite functions} + Addition of linear piecewise graphs: add \(y\)-values at key points -\((f \circ g)(x)\) is defined iff -\(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\) + Product functions: + \begin{itemize} + \tightlist + \item + product will equal 0 if \(f=0\) or \(g=0\) + \item + \(f^\prime(x)=0 \veebar g^\prime(x)=0 \not\Rightarrow (f \times g)^\prime(x)=0\) + \end{itemize} - \pgfplotsset{every axis/.append style={ ticks=none, xlabel=, ylabel=, }} % remove axis labels & ticks - \begin{table*}[ht] + \subsection*{Composite functions} + + \((f \circ g)(x)\) is defined iff + \(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\) + + \pgfplotsset{ + blank/.append style={% + enlargelimits=true, + ticks=none, + yticklabels={,,}, xticklabels={,,}, + xlabel=, ylabel=, + scale=0.4, + samples=100, smooth, unbounded coords=jump + } + } + \tikzset{ + blankplot/.append style={orange, mark=none} + } + + \begin{figure*}[ht] \centering - \begin{tabu} to \textwidth {@{} X[0.3,r] *2{|X[c,m]}@{}} - & \(n\) is even & \(n\) is odd \\ \tabucline{1pt} - \(x^n, n \in \mathbb{Z}^+\) & - \vspace{1em}\begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[orange, mark=none] {(x^2)}; \end{axis}\end{tikzpicture} & - \begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[orange, mark=none] {(x^3)}; \end{axis}\end{tikzpicture} \\ - \(x^n, n \in \mathbb{Z}^-\) & - \begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-4, xmax=4, ymax=8, ymin=-0, scale=0.4, smooth] \addplot[orange, mark=none, samples=100] {(x^(-2))}; \end{axis}\end{tikzpicture} & - \begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[orange, mark=none, domain=-3:-0.1] {(x^(-1))}; \addplot[orange, mark=none, domain=0.1:3] {(x^(-1))}; \end{axis}\end{tikzpicture} \\ - \(x^{\frac{1}{n}}, n \in \mathbb{Z}^-\) & - \begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-1, xmax=5, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[orange, mark=none] {(x^(1/2))}; \end{axis}\end{tikzpicture} & - \begin{tikzpicture} - \begin{axis}[enlargelimits=false, yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, ymin=-3, ymax=3, smooth, scale=0.4] - \addplot [orange,domain=-2:2,samples=1000,no markers] gnuplot[id=poly]{sgn(x)*(abs(x)**(1./3)) }; - \end{axis} - \end{tikzpicture} - \end{tabu} - \hrule - \end{table*} - \pgfplotsset{every axis/.append style={ xlabel=\(x\), ylabel=\(y\) }} % put axis labels back + + \begin{tabularx}{\textwidth}{r|Y|Y} + + & \(n\) is even & \(n\) is odd \\ \hline + + \centering \(x^n, n \in \mathbb{Z}^+\) & + + \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture} + \begin{axis}[blank, xmin=-3, xmax=3] + \addplot[blankplot] {(x^2)}; + \end{axis} + \end{tikzpicture}} & + + \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture} + \begin{axis}[blank, xmin=-3, xmax=3] + \addplot[blankplot, domain=-3:3] {(x^3)}; + \end{axis} + \end{tikzpicture}} \\ \hline + + \centering \(x^n, n \in \mathbb{Z}^-\) & + + \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture} + \begin{axis}[blank, xmin=-4, xmax=4, ymax=8, ymin=-0] + \addplot[blankplot, samples=100] {(x^(-2))}; + \end{axis} + \end{tikzpicture}} & + + \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture} + \begin{axis}[blank, xmin=-3, xmax=3] + \addplot[blankplot, domain=-3:-0.1] {(x^(-1))}; + \addplot[blankplot, domain=0.1:3] {(x^(-1))}; + \end{axis} + \end{tikzpicture}} \\ \hline + + \centering \(x^{\frac{1}{n}}, n \in \mathbb{Z}^-\) & + + \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture} + \begin{axis}[blank, xmin=-1, xmax=5] + \addplot[blankplot] {(x^(1/2))}; + \end{axis} + \end{tikzpicture}} & + + \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture} + \begin{axis}[blank, xmin=-3, xmax=3, ymin=-3, ymax=3] + \addplot [blankplot, domain=-2:2] gnuplot[id=poly]{sgn(x)*(abs(x)**(1./3)) }; + \end{axis} + \end{tikzpicture}} \\ \hline + + \end{tabularx} + \end{figure*} \section{Polynomials} @@ -292,5 +365,281 @@ Product functions: \input{circ-functions} \input{calculus} - \end{multicols} - \end{document} + + + \section{Statistics} + + \subsection*{Probability} + + \begin{align*} + \Pr(A \cup B) &= \Pr(A) + \Pr(B) - \Pr(A \cap B) \\ + \Pr(A \cap B) &= \Pr(A|B) \times \Pr(B) \\ + \Pr(A|B) &= \frac{\Pr(A \cap B)}{\Pr(B)} \\ + \Pr(A) &= \Pr(A|B) \cdot \Pr(B) + \Pr(A|B^{\prime}) \cdot \Pr(B^{\prime}) + \end{align*} + + Mutually exclusive \(\implies \Pr(A \cup B) = 0\) \\ + + Independent events: + \begin{flalign*} + \quad \Pr(A \cap B) &= \Pr(A) \times \Pr(B)& \\ + \Pr(A|B) &= \Pr(A) \\ + \Pr(B|A) &= \Pr(B) + \end{flalign*} + + \subsection*{Combinatorics} + + \begin{itemize} + \item Arrangements \({n \choose k} = \frac{n!}{(n-k)}\) + \item \colorbox{important}{Combinations} \({n \choose k} = \frac{n!}{k!(n-k)!}\) + \item Note \({n \choose k} = {n \choose k-1}\) + \end{itemize} + + \subsection*{Distributions} + + \subsubsection*{Mean \(\mu\)} + + \textbf{Mean} \(\mu\) or \textbf{expected value} \(E(X)\) + + \begin{align*} + E(X) &= \frac{\Sigma \left[ x \cdot f(x) \right]}{\Sigma f} \tag{\(f =\) absolute frequency} \\ + &= \sum_{i=1}^n \left[ x_i \cdot \Pr(X=x_i) \right] \tag{discrete}\\ + &= \int_\textbf{X} (x \cdot f(x)) \> dx + \end{align*} + + \subsubsection*{Mode} + + Most popular value (has highest probability of all \(X\) values). Multiple modes can exist if \(>1 \> X\) value have equal-highest probability. Number must exist in distribution. + + \subsubsection*{Median} + + If \(m > 0.5\), then value of \(X\) that is reached is the median of \(X\). If \(m = 0.5 = 0.5\), then \(m\) is halfway between this value and the next. To find \(m\), add values of \(X\) from smallest to alrgest until the sum reaches 0.5. + + \[ m = X \> \text{such that} \> \int_{-\infty}^{m} f(x) dx = 0.5 \] + + \subsubsection*{Variance \(\sigma^2\)} + + \begin{align*} + \operatorname{Var}(x) &= \sum_{i=1}^n p_i (x_i-\mu)^2 \\ + &= \sum (x-\mu)^2 \times \Pr(X=x) \\ + &= \sum x^2 \times p(x) - \mu^2 \\ + &= \operatorname{E}(X^2) - [\operatorname{E}(X)]^2 + &= E\left[(X-\mu)^2\right] + \end{align*} + + \subsubsection*{Standard deviation \(\sigma\)} + + \begin{align*} + \sigma &= \operatorname{sd}(X) \\ + &= \sqrt{\operatorname{Var}(X)} + \end{align*} + + \subsection*{Binomial distributions} + + Conditions for a \textit{binomial distribution}: + \begin{enumerate} + \item Two possible outcomes: \textbf{success} or \textbf{failure} + \item \(\Pr(\text{success})\) is constant across trials (also denoted \(p\)) + \item Finite number \(n\) of independent trials + \end{enumerate} + + + \subsubsection*{Properties of \(X \sim \operatorname{Bi}(n,p)\)} + + \begin{align*} + \mu(X) &= np \\ + \operatorname{Var}(X) &= np(1-p) \\ + \sigma(X) &= \sqrt{np(1-p)} \\ + \Pr(X=x) &= {n \choose x} \cdot p^x \cdot (1-p)^{n-x} + \end{align*} + + \begin{cas} + Interactive \(\rightarrow\) Distribution \(\rightarrow\) \verb;binomialPdf; then input + \begin{description}[nosep, style=multiline, labelindent=0.5cm, leftmargin=3cm, font=\normalfont] + \item [x:] no. of successes + \item [numtrial:] no. of trials + \item [pos:] probability of success + \end{description} + \end{cas} + + \subsection*{Continuous random variables} + + A continuous random variable \(X\) has a pdf \(f\) such that: + + \begin{enumerate} + \item \(f(x) \ge 0 \forall x \) + \item \(\int^\infty_{-\infty} f(x) \> dx = 1\) + \end{enumerate} + + \begin{align*} + E(X) &= \int_\textbf{X} (x \cdot f(x)) \> dx \\ + \operatorname{Var}(X) &= E\left[(X-\mu)^2\right] + \end{align*} + + \[ \Pr(X \le c) = \int^c_{-\infty} f(x) \> dx \] + + + \subsection*{Two random variables \(X, Y\)} + + If \(X\) and \(Y\) are independent: + \begin{align*} + \operatorname{E}(aX+bY) & = a\operatorname{E}(X)+b\operatorname{E}(Y) \\ + \operatorname{Var}(aX \pm bY \pm c) &= a^2 \operatorname{Var}(X) + b^2 \operatorname{Var}(Y) + \end{align*} + + \subsection*{Linear functions \(X \rightarrow aX+b\)} + + \begin{align*} + \Pr(Y \le y) &= \Pr(aX+b \le y) \\ + &= \Pr\left(X \le \dfrac{y-b}{a}\right) \\ + &= \int^{\frac{y-b}{a}}_{-\infty} f(x) \> dx + \end{align*} + + \begin{align*} + \textbf{Mean:} && \operatorname{E}(aX+b) & = a\operatorname{E}(X)+b \\ + \textbf{Variance:} && \operatorname{Var}(aX+b) &= a^2 \operatorname{Var}(X) \\ + \end{align*} + + \subsection*{Expectation theorems} + + For some non-linear function \(g\), the expected value \(E(g(X))\) is not equal to \(g(E(X))\). + + \begin{align*} + E(X^2) &= \operatorname{Var}(X) - \left[E(X)\right]^2 \\ + E(X^n) &= \Sigma x^n \cdot p(x) \tag{non-linear} \\ + &\ne [E(X)]^n \\ + E(aX \pm b) &= aE(X) \pm b \tag{linear} \\ + E(b) &= b \tag{\(\forall b \in \mathbb{R}\)}\\ + E(X+Y) &= E(X) + E(Y) \tag{two variables} + \end{align*} + + \subsection*{Sample mean} + + Approximation of the \textbf{population mean} determined experimentally. + + \[ \overline{x} = \dfrac{\Sigma x}{n} \] + + where + \begin{description}[nosep, labelindent=0.5cm] + \item \(n\) is the size of the sample (number of sample points) + \item \(x\) is the value of a sample point + \end{description} + + \begin{cas} + \begin{enumerate}[leftmargin=3mm] + \item Spreadsheet + \item In cell A1:\\ \path{mean(randNorm(sd, mean, sample size))} + \item Edit \(\rightarrow\) Fill \(\rightarrow\) Fill Range + \item Input range as A1:An where \(n\) is the number of samples + \item Graph \(\rightarrow\) Histogram + \end{enumerate} + \end{cas} + + \subsubsection*{Sample size of \(n\)} + + \[ \overline{X} = \sum_{i=1}^n \frac{x_i}{n} = \dfrac{\sum x}{n} \] + + Sample mean is distributed with mean \(\mu\) and sd \(\frac{\sigma}{\sqrt{n}}\) (approaches these values for increasing sample size \(n\)). + + For a new distribution with mean of \(n\) trials, \(\operatorname{E}(X^\prime) = \operatorname{E}(X), \quad \operatorname{sd}(X^\prime) = \dfrac{\operatorname{sd}(X)}{\sqrt{n}}\) + + \begin{cas} + + \begin{itemize} + \item Spreadsheet \(\rightarrow\) Catalog \(\rightarrow\) \verb;randNorm(sd, mean, n); where \verb;n; is the number of samples. Show histogram with Histogram key in top left + \item To calculate parameters of a dataset: Calc \(\rightarrow\) One-variable + \end{itemize} + + \end{cas} + + \subsection*{Normal distributions} + + + \[ Z = \frac{X - \mu}{\sigma} \] + + Normal distributions must have area (total prob.) of 1 \(\implies \int^\infty_{-\infty} f(x) \> dx = 1\) \\ + \(\text{mean} = \text{mode} = \text{median}\) + + \begin{warning} + Always express \(z\) as +ve. 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