-\PassOptionsToPackage{unicode=true}{hyperref} % options for packages loaded elsewhere
-\PassOptionsToPackage{hyphens}{url}
-%
-\documentclass[]{article}
-\usepackage{lmodern}
-\usepackage{amssymb,amsmath}
-\usepackage{ifxetex,ifluatex}
-\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{
- pdfborder={0 0 0},
- breaklinks=true}
-\urlstyle{same} % don't use monospace font for urls
-\usepackage[margin=2cm]{geometry}
-\usepackage{multicol}
-\newcommand{\columnsbegin}{\begin{multicols}{2}}
-\newcommand{\columnsend}{\end{multicols}}
-\setlength\columnsep{20pt}
-\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
-
-
-\date{}
-
-\begin{document}
-
-\columnsbegin
-\hypertarget{circular-functions}{%
-\section{Circular functions}\label{circular-functions}}
-
-\hypertarget{radians-and-degrees}{%
-\subsection{Radians and degrees}\label{radians-and-degrees}}
+\section{Circular functions}
-\[1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi}\]
-
-\hypertarget{exact-values}{%
-\subsection{Exact values}\label{exact-values}}
+\subsection*{Radians and degrees}
-\includegraphics[scale=0.5]{./graphics/exact-values-1.png}
+\[1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi}\]
-\hypertarget{sin-and-cos-graphs}{%
-\subsection{\texorpdfstring{\(\sin\) and \(\cos\)
-graphs}{\textbackslash{}sin and \textbackslash{}cos graphs}}\label{sin-and-cos-graphs}}
+\subsection*{Exact values}
-\[f(x)=a \sin(bx-c)+d\] \[f(x)=a \cos(bx-c)+d\]
-where
+ \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}
-\begin{itemize}
-\tightlist
-\item
- \(a\) is the \(y\)-dilation (amplitude)
-\item
- \(b\) is the \(x\)-dilation (period)
-\item
- \(c\) is the \(x\)-shift (phase)
-\item
- \(d\) is the \(y\)-shift (equilibrium position)
-\end{itemize}
+ \subsection*{Compound angle formulas}
-Domain is \(\mathbb{R}\)
-Range is \([-b+c, b+c]\);
+ \begin{align*}
+ \cos(x \pm y) &= \cos x + \cos y \mp \sin x \sin y \\
+ \sin(x \pm y) &= \sin x \cos y \pm \cos x \sin y \\
+ \tan(x \pm y) &= {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}
+ \end{align*}
-Graph of \(\cos(x)\) starts at \((0,1)\). Graph of \(\sin(x)\) starts at
-\((0,0)\).
+ \subsection*{Double angle formulas}
-\textbf{Mean / equilibrium:} line that the graph oscillates around
-(\(y=d\))
+ \begin{align*}
+ \cos 2x &= \cos^2 x - \sin^2 x \\
+ & = 1 - 2\sin^2 x \\
+ & = 2 \cos^2 x -1 \\
+ \sin 2x &= 2 \sin x \cos x \\
+ \tan 2x &= \dfrac{2 \tan x}{1 - \tan^2 x}
+ \end{align*}
-\hypertarget{amplitude}{%
-\subsubsection{Amplitude}\label{amplitude}}
-Amplitude of \(a\) means graph oscillates between \(+a\) and \(-a\) in
-\(y\)-axis
-\(a=0\) produces straight line
+\subsection*{Symmetry}
-\(a < 0\) inverts the phase (\(\sin\) becomes \(\cos\), vice vera)
+\begin{align*}
+ \sin(\theta+\frac{\pi}{2}) &= \sin\theta \\
+ \sin(\theta+\pi) &= -\sin\theta \\ \\
+ \cos(\theta+\frac{\pi}{2}) &= -\cos\theta \\
+ \cos(\theta+\pi) &= -\cos(\theta+\frac{3\pi}{2}) \\
+ &= \cos(-\theta)
+\end{align*}
-\hypertarget{period}{%
-\subsubsection{Period}\label{period}}
+\subsection*{Complementary relationships}
-Period \(T\) is \({2 \pi}\over b\)
+\begin{align*}
+ \sin \theta &= \cos(\frac{\pi}{2} - \theta) \\
+ &= -\cos(\theta+\frac{\pi}{2}) \\
+ \cos\theta &= \sin(\frac{\pi}{2} - \theta) \\
+ &= \sin(\theta+\frac{\pi}{2})
+\end{align*}
-\(b=0\) produces straight line
+\subsection*{Pythagorean identity}
-\(b<0\) inverts the phase
+\[\cos^2\theta+\sin^2\theta=1\]
-\hypertarget{phase}{%
-\subsubsection{Phase}\label{phase}}
+ \subsection*{Inverse circular functions}
-\(c\) moves the graph left-right in the \(x\) axis.
+ \begin{tikzpicture}
+ \begin{axis}[ymin=-2, ymax=4, xmin=-1.1, xmax=1.1, ytick={-1.5708, 1.5708, 3.14159},yticklabels={$-\frac{\pi}{2}$, $\frac{\pi}{2}$, $\pi$}]
+ \addplot[color=red, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {asin(x)} node [pos=0.25, below right] {\(\sin^{-1}x\)};
+ \addplot[color=blue, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {acos(x)} node [pos=0.25, below left] {\(\cos^{-1}x\)};
+ \addplot[mark=*, red] coordinates {(-1,-1.5708)} node[right, font=\footnotesize]{\((-1,-\frac{\pi}{2})\)} ;
+ \addplot[mark=*, red] coordinates {(1,1.5708)} node[left, font=\footnotesize]{\((1,\frac{\pi}{2})\)} ;
+ \addplot[mark=*, blue] coordinates {(1,0)};
+ \addplot[mark=*, blue] coordinates {(-1,3.1415)} node[right, font=\footnotesize]{\((-1,\pi)\)} ;
+ \end{axis}
+ \end{tikzpicture}\\
-If \(c=T={{2\pi}\over b}\), the graph has no actual phase shift.
+ Inverse functions: \(f(f^{-1}(x)) = x\) (restrict domain)
-\hypertarget{symmetry}{%
-\subsection{Symmetry}\label{symmetry}}
+ \[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y\]
+ \hfill where \(\sin y = x, \> y \in [{-\pi \over 2}, {\pi \over 2}]\)
-\[\sin(\theta+{\pi\over 2})=\sin\theta\]
-\[\sin(\theta+\pi)=-\sin\theta\]
+ \[\cos^{-1}: [-1,1] \rightarrow \mathbb{R}, \quad \cos^{-1} x = y\]
+ \hfill where \(\cos y = x, \> y \in [0, \pi]\)
-\[\cos(\theta+{\pi \over 2})=-\cos\theta\]
-\[\cos(\theta+\pi)=-cos(\theta+{3\pi \over 2})=\cos(-\theta)\]
+ \[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\]
+ \hfill where \(\tan y = x, \> y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\)
-\hypertarget{pythagorean-identity}{%
-\subsection{Pythagorean identity}\label{pythagorean-identity}}
+ \begin{tikzpicture}
+ \begin{axis}[yticklabel style={yshift=1.0pt, anchor=north east},x=0.1cm, y=1cm, ymax=2, ymin=-2, xticklabels={}, ytick={-1.5708,1.5708},yticklabels={\(-\frac{\pi}{2}\),\(\frac{\pi}{2}\)}]
+ \addplot[color=orange, smooth] gnuplot [domain=-35:35, unbounded coords=jump,samples=350] {atan(x)} node [pos=0.5, above left] {\(\tan^{-1}x\)};
+ \addplot[->, gray, dotted, thick, domain=-35:35] {1.5708};
+ \addplot[->, gray, dotted, thick, domain=-35:35] {-1.5708};
+ \end{axis}
+ \end{tikzpicture}
-\[\cos^2\theta+\sin^2\theta=1\]
+\subsection*{\(\sin\) and \(\cos\) graphs}
-\hypertarget{complementary-relationships}{%
-\subsection{Complementary
-relationships}\label{complementary-relationships}}
+\[ f(x)=a\sin(bx-c)+d \]
-\[\sin({\pi \over 2} - \theta)=\cos\theta\]
-\[\cos({\pi \over 2} - \theta)=\sin\theta\]
+where:
+\begin{description}
+ \item Period \(=\frac{2\pi}{n}\)
+ \item dom \(= \mathbb{R}\)
+ \item ran \(= [-b+c, b+c]\);
+ \item \(\cos(x)\) starts at \((0,1)\), \(\sin(x)\) starts at \((0,0)\)
+ \item 0 amplitidue \(\implies\) straight line
+ \item \(a<0\) or \(b<0\) inverts phase (swap \(\sin\) and \(\cos\))
+ \item \(c=T={{2\pi}\over b} \implies\) no net phase shift
+\end{description}
-\[\sin\theta=-\cos(\theta+{\pi \over 2})\]
-\[\cos\theta=\sin(\theta+{\pi \over 2})\]
-
-\hypertarget{tan-graph}{%
-\subsection{\texorpdfstring{\(\tan\)
-graph}{\textbackslash{}tan graph}}\label{tan-graph}}
+\subsection*{\(\tan\) graphs}
\[y=a\tan(nx)\]
-where
-
-\begin{itemize}
-\tightlist
-\item
- \(a\) is \(x\)-dilation (period)
-\item
- \(n\) is \(y\)-dilation (\(\equiv\) amplitude)
-\item
- period \(T\) is \(\pi \over n\)
-\item
- range is \(R\)
-\item
- roots at \(x={k\pi \over n}\)
-\item
- asymptotes at \(x={{(2k+1)\pi}\over 2n},\quad k \in \mathbb{Z}\)
-\end{itemize}
+\begin{description}
+ \item Period \(= \dfrac{\pi}{n}\)
+ \item Range is \(\mathbb{R}\)
+ \item Roots at \(x={\dfrac{k\pi}{n}}\) where \(k \in \mathbb{Z}\)
+ \item Asymptotes at \(x=\frac{(2k+1)\pi}{2n}\)
+\end{description}
\textbf{Asymptotes should always have equations and arrow pointing up}
-\hypertarget{solving-trig-equations}{%
-\subsection{Solving trig equations}\label{solving-trig-equations}}
+\subsection*{Solving trig equations}
\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\(2\theta={\pi\over 3}, {2\pi \over 3}, {7\pi \over 3}, {8\pi \over 3}\)
\(\therefore \theta = {\pi \over 6}, {\pi \over 3}, {7 \pi \over 6}, {4\pi \over 3}\)
-\columnsend
-\end{document}
--- /dev/null
+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)) ;
--- /dev/null
+
+# Curve 0 of 1, 1000 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 -1.1626386e+00 i
+-1.5675676e+00 -1.1616503e+00 i
+-1.5635636e+00 -1.1606604e+00 i
+-1.5595596e+00 -1.1596688e+00 i
+-1.5555556e+00 -1.1586755e+00 i
+-1.5515516e+00 -1.1576805e+00 i
+-1.5475475e+00 -1.1566838e+00 i
+-1.5435435e+00 -1.1556854e+00 i
+-1.5395395e+00 -1.1546852e+00 i
+-1.5355355e+00 -1.1536833e+00 i
+-1.5315315e+00 -1.1526797e+00 i
+-1.5275275e+00 -1.1516743e+00 i
+-1.5235235e+00 -1.1506672e+00 i
+-1.5195195e+00 -1.1496583e+00 i
+-1.5155155e+00 -1.1486476e+00 i
+-1.5115115e+00 -1.1476351e+00 i
+-1.5075075e+00 -1.1466208e+00 i
+-1.5035035e+00 -1.1456048e+00 i
+-1.4994995e+00 -1.1445869e+00 i
+-1.4954955e+00 -1.1435672e+00 i
+-1.4914915e+00 -1.1425457e+00 i
+-1.4874875e+00 -1.1415224e+00 i
+-1.4834835e+00 -1.1404972e+00 i
+-1.4794795e+00 -1.1394702e+00 i
+-1.4754755e+00 -1.1384414e+00 i
+-1.4714715e+00 -1.1374106e+00 i
+-1.4674675e+00 -1.1363780e+00 i
+-1.4634635e+00 -1.1353435e+00 i
+-1.4594595e+00 -1.1343072e+00 i
+-1.4554555e+00 -1.1332689e+00 i
+-1.4514515e+00 -1.1322287e+00 i
+-1.4474474e+00 -1.1311866e+00 i
+-1.4434434e+00 -1.1301426e+00 i
+-1.4394394e+00 -1.1290967e+00 i
+-1.4354354e+00 -1.1280488e+00 i
+-1.4314314e+00 -1.1269990e+00 i
+-1.4274274e+00 -1.1259472e+00 i
+-1.4234234e+00 -1.1248934e+00 i
+-1.4194194e+00 -1.1238377e+00 i
+-1.4154154e+00 -1.1227799e+00 i
+-1.4114114e+00 -1.1217202e+00 i
+-1.4074074e+00 -1.1206585e+00 i
+-1.4034034e+00 -1.1195947e+00 i
+-1.3993994e+00 -1.1185289e+00 i
+-1.3953954e+00 -1.1174611e+00 i
+-1.3913914e+00 -1.1163913e+00 i
+-1.3873874e+00 -1.1153194e+00 i
+-1.3833834e+00 -1.1142454e+00 i
+-1.3793794e+00 -1.1131694e+00 i
+-1.3753754e+00 -1.1120912e+00 i
+-1.3713714e+00 -1.1110110e+00 i
+-1.3673674e+00 -1.1099287e+00 i
+-1.3633634e+00 -1.1088442e+00 i
+-1.3593594e+00 -1.1077577e+00 i
+-1.3553554e+00 -1.1066690e+00 i
+-1.3513514e+00 -1.1055781e+00 i
+-1.3473473e+00 -1.1044851e+00 i
+-1.3433433e+00 -1.1033899e+00 i
+-1.3393393e+00 -1.1022926e+00 i
+-1.3353353e+00 -1.1011930e+00 i
+-1.3313313e+00 -1.1000913e+00 i
+-1.3273273e+00 -1.0989873e+00 i
+-1.3233233e+00 -1.0978811e+00 i
+-1.3193193e+00 -1.0967727e+00 i
+-1.3153153e+00 -1.0956621e+00 i
+-1.3113113e+00 -1.0945492e+00 i
+-1.3073073e+00 -1.0934340e+00 i
+-1.3033033e+00 -1.0923165e+00 i
+-1.2992993e+00 -1.0911968e+00 i
+-1.2952953e+00 -1.0900747e+00 i
+-1.2912913e+00 -1.0889503e+00 i
+-1.2872873e+00 -1.0878236e+00 i
+-1.2832833e+00 -1.0866946e+00 i
+-1.2792793e+00 -1.0855632e+00 i
+-1.2752753e+00 -1.0844295e+00 i
+-1.2712713e+00 -1.0832934e+00 i
+-1.2672673e+00 -1.0821548e+00 i
+-1.2632633e+00 -1.0810139e+00 i
+-1.2592593e+00 -1.0798706e+00 i
+-1.2552553e+00 -1.0787248e+00 i
+-1.2512513e+00 -1.0775767e+00 i
+-1.2472472e+00 -1.0764260e+00 i
+-1.2432432e+00 -1.0752729e+00 i
+-1.2392392e+00 -1.0741173e+00 i
+-1.2352352e+00 -1.0729592e+00 i
+-1.2312312e+00 -1.0717987e+00 i
+-1.2272272e+00 -1.0706356e+00 i
+-1.2232232e+00 -1.0694699e+00 i
+-1.2192192e+00 -1.0683017e+00 i
+-1.2152152e+00 -1.0671310e+00 i
+-1.2112112e+00 -1.0659577e+00 i
+-1.2072072e+00 -1.0647818e+00 i
+-1.2032032e+00 -1.0636033e+00 i
+-1.1991992e+00 -1.0624221e+00 i
+-1.1951952e+00 -1.0612384e+00 i
+-1.1911912e+00 -1.0600520e+00 i
+-1.1871872e+00 -1.0588629e+00 i
+-1.1831832e+00 -1.0576712e+00 i
+-1.1791792e+00 -1.0564767e+00 i
+-1.1751752e+00 -1.0552796e+00 i
+-1.1711712e+00 -1.0540797e+00 i
+-1.1671672e+00 -1.0528771e+00 i
+-1.1631632e+00 -1.0516718e+00 i
+-1.1591592e+00 -1.0504636e+00 i
+-1.1551552e+00 -1.0492527e+00 i
+-1.1511512e+00 -1.0480390e+00 i
+-1.1471471e+00 -1.0468225e+00 i
+-1.1431431e+00 -1.0456031e+00 i
+-1.1391391e+00 -1.0443809e+00 i
+-1.1351351e+00 -1.0431558e+00 i
+-1.1311311e+00 -1.0419279e+00 i
+-1.1271271e+00 -1.0406970e+00 i
+-1.1231231e+00 -1.0394632e+00 i
+-1.1191191e+00 -1.0382265e+00 i
+-1.1151151e+00 -1.0369868e+00 i
+-1.1111111e+00 -1.0357442e+00 i
+-1.1071071e+00 -1.0344985e+00 i
+-1.1031031e+00 -1.0332499e+00 i
+-1.0990991e+00 -1.0319982e+00 i
+-1.0950951e+00 -1.0307435e+00 i
+-1.0910911e+00 -1.0294857e+00 i
+-1.0870871e+00 -1.0282249e+00 i
+-1.0830831e+00 -1.0269609e+00 i
+-1.0790791e+00 -1.0256939e+00 i
+-1.0750751e+00 -1.0244237e+00 i
+-1.0710711e+00 -1.0231503e+00 i
+-1.0670671e+00 -1.0218737e+00 i
+-1.0630631e+00 -1.0205940e+00 i
+-1.0590591e+00 -1.0193110e+00 i
+-1.0550551e+00 -1.0180248e+00 i
+-1.0510511e+00 -1.0167354e+00 i
+-1.0470470e+00 -1.0154426e+00 i
+-1.0430430e+00 -1.0141466e+00 i
+-1.0390390e+00 -1.0128473e+00 i
+-1.0350350e+00 -1.0115446e+00 i
+-1.0310310e+00 -1.0102385e+00 i
+-1.0270270e+00 -1.0089290e+00 i
+-1.0230230e+00 -1.0076162e+00 i
+-1.0190190e+00 -1.0062999e+00 i
+-1.0150150e+00 -1.0049802e+00 i
+-1.0110110e+00 -1.0036569e+00 i
+-1.0070070e+00 -1.0023302e+00 i
+-1.0030030e+00 -1.0010000e+00 i
+-9.9899900e-01 -9.9966622e-01 i
+-9.9499499e-01 -9.9832887e-01 i
+-9.9099099e-01 -9.9698793e-01 i
+-9.8698699e-01 -9.9564338e-01 i
+-9.8298298e-01 -9.9429518e-01 i
+-9.7897898e-01 -9.9294331e-01 i
+-9.7497497e-01 -9.9158776e-01 i
+-9.7097097e-01 -9.9022849e-01 i
+-9.6696697e-01 -9.8886547e-01 i
+-9.6296296e-01 -9.8749869e-01 i
+-9.5895896e-01 -9.8612811e-01 i
+-9.5495495e-01 -9.8475372e-01 i
+-9.5095095e-01 -9.8337547e-01 i
+-9.4694695e-01 -9.8199336e-01 i
+-9.4294294e-01 -9.8060734e-01 i
+-9.3893894e-01 -9.7921739e-01 i
+-9.3493493e-01 -9.7782348e-01 i
+-9.3093093e-01 -9.7642559e-01 i
+-9.2692693e-01 -9.7502369e-01 i
+-9.2292292e-01 -9.7361774e-01 i
+-9.1891892e-01 -9.7220772e-01 i
+-9.1491491e-01 -9.7079360e-01 i
+-9.1091091e-01 -9.6937534e-01 i
+-9.0690691e-01 -9.6795292e-01 i
+-9.0290290e-01 -9.6652632e-01 i
+-8.9889890e-01 -9.6509548e-01 i
+-8.9489489e-01 -9.6366039e-01 i
+-8.9089089e-01 -9.6222102e-01 i
+-8.8688689e-01 -9.6077732e-01 i
+-8.8288288e-01 -9.5932928e-01 i
+-8.7887888e-01 -9.5787685e-01 i
+-8.7487487e-01 -9.5642000e-01 i
+-8.7087087e-01 -9.5495870e-01 i
+-8.6686687e-01 -9.5349291e-01 i
+-8.6286286e-01 -9.5202260e-01 i
+-8.5885886e-01 -9.5054774e-01 i
+-8.5485485e-01 -9.4906828e-01 i
+-8.5085085e-01 -9.4758420e-01 i
+-8.4684685e-01 -9.4609546e-01 i
+-8.4284284e-01 -9.4460202e-01 i
+-8.3883884e-01 -9.4310383e-01 i
+-8.3483483e-01 -9.4160088e-01 i
+-8.3083083e-01 -9.4009311e-01 i
+-8.2682683e-01 -9.3858048e-01 i
+-8.2282282e-01 -9.3706297e-01 i
+-8.1881882e-01 -9.3554053e-01 i
+-8.1481481e-01 -9.3401311e-01 i
+-8.1081081e-01 -9.3248068e-01 i
+-8.0680681e-01 -9.3094320e-01 i
+-8.0280280e-01 -9.2940062e-01 i
+-7.9879880e-01 -9.2785291e-01 i
+-7.9479479e-01 -9.2630002e-01 i
+-7.9079079e-01 -9.2474190e-01 i
+-7.8678679e-01 -9.2317851e-01 i
+-7.8278278e-01 -9.2160981e-01 i
+-7.7877878e-01 -9.2003575e-01 i
+-7.7477477e-01 -9.1845629e-01 i
+-7.7077077e-01 -9.1687137e-01 i
+-7.6676677e-01 -9.1528096e-01 i
+-7.6276276e-01 -9.1368500e-01 i
+-7.5875876e-01 -9.1208344e-01 i
+-7.5475475e-01 -9.1047625e-01 i
+-7.5075075e-01 -9.0886335e-01 i
+-7.4674675e-01 -9.0724471e-01 i
+-7.4274274e-01 -9.0562028e-01 i
+-7.3873874e-01 -9.0399000e-01 i
+-7.3473473e-01 -9.0235381e-01 i
+-7.3073073e-01 -9.0071167e-01 i
+-7.2672673e-01 -8.9906352e-01 i
+-7.2272272e-01 -8.9740931e-01 i
+-7.1871872e-01 -8.9574897e-01 i
+-7.1471471e-01 -8.9408246e-01 i
+-7.1071071e-01 -8.9240971e-01 i
+-7.0670671e-01 -8.9073067e-01 i
+-7.0270270e-01 -8.8904527e-01 i
+-6.9869870e-01 -8.8735346e-01 i
+-6.9469469e-01 -8.8565517e-01 i
+-6.9069069e-01 -8.8395034e-01 i
+-6.8668669e-01 -8.8223891e-01 i
+-6.8268268e-01 -8.8052082e-01 i
+-6.7867868e-01 -8.7879599e-01 i
+-6.7467467e-01 -8.7706437e-01 i
+-6.7067067e-01 -8.7532588e-01 i
+-6.6666667e-01 -8.7358046e-01 i
+-6.6266266e-01 -8.7182804e-01 i
+-6.5865866e-01 -8.7006855e-01 i
+-6.5465465e-01 -8.6830190e-01 i
+-6.5065065e-01 -8.6652804e-01 i
+-6.4664665e-01 -8.6474689e-01 i
+-6.4264264e-01 -8.6295837e-01 i
+-6.3863864e-01 -8.6116241e-01 i
+-6.3463463e-01 -8.5935892e-01 i
+-6.3063063e-01 -8.5754783e-01 i
+-6.2662663e-01 -8.5572906e-01 i
+-6.2262262e-01 -8.5390253e-01 i
+-6.1861862e-01 -8.5206814e-01 i
+-6.1461461e-01 -8.5022583e-01 i
+-6.1061061e-01 -8.4837549e-01 i
+-6.0660661e-01 -8.4651705e-01 i
+-6.0260260e-01 -8.4465042e-01 i
+-5.9859860e-01 -8.4277549e-01 i
+-5.9459459e-01 -8.4089219e-01 i
+-5.9059059e-01 -8.3900041e-01 i
+-5.8658659e-01 -8.3710007e-01 i
+-5.8258258e-01 -8.3519105e-01 i
+-5.7857858e-01 -8.3327327e-01 i
+-5.7457457e-01 -8.3134662e-01 i
+-5.7057057e-01 -8.2941100e-01 i
+-5.6656657e-01 -8.2746630e-01 i
+-5.6256256e-01 -8.2551242e-01 i
+-5.5855856e-01 -8.2354924e-01 i
+-5.5455455e-01 -8.2157666e-01 i
+-5.5055055e-01 -8.1959456e-01 i
+-5.4654655e-01 -8.1760283e-01 i
+-5.4254254e-01 -8.1560134e-01 i
+-5.3853854e-01 -8.1358999e-01 i
+-5.3453453e-01 -8.1156864e-01 i
+-5.3053053e-01 -8.0953717e-01 i
+-5.2652653e-01 -8.0749545e-01 i
+-5.2252252e-01 -8.0544336e-01 i
+-5.1851852e-01 -8.0338075e-01 i
+-5.1451451e-01 -8.0130750e-01 i
+-5.1051051e-01 -7.9922347e-01 i
+-5.0650651e-01 -7.9712851e-01 i
+-5.0250250e-01 -7.9502248e-01 i
+-4.9849850e-01 -7.9290523e-01 i
+-4.9449449e-01 -7.9077662e-01 i
+-4.9049049e-01 -7.8863648e-01 i
+-4.8648649e-01 -7.8648467e-01 i
+-4.8248248e-01 -7.8432101e-01 i
+-4.7847848e-01 -7.8214535e-01 i
+-4.7447447e-01 -7.7995752e-01 i
+-4.7047047e-01 -7.7775735e-01 i
+-4.6646647e-01 -7.7554466e-01 i
+-4.6246246e-01 -7.7331926e-01 i
+-4.5845846e-01 -7.7108099e-01 i
+-4.5445445e-01 -7.6882965e-01 i
+-4.5045045e-01 -7.6656504e-01 i
+-4.4644645e-01 -7.6428697e-01 i
+-4.4244244e-01 -7.6199524e-01 i
+-4.3843844e-01 -7.5968965e-01 i
+-4.3443443e-01 -7.5736997e-01 i
+-4.3043043e-01 -7.5503599e-01 i
+-4.2642643e-01 -7.5268750e-01 i
+-4.2242242e-01 -7.5032426e-01 i
+-4.1841842e-01 -7.4794604e-01 i
+-4.1441441e-01 -7.4555259e-01 i
+-4.1041041e-01 -7.4314368e-01 i
+-4.0640641e-01 -7.4071905e-01 i
+-4.0240240e-01 -7.3827844e-01 i
+-3.9839840e-01 -7.3582159e-01 i
+-3.9439439e-01 -7.3334822e-01 i
+-3.9039039e-01 -7.3085806e-01 i
+-3.8638639e-01 -7.2835081e-01 i
+-3.8238238e-01 -7.2582617e-01 i
+-3.7837838e-01 -7.2328385e-01 i
+-3.7437437e-01 -7.2072354e-01 i
+-3.7037037e-01 -7.1814490e-01 i
+-3.6636637e-01 -7.1554760e-01 i
+-3.6236236e-01 -7.1293132e-01 i
+-3.5835836e-01 -7.1029569e-01 i
+-3.5435435e-01 -7.0764035e-01 i
+-3.5035035e-01 -7.0496494e-01 i
+-3.4634635e-01 -7.0226906e-01 i
+-3.4234234e-01 -6.9955233e-01 i
+-3.3833834e-01 -6.9681433e-01 i
+-3.3433433e-01 -6.9405464e-01 i
+-3.3033033e-01 -6.9127282e-01 i
+-3.2632633e-01 -6.8846844e-01 i
+-3.2232232e-01 -6.8564102e-01 i
+-3.1831832e-01 -6.8279009e-01 i
+-3.1431431e-01 -6.7991515e-01 i
+-3.1031031e-01 -6.7701569e-01 i
+-3.0630631e-01 -6.7409118e-01 i
+-3.0230230e-01 -6.7114107e-01 i
+-2.9829830e-01 -6.6816480e-01 i
+-2.9429429e-01 -6.6516177e-01 i
+-2.9029029e-01 -6.6213138e-01 i
+-2.8628629e-01 -6.5907299e-01 i
+-2.8228228e-01 -6.5598595e-01 i
+-2.7827828e-01 -6.5286958e-01 i
+-2.7427427e-01 -6.4972318e-01 i
+-2.7027027e-01 -6.4654599e-01 i
+-2.6626627e-01 -6.4333728e-01 i
+-2.6226226e-01 -6.4009623e-01 i
+-2.5825826e-01 -6.3682202e-01 i
+-2.5425425e-01 -6.3351379e-01 i
+-2.5025025e-01 -6.3017065e-01 i
+-2.4624625e-01 -6.2679166e-01 i
+-2.4224224e-01 -6.2337583e-01 i
+-2.3823824e-01 -6.1992215e-01 i
+-2.3423423e-01 -6.1642956e-01 i
+-2.3023023e-01 -6.1289693e-01 i
+-2.2622623e-01 -6.0932311e-01 i
+-2.2222222e-01 -6.0570686e-01 i
+-2.1821822e-01 -6.0204691e-01 i
+-2.1421421e-01 -5.9834192e-01 i
+-2.1021021e-01 -5.9459046e-01 i
+-2.0620621e-01 -5.9079105e-01 i
+-2.0220220e-01 -5.8694214e-01 i
+-1.9819820e-01 -5.8304208e-01 i
+-1.9419419e-01 -5.7908913e-01 i
+-1.9019019e-01 -5.7508147e-01 i
+-1.8618619e-01 -5.7101715e-01 i
+-1.8218218e-01 -5.6689414e-01 i
+-1.7817818e-01 -5.6271027e-01 i
+-1.7417417e-01 -5.5846323e-01 i
+-1.7017017e-01 -5.5415060e-01 i
+-1.6616617e-01 -5.4976978e-01 i
+-1.6216216e-01 -5.4531801e-01 i
+-1.5815816e-01 -5.4079234e-01 i
+-1.5415415e-01 -5.3618963e-01 i
+-1.5015015e-01 -5.3150651e-01 i
+-1.4614615e-01 -5.2673938e-01 i
+-1.4214214e-01 -5.2188436e-01 i
+-1.3813814e-01 -5.1693730e-01 i
+-1.3413413e-01 -5.1189368e-01 i
+-1.3013013e-01 -5.0674867e-01 i
+-1.2612613e-01 -5.0149701e-01 i
+-1.2212212e-01 -4.9613300e-01 i
+-1.1811812e-01 -4.9065042e-01 i
+-1.1411411e-01 -4.8504249e-01 i
+-1.1011011e-01 -4.7930181e-01 i
+-1.0610611e-01 -4.7342021e-01 i
+-1.0210210e-01 -4.6738872e-01 i
+-9.8098098e-02 -4.6119741e-01 i
+-9.4094094e-02 -4.5483526e-01 i
+-9.0090090e-02 -4.4828995e-01 i
+-8.6086086e-02 -4.4154773e-01 i
+-8.2082082e-02 -4.3459306e-01 i
+-7.8078078e-02 -4.2740839e-01 i
+-7.4074074e-02 -4.1997368e-01 i
+-7.0070070e-02 -4.1226600e-01 i
+-6.6066066e-02 -4.0425880e-01 i
+-6.2062062e-02 -3.9592118e-01 i
+-5.8058058e-02 -3.8721678e-01 i
+-5.4054054e-02 -3.7810239e-01 i
+-5.0050050e-02 -3.6852603e-01 i
+-4.6046046e-02 -3.5842430e-01 i
+-4.2042042e-02 -3.4771861e-01 i
+-3.8038038e-02 -3.3630968e-01 i
+-3.4034034e-02 -3.2406924e-01 i
+-3.0030030e-02 -3.1082689e-01 i
+-2.6026026e-02 -2.9634842e-01 i
+-2.2022022e-02 -2.8029740e-01 i
+-1.8018018e-02 -2.6216156e-01 i
+-1.4014014e-02 -2.4109462e-01 i
+-1.0010010e-02 -2.1551533e-01 i
+-6.0060060e-03 -1.8177267e-01 i
+-2.0020020e-03 -1.2603413e-01 i
+2.0020020e-03 1.2603413e-01 i
+6.0060060e-03 1.8177267e-01 i
+1.0010010e-02 2.1551533e-01 i
+1.4014014e-02 2.4109462e-01 i
+1.8018018e-02 2.6216156e-01 i
+2.2022022e-02 2.8029740e-01 i
+2.6026026e-02 2.9634842e-01 i
+3.0030030e-02 3.1082689e-01 i
+3.4034034e-02 3.2406924e-01 i
+3.8038038e-02 3.3630968e-01 i
+4.2042042e-02 3.4771861e-01 i
+4.6046046e-02 3.5842430e-01 i
+5.0050050e-02 3.6852603e-01 i
+5.4054054e-02 3.7810239e-01 i
+5.8058058e-02 3.8721678e-01 i
+6.2062062e-02 3.9592118e-01 i
+6.6066066e-02 4.0425880e-01 i
+7.0070070e-02 4.1226600e-01 i
+7.4074074e-02 4.1997368e-01 i
+7.8078078e-02 4.2740839e-01 i
+8.2082082e-02 4.3459306e-01 i
+8.6086086e-02 4.4154773e-01 i
+9.0090090e-02 4.4828995e-01 i
+9.4094094e-02 4.5483526e-01 i
+9.8098098e-02 4.6119741e-01 i
+1.0210210e-01 4.6738872e-01 i
+1.0610611e-01 4.7342021e-01 i
+1.1011011e-01 4.7930181e-01 i
+1.1411411e-01 4.8504249e-01 i
+1.1811812e-01 4.9065042e-01 i
+1.2212212e-01 4.9613300e-01 i
+1.2612613e-01 5.0149701e-01 i
+1.3013013e-01 5.0674867e-01 i
+1.3413413e-01 5.1189368e-01 i
+1.3813814e-01 5.1693730e-01 i
+1.4214214e-01 5.2188436e-01 i
+1.4614615e-01 5.2673938e-01 i
+1.5015015e-01 5.3150651e-01 i
+1.5415415e-01 5.3618963e-01 i
+1.5815816e-01 5.4079234e-01 i
+1.6216216e-01 5.4531801e-01 i
+1.6616617e-01 5.4976978e-01 i
+1.7017017e-01 5.5415060e-01 i
+1.7417417e-01 5.5846323e-01 i
+1.7817818e-01 5.6271027e-01 i
+1.8218218e-01 5.6689414e-01 i
+1.8618619e-01 5.7101715e-01 i
+1.9019019e-01 5.7508147e-01 i
+1.9419419e-01 5.7908913e-01 i
+1.9819820e-01 5.8304208e-01 i
+2.0220220e-01 5.8694214e-01 i
+2.0620621e-01 5.9079105e-01 i
+2.1021021e-01 5.9459046e-01 i
+2.1421421e-01 5.9834192e-01 i
+2.1821822e-01 6.0204691e-01 i
+2.2222222e-01 6.0570686e-01 i
+2.2622623e-01 6.0932311e-01 i
+2.3023023e-01 6.1289693e-01 i
+2.3423423e-01 6.1642956e-01 i
+2.3823824e-01 6.1992215e-01 i
+2.4224224e-01 6.2337583e-01 i
+2.4624625e-01 6.2679166e-01 i
+2.5025025e-01 6.3017065e-01 i
+2.5425425e-01 6.3351379e-01 i
+2.5825826e-01 6.3682202e-01 i
+2.6226226e-01 6.4009623e-01 i
+2.6626627e-01 6.4333728e-01 i
+2.7027027e-01 6.4654599e-01 i
+2.7427427e-01 6.4972318e-01 i
+2.7827828e-01 6.5286958e-01 i
+2.8228228e-01 6.5598595e-01 i
+2.8628629e-01 6.5907299e-01 i
+2.9029029e-01 6.6213138e-01 i
+2.9429429e-01 6.6516177e-01 i
+2.9829830e-01 6.6816480e-01 i
+3.0230230e-01 6.7114107e-01 i
+3.0630631e-01 6.7409118e-01 i
+3.1031031e-01 6.7701569e-01 i
+3.1431431e-01 6.7991515e-01 i
+3.1831832e-01 6.8279009e-01 i
+3.2232232e-01 6.8564102e-01 i
+3.2632633e-01 6.8846844e-01 i
+3.3033033e-01 6.9127282e-01 i
+3.3433433e-01 6.9405464e-01 i
+3.3833834e-01 6.9681433e-01 i
+3.4234234e-01 6.9955233e-01 i
+3.4634635e-01 7.0226906e-01 i
+3.5035035e-01 7.0496494e-01 i
+3.5435435e-01 7.0764035e-01 i
+3.5835836e-01 7.1029569e-01 i
+3.6236236e-01 7.1293132e-01 i
+3.6636637e-01 7.1554760e-01 i
+3.7037037e-01 7.1814490e-01 i
+3.7437437e-01 7.2072354e-01 i
+3.7837838e-01 7.2328385e-01 i
+3.8238238e-01 7.2582617e-01 i
+3.8638639e-01 7.2835081e-01 i
+3.9039039e-01 7.3085806e-01 i
+3.9439439e-01 7.3334822e-01 i
+3.9839840e-01 7.3582159e-01 i
+4.0240240e-01 7.3827844e-01 i
+4.0640641e-01 7.4071905e-01 i
+4.1041041e-01 7.4314368e-01 i
+4.1441441e-01 7.4555259e-01 i
+4.1841842e-01 7.4794604e-01 i
+4.2242242e-01 7.5032426e-01 i
+4.2642643e-01 7.5268750e-01 i
+4.3043043e-01 7.5503599e-01 i
+4.3443443e-01 7.5736997e-01 i
+4.3843844e-01 7.5968965e-01 i
+4.4244244e-01 7.6199524e-01 i
+4.4644645e-01 7.6428697e-01 i
+4.5045045e-01 7.6656504e-01 i
+4.5445445e-01 7.6882965e-01 i
+4.5845846e-01 7.7108099e-01 i
+4.6246246e-01 7.7331926e-01 i
+4.6646647e-01 7.7554466e-01 i
+4.7047047e-01 7.7775735e-01 i
+4.7447447e-01 7.7995752e-01 i
+4.7847848e-01 7.8214535e-01 i
+4.8248248e-01 7.8432101e-01 i
+4.8648649e-01 7.8648467e-01 i
+4.9049049e-01 7.8863648e-01 i
+4.9449449e-01 7.9077662e-01 i
+4.9849850e-01 7.9290523e-01 i
+5.0250250e-01 7.9502248e-01 i
+5.0650651e-01 7.9712851e-01 i
+5.1051051e-01 7.9922347e-01 i
+5.1451451e-01 8.0130750e-01 i
+5.1851852e-01 8.0338075e-01 i
+5.2252252e-01 8.0544336e-01 i
+5.2652653e-01 8.0749545e-01 i
+5.3053053e-01 8.0953717e-01 i
+5.3453453e-01 8.1156864e-01 i
+5.3853854e-01 8.1358999e-01 i
+5.4254254e-01 8.1560134e-01 i
+5.4654655e-01 8.1760283e-01 i
+5.5055055e-01 8.1959456e-01 i
+5.5455455e-01 8.2157666e-01 i
+5.5855856e-01 8.2354924e-01 i
+5.6256256e-01 8.2551242e-01 i
+5.6656657e-01 8.2746630e-01 i
+5.7057057e-01 8.2941100e-01 i
+5.7457457e-01 8.3134662e-01 i
+5.7857858e-01 8.3327327e-01 i
+5.8258258e-01 8.3519105e-01 i
+5.8658659e-01 8.3710007e-01 i
+5.9059059e-01 8.3900041e-01 i
+5.9459459e-01 8.4089219e-01 i
+5.9859860e-01 8.4277549e-01 i
+6.0260260e-01 8.4465042e-01 i
+6.0660661e-01 8.4651705e-01 i
+6.1061061e-01 8.4837549e-01 i
+6.1461461e-01 8.5022583e-01 i
+6.1861862e-01 8.5206814e-01 i
+6.2262262e-01 8.5390253e-01 i
+6.2662663e-01 8.5572906e-01 i
+6.3063063e-01 8.5754783e-01 i
+6.3463463e-01 8.5935892e-01 i
+6.3863864e-01 8.6116241e-01 i
+6.4264264e-01 8.6295837e-01 i
+6.4664665e-01 8.6474689e-01 i
+6.5065065e-01 8.6652804e-01 i
+6.5465465e-01 8.6830190e-01 i
+6.5865866e-01 8.7006855e-01 i
+6.6266266e-01 8.7182804e-01 i
+6.6666667e-01 8.7358046e-01 i
+6.7067067e-01 8.7532588e-01 i
+6.7467467e-01 8.7706437e-01 i
+6.7867868e-01 8.7879599e-01 i
+6.8268268e-01 8.8052082e-01 i
+6.8668669e-01 8.8223891e-01 i
+6.9069069e-01 8.8395034e-01 i
+6.9469469e-01 8.8565517e-01 i
+6.9869870e-01 8.8735346e-01 i
+7.0270270e-01 8.8904527e-01 i
+7.0670671e-01 8.9073067e-01 i
+7.1071071e-01 8.9240971e-01 i
+7.1471471e-01 8.9408246e-01 i
+7.1871872e-01 8.9574897e-01 i
+7.2272272e-01 8.9740931e-01 i
+7.2672673e-01 8.9906352e-01 i
+7.3073073e-01 9.0071167e-01 i
+7.3473473e-01 9.0235381e-01 i
+7.3873874e-01 9.0399000e-01 i
+7.4274274e-01 9.0562028e-01 i
+7.4674675e-01 9.0724471e-01 i
+7.5075075e-01 9.0886335e-01 i
+7.5475475e-01 9.1047625e-01 i
+7.5875876e-01 9.1208344e-01 i
+7.6276276e-01 9.1368500e-01 i
+7.6676677e-01 9.1528096e-01 i
+7.7077077e-01 9.1687137e-01 i
+7.7477477e-01 9.1845629e-01 i
+7.7877878e-01 9.2003575e-01 i
+7.8278278e-01 9.2160981e-01 i
+7.8678679e-01 9.2317851e-01 i
+7.9079079e-01 9.2474190e-01 i
+7.9479479e-01 9.2630002e-01 i
+7.9879880e-01 9.2785291e-01 i
+8.0280280e-01 9.2940062e-01 i
+8.0680681e-01 9.3094320e-01 i
+8.1081081e-01 9.3248068e-01 i
+8.1481481e-01 9.3401311e-01 i
+8.1881882e-01 9.3554053e-01 i
+8.2282282e-01 9.3706297e-01 i
+8.2682683e-01 9.3858048e-01 i
+8.3083083e-01 9.4009311e-01 i
+8.3483483e-01 9.4160088e-01 i
+8.3883884e-01 9.4310383e-01 i
+8.4284284e-01 9.4460202e-01 i
+8.4684685e-01 9.4609546e-01 i
+8.5085085e-01 9.4758420e-01 i
+8.5485485e-01 9.4906828e-01 i
+8.5885886e-01 9.5054774e-01 i
+8.6286286e-01 9.5202260e-01 i
+8.6686687e-01 9.5349291e-01 i
+8.7087087e-01 9.5495870e-01 i
+8.7487487e-01 9.5642000e-01 i
+8.7887888e-01 9.5787685e-01 i
+8.8288288e-01 9.5932928e-01 i
+8.8688689e-01 9.6077732e-01 i
+8.9089089e-01 9.6222102e-01 i
+8.9489489e-01 9.6366039e-01 i
+8.9889890e-01 9.6509548e-01 i
+9.0290290e-01 9.6652632e-01 i
+9.0690691e-01 9.6795292e-01 i
+9.1091091e-01 9.6937534e-01 i
+9.1491491e-01 9.7079360e-01 i
+9.1891892e-01 9.7220772e-01 i
+9.2292292e-01 9.7361774e-01 i
+9.2692693e-01 9.7502369e-01 i
+9.3093093e-01 9.7642559e-01 i
+9.3493493e-01 9.7782348e-01 i
+9.3893894e-01 9.7921739e-01 i
+9.4294294e-01 9.8060734e-01 i
+9.4694695e-01 9.8199336e-01 i
+9.5095095e-01 9.8337547e-01 i
+9.5495495e-01 9.8475372e-01 i
+9.5895896e-01 9.8612811e-01 i
+9.6296296e-01 9.8749869e-01 i
+9.6696697e-01 9.8886547e-01 i
+9.7097097e-01 9.9022849e-01 i
+9.7497497e-01 9.9158776e-01 i
+9.7897898e-01 9.9294331e-01 i
+9.8298298e-01 9.9429518e-01 i
+9.8698699e-01 9.9564338e-01 i
+9.9099099e-01 9.9698793e-01 i
+9.9499499e-01 9.9832887e-01 i
+9.9899900e-01 9.9966622e-01 i
+1.0030030e+00 1.0010000e+00 i
+1.0070070e+00 1.0023302e+00 i
+1.0110110e+00 1.0036569e+00 i
+1.0150150e+00 1.0049802e+00 i
+1.0190190e+00 1.0062999e+00 i
+1.0230230e+00 1.0076162e+00 i
+1.0270270e+00 1.0089290e+00 i
+1.0310310e+00 1.0102385e+00 i
+1.0350350e+00 1.0115446e+00 i
+1.0390390e+00 1.0128473e+00 i
+1.0430430e+00 1.0141466e+00 i
+1.0470470e+00 1.0154426e+00 i
+1.0510511e+00 1.0167354e+00 i
+1.0550551e+00 1.0180248e+00 i
+1.0590591e+00 1.0193110e+00 i
+1.0630631e+00 1.0205940e+00 i
+1.0670671e+00 1.0218737e+00 i
+1.0710711e+00 1.0231503e+00 i
+1.0750751e+00 1.0244237e+00 i
+1.0790791e+00 1.0256939e+00 i
+1.0830831e+00 1.0269609e+00 i
+1.0870871e+00 1.0282249e+00 i
+1.0910911e+00 1.0294857e+00 i
+1.0950951e+00 1.0307435e+00 i
+1.0990991e+00 1.0319982e+00 i
+1.1031031e+00 1.0332499e+00 i
+1.1071071e+00 1.0344985e+00 i
+1.1111111e+00 1.0357442e+00 i
+1.1151151e+00 1.0369868e+00 i
+1.1191191e+00 1.0382265e+00 i
+1.1231231e+00 1.0394632e+00 i
+1.1271271e+00 1.0406970e+00 i
+1.1311311e+00 1.0419279e+00 i
+1.1351351e+00 1.0431558e+00 i
+1.1391391e+00 1.0443809e+00 i
+1.1431431e+00 1.0456031e+00 i
+1.1471471e+00 1.0468225e+00 i
+1.1511512e+00 1.0480390e+00 i
+1.1551552e+00 1.0492527e+00 i
+1.1591592e+00 1.0504636e+00 i
+1.1631632e+00 1.0516718e+00 i
+1.1671672e+00 1.0528771e+00 i
+1.1711712e+00 1.0540797e+00 i
+1.1751752e+00 1.0552796e+00 i
+1.1791792e+00 1.0564767e+00 i
+1.1831832e+00 1.0576712e+00 i
+1.1871872e+00 1.0588629e+00 i
+1.1911912e+00 1.0600520e+00 i
+1.1951952e+00 1.0612384e+00 i
+1.1991992e+00 1.0624221e+00 i
+1.2032032e+00 1.0636033e+00 i
+1.2072072e+00 1.0647818e+00 i
+1.2112112e+00 1.0659577e+00 i
+1.2152152e+00 1.0671310e+00 i
+1.2192192e+00 1.0683017e+00 i
+1.2232232e+00 1.0694699e+00 i
+1.2272272e+00 1.0706356e+00 i
+1.2312312e+00 1.0717987e+00 i
+1.2352352e+00 1.0729592e+00 i
+1.2392392e+00 1.0741173e+00 i
+1.2432432e+00 1.0752729e+00 i
+1.2472472e+00 1.0764260e+00 i
+1.2512513e+00 1.0775767e+00 i
+1.2552553e+00 1.0787248e+00 i
+1.2592593e+00 1.0798706e+00 i
+1.2632633e+00 1.0810139e+00 i
+1.2672673e+00 1.0821548e+00 i
+1.2712713e+00 1.0832934e+00 i
+1.2752753e+00 1.0844295e+00 i
+1.2792793e+00 1.0855632e+00 i
+1.2832833e+00 1.0866946e+00 i
+1.2872873e+00 1.0878236e+00 i
+1.2912913e+00 1.0889503e+00 i
+1.2952953e+00 1.0900747e+00 i
+1.2992993e+00 1.0911968e+00 i
+1.3033033e+00 1.0923165e+00 i
+1.3073073e+00 1.0934340e+00 i
+1.3113113e+00 1.0945492e+00 i
+1.3153153e+00 1.0956621e+00 i
+1.3193193e+00 1.0967727e+00 i
+1.3233233e+00 1.0978811e+00 i
+1.3273273e+00 1.0989873e+00 i
+1.3313313e+00 1.1000913e+00 i
+1.3353353e+00 1.1011930e+00 i
+1.3393393e+00 1.1022926e+00 i
+1.3433433e+00 1.1033899e+00 i
+1.3473473e+00 1.1044851e+00 i
+1.3513514e+00 1.1055781e+00 i
+1.3553554e+00 1.1066690e+00 i
+1.3593594e+00 1.1077577e+00 i
+1.3633634e+00 1.1088442e+00 i
+1.3673674e+00 1.1099287e+00 i
+1.3713714e+00 1.1110110e+00 i
+1.3753754e+00 1.1120912e+00 i
+1.3793794e+00 1.1131694e+00 i
+1.3833834e+00 1.1142454e+00 i
+1.3873874e+00 1.1153194e+00 i
+1.3913914e+00 1.1163913e+00 i
+1.3953954e+00 1.1174611e+00 i
+1.3993994e+00 1.1185289e+00 i
+1.4034034e+00 1.1195947e+00 i
+1.4074074e+00 1.1206585e+00 i
+1.4114114e+00 1.1217202e+00 i
+1.4154154e+00 1.1227799e+00 i
+1.4194194e+00 1.1238377e+00 i
+1.4234234e+00 1.1248934e+00 i
+1.4274274e+00 1.1259472e+00 i
+1.4314314e+00 1.1269990e+00 i
+1.4354354e+00 1.1280488e+00 i
+1.4394394e+00 1.1290967e+00 i
+1.4434434e+00 1.1301426e+00 i
+1.4474474e+00 1.1311866e+00 i
+1.4514515e+00 1.1322287e+00 i
+1.4554555e+00 1.1332689e+00 i
+1.4594595e+00 1.1343072e+00 i
+1.4634635e+00 1.1353435e+00 i
+1.4674675e+00 1.1363780e+00 i
+1.4714715e+00 1.1374106e+00 i
+1.4754755e+00 1.1384414e+00 i
+1.4794795e+00 1.1394702e+00 i
+1.4834835e+00 1.1404972e+00 i
+1.4874875e+00 1.1415224e+00 i
+1.4914915e+00 1.1425457e+00 i
+1.4954955e+00 1.1435672e+00 i
+1.4994995e+00 1.1445869e+00 i
+1.5035035e+00 1.1456048e+00 i
+1.5075075e+00 1.1466208e+00 i
+1.5115115e+00 1.1476351e+00 i
+1.5155155e+00 1.1486476e+00 i
+1.5195195e+00 1.1496583e+00 i
+1.5235235e+00 1.1506672e+00 i
+1.5275275e+00 1.1516743e+00 i
+1.5315315e+00 1.1526797e+00 i
+1.5355355e+00 1.1536833e+00 i
+1.5395395e+00 1.1546852e+00 i
+1.5435435e+00 1.1556854e+00 i
+1.5475475e+00 1.1566838e+00 i
+1.5515516e+00 1.1576805e+00 i
+1.5555556e+00 1.1586755e+00 i
+1.5595596e+00 1.1596688e+00 i
+1.5635636e+00 1.1606604e+00 i
+1.5675676e+00 1.1616503e+00 i
+1.5715716e+00 1.1626386e+00 i
+1.5755756e+00 1.1636251e+00 i
+1.5795796e+00 1.1646100e+00 i
+1.5835836e+00 1.1655932e+00 i
+1.5875876e+00 1.1665747e+00 i
+1.5915916e+00 1.1675546e+00 i
+1.5955956e+00 1.1685329e+00 i
+1.5995996e+00 1.1695095e+00 i
+1.6036036e+00 1.1704845e+00 i
+1.6076076e+00 1.1714579e+00 i
+1.6116116e+00 1.1724297e+00 i
+1.6156156e+00 1.1733998e+00 i
+1.6196196e+00 1.1743684e+00 i
+1.6236236e+00 1.1753353e+00 i
+1.6276276e+00 1.1763007e+00 i
+1.6316316e+00 1.1772645e+00 i
+1.6356356e+00 1.1782267e+00 i
+1.6396396e+00 1.1791873e+00 i
+1.6436436e+00 1.1801464e+00 i
+1.6476476e+00 1.1811039e+00 i
+1.6516517e+00 1.1820599e+00 i
+1.6556557e+00 1.1830143e+00 i
+1.6596597e+00 1.1839672e+00 i
+1.6636637e+00 1.1849186e+00 i
+1.6676677e+00 1.1858684e+00 i
+1.6716717e+00 1.1868167e+00 i
+1.6756757e+00 1.1877635e+00 i
+1.6796797e+00 1.1887088e+00 i
+1.6836837e+00 1.1896526e+00 i
+1.6876877e+00 1.1905949e+00 i
+1.6916917e+00 1.1915357e+00 i
+1.6956957e+00 1.1924751e+00 i
+1.6996997e+00 1.1934129e+00 i
+1.7037037e+00 1.1943493e+00 i
+1.7077077e+00 1.1952842e+00 i
+1.7117117e+00 1.1962177e+00 i
+1.7157157e+00 1.1971497e+00 i
+1.7197197e+00 1.1980802e+00 i
+1.7237237e+00 1.1990093e+00 i
+1.7277277e+00 1.1999370e+00 i
+1.7317317e+00 1.2008632e+00 i
+1.7357357e+00 1.2017880e+00 i
+1.7397397e+00 1.2027114e+00 i
+1.7437437e+00 1.2036334e+00 i
+1.7477477e+00 1.2045539e+00 i
+1.7517518e+00 1.2054731e+00 i
+1.7557558e+00 1.2063908e+00 i
+1.7597598e+00 1.2073072e+00 i
+1.7637638e+00 1.2082222e+00 i
+1.7677678e+00 1.2091358e+00 i
+1.7717718e+00 1.2100480e+00 i
+1.7757758e+00 1.2109588e+00 i
+1.7797798e+00 1.2118683e+00 i
+1.7837838e+00 1.2127764e+00 i
+1.7877878e+00 1.2136831e+00 i
+1.7917918e+00 1.2145885e+00 i
+1.7957958e+00 1.2154926e+00 i
+1.7997998e+00 1.2163953e+00 i
+1.8038038e+00 1.2172967e+00 i
+1.8078078e+00 1.2181967e+00 i
+1.8118118e+00 1.2190954e+00 i
+1.8158158e+00 1.2199928e+00 i
+1.8198198e+00 1.2208889e+00 i
+1.8238238e+00 1.2217836e+00 i
+1.8278278e+00 1.2226771e+00 i
+1.8318318e+00 1.2235692e+00 i
+1.8358358e+00 1.2244600e+00 i
+1.8398398e+00 1.2253496e+00 i
+1.8438438e+00 1.2262378e+00 i
+1.8478478e+00 1.2271248e+00 i
+1.8518519e+00 1.2280105e+00 i
+1.8558559e+00 1.2288949e+00 i
+1.8598599e+00 1.2297781e+00 i
+1.8638639e+00 1.2306599e+00 i
+1.8678679e+00 1.2315406e+00 i
+1.8718719e+00 1.2324199e+00 i
+1.8758759e+00 1.2332980e+00 i
+1.8798799e+00 1.2341749e+00 i
+1.8838839e+00 1.2350505e+00 i
+1.8878879e+00 1.2359249e+00 i
+1.8918919e+00 1.2367980e+00 i
+1.8958959e+00 1.2376699e+00 i
+1.8998999e+00 1.2385406e+00 i
+1.9039039e+00 1.2394100e+00 i
+1.9079079e+00 1.2402783e+00 i
+1.9119119e+00 1.2411453e+00 i
+1.9159159e+00 1.2420111e+00 i
+1.9199199e+00 1.2428757e+00 i
+1.9239239e+00 1.2437391e+00 i
+1.9279279e+00 1.2446013e+00 i
+1.9319319e+00 1.2454624e+00 i
+1.9359359e+00 1.2463222e+00 i
+1.9399399e+00 1.2471808e+00 i
+1.9439439e+00 1.2480383e+00 i
+1.9479479e+00 1.2488946e+00 i
+1.9519520e+00 1.2497497e+00 i
+1.9559560e+00 1.2506036e+00 i
+1.9599600e+00 1.2514564e+00 i
+1.9639640e+00 1.2523080e+00 i
+1.9679680e+00 1.2531585e+00 i
+1.9719720e+00 1.2540078e+00 i
+1.9759760e+00 1.2548560e+00 i
+1.9799800e+00 1.2557030e+00 i
+1.9839840e+00 1.2565489e+00 i
+1.9879880e+00 1.2573936e+00 i
+1.9919920e+00 1.2582372e+00 i
+1.9959960e+00 1.2590797e+00 i
+2.0000000e+00 1.2599210e+00 i
+
--- /dev/null
+\documentclass[a4paper]{article}
+\usepackage{standalone}
+\usepackage{newclude}
+\usepackage[a4paper,margin=2cm]{geometry}
+\usepackage{multicol}
+\usepackage{multirow}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{harpoon}
+\usepackage{tabularx}
+\usepackage{makecell}
+\usepackage[dvipsnames, table]{xcolor}
+\usepackage{blindtext}
+\usepackage{graphicx}
+\usepackage{wrapfig}
+\usepackage{tikz}
+\usepackage{tikz-3dplot}
+\usepackage{pgfplots}
+\pgfplotsset{compat=1.8}
+\usepackage{mathtools}
+\usetikzlibrary{calc}
+\usetikzlibrary{angles}
+\usetikzlibrary{datavisualization.formats.functions}
+\usetikzlibrary{decorations.markings}
+\usepgflibrary{arrows.meta}
+\usepackage{longtable}
+\usepackage{fancyhdr}
+\pagestyle{fancy}
+\fancyhead[LO,LE]{Year 12 Methods}
+\fancyhead[CO,CE]{Andrew Lorimer}
+\providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
+\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}
+\definecolor{cas}{HTML}{e6f0fe}
+\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={$y$},
+}}
+\begin{document}
+
+\title{\vspace{-2cm}\hrule\vspace{0.4cm} Year 12 Methods}
+\author{Andrew Lorimer}
+\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}
+
+\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}
+
+\subsection*{Finding inverse functions \(f^{-1}\)}
+
+\begin{itemize}
+\tightlist
+\item
+ if \(f(g(x)) = x\), then \(g\) is the inverse of \(f\)
+\item
+ reflection across \(y-x\)
+\item
+ \(\operatorname{ran} \> f = \operatorname{dom} \> f^{-1}, \quad \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}\)
+\item
+ inverse \(\ne\) inverse \emph{function} (i.e.~inverse must pass
+ vertical line test)\\
+ \(\implies f^{-1}(x)\) exists \(\iff f(x)\) is one to one
+\item
+ \(f^{-1}(x)=f(x)\) intersections may lie on line \(y=x\)
+\end{itemize}
+
+\subsubsection*{Requirements for showing working for \(f^{-1}\)}
+
+\begin{enumerate}
+\def\labelenumi{\arabic{enumi}.}
+\tightlist
+\item
+ start with \emph{``let \(y=f(x)\)''}
+\item
+ must state \emph{``take inverse''} for line where \(y\) and \(x\) are
+ swapped
+\item
+ do all working in terms of \(y=\dots\)
+\item
+ for sqrt, state \(\pm\) solutions then show restricted
+\item
+ for inverse \emph{function}, state in function notation
+\end{enumerate}
+\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*{Odd and even functions}
+
+Even when \(f(x) = -f(x)\)\\
+Odd when \(-f(x) = f(-x)\)
+
+Function is even if it is symmetrical across \(y\)-axis
+\hspace{5em}\(\implies f(x)=f(-x)\)\\
+Function \(x^{\pm {p \over q}}\) is odd if \(q\) is odd\\
+
+\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}
+\pagebreak
+ \pgfplotsset{every axis/.append style={
+ xlabel=, % put the x axis in the middle
+ ylabel=, % put the y axis in the middle
+ }}
+ \begin{table*}[ht]
+ \centering
+ \begin{tabularx}{\textwidth}{r|X|X}
+ & \(n\) is even & \(n\) is odd \\ \hline
+ \(x^n, n \in \mathbb{Z}^+\) &
+ \makecell{\\\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}} &
+ \makecell{\\\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}} \\
+ \(x^n, n \in \mathbb{Z}^-\) &
+ \makecell{\\\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}} &
+ \makecell{\\\begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, scale=0.4, samples=100, smooth] \addplot[orange, mark=none] {(x^(-1))}; \end{axis}\end{tikzpicture}} \\
+ \(x^{\frac{1}{n}}, n \in \mathbb{Z}^-\) &
+ \makecell{\\\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}} &
+ \makecell{\\\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{tabularx}
+ \end{table*}
+ \pgfplotsset{every axis/.append style={
+ xlabel=\(x\), % put the x axis in the middle
+ ylabel=\(y\), % put the y axis in the middle
+ }}
+
+\section{Polynomials}
+
+\subsection*{Quadratics}
+
+\[ x^2 + bx + c = (x+m)(x+n) \]
+\hfill where \(mn=c, \> m+n=b\)
+
+\begin{align*}
+ \hline
+ \textbf{Difference} && a^2 - b^2 &= (a-b)(a+b) \\[2ex]
+ \textbf{Perfect sq.} && a^2 \pm 2ab + b^2 &= (a \pm b^2) \\[2ex]
+ \textbf{Completing} && x^2+bx+c &= (x+\frac{b}{2})^2+c-\frac{b^2}{4} \\
+ && ax^2+bx+c &= a(x-\frac{b}{2a})^2+c-\frac{b^2}{4a} \\[2ex]
+ \textbf{Quadratic} && x &= \dfrac{-b\pm\sqrt{b^2-4ac}}{2a} \\
+ && & \text{where} \Delta=b^2-4ac \\
+ \hline
+\end{align*}
+
+\subsection*{Cubics}
+
+\textbf{Difference of cubes:} \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)\\
+\textbf{Sum of cubes:} \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)\\
+\textbf{Perfect cubes:} \(a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3\)
+
+\[ y=a(bx-h)^3 + c \]
+
+\begin{itemize}
+\tightlist
+\item
+ \(m=0\) at \emph{stationary point of inflection}
+ (i.e.~(\({h \over b}, k)\))
+\item
+ in form \(y=(x-a)^2(x-b)\), local max at \(x=a\), local min at \(x=b\)
+\item
+ in form \(y=a(x-b)(x-c)(x-d)\): \(x\)-intercepts at \(b, c, d\)
+\item
+ in form \(y=a(x-b)^2(x-c)\), touches \(x\)-axis at \(b\), intercept at
+ \(c\)
+\end{itemize}
+
+\subsection*{Linear and quadratic
+graphs}
+
+\subsubsection*{Forms of linear
+equations}
+
+\begin{itemize}
+\tightlist
+ \item \(y=mx+c\)
+ \item \(\frac{x}{a} + \frac{y}{b}=1\) where \((x_1, y_1)\) lies on the graph
+ \item \(y-y_1 = m(x-x_1)\) where \((a,0)\) and \((0,b)\) are \(x\)- and \(y\)-intercepts
+\end{itemize}
+
+\subsection*{Line properties}
+
+Parallel lines: \(m_1 = m_2\)\\
+Perpendicular lines: \(m_1 \times m_2 = -1\)\\
+Distance: \(|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
+
+\subsection*{Quartic graphs}
+
+\subsubsection*{Forms of quartic
+equations}
+
+\(y=ax^4\)\\
+\(y=a(x-b)(x-c)(x-d)(x-e)\)\\
+\(y=ax^4+cd^2 (c \ge 0)\)\\
+\(y=ax^2(x-b)(x-c)\)\\
+\(y=a(x-b)^2(x-c)^2\)\\
+\(y=a(x-b)(x-c)^3\)
+
+\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}
+
+
+\input{temp/transformations}
+\input{temp/stuff}
+\input{circ-functions}
+\input{temp/calculus}
+
+\end{multicols}
+\end{document}
--- /dev/null
+\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}
+
+\emph{Stationary point} - i.e.
+\(f^\prime(x)=0\)\\
+\emph{Point of inflection} - max \(|\)gradient\(|\) (i.e.
+\(f^{\prime\prime} = 0\))
+
+ \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}
+
--- /dev/null
+\setstretch{1.3}
+\pagenumbering{gobble}
+
+\hypertarget{inverse-functions}{%
+\section{Inverse functions}\label{inverse-functions}}
+
+\hypertarget{functions}{%
+\subsection{Functions}\label{functions}}
+
+\begin{itemize}
+\tightlist
+\item
+ vertical line test
+\item
+ each \(x\) value produces only one \(y\) value
+\end{itemize}
+
+\hypertarget{one-to-one-functions}{%
+\subsection{One to one functions}\label{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}
+
+\hypertarget{deriving-f-1}{%
+\subsection{\texorpdfstring{Deriving
+\(f^{-1}\)}{Deriving f\^{}\{-1\}}}\label{deriving-f-1}}
+
+\begin{itemize}
+\tightlist
+\item
+ if \(f(g(x)) = x\), then \(g\) is the inverse of \(f\)
+\item
+ reflection across \(y-x\)
+\item
+ \(\operatorname{ran} \> f = \operatorname{dom} \> f^{-1}, \quad \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}\)
+\item
+ inverse \(\ne\) inverse \emph{function} (i.e.~inverse must pass
+ vertical line test)\\
+ \(\implies f^{-1}(x)\) exists \(\iff f(x)\) is one to one
+\item
+ \(f^{-1}(x)=f(x)\) intersections may lie on line \(y=x\)
+\end{itemize}
+
+\hypertarget{requirements-for-showing-working-for-f-1}{%
+\subsubsection{\texorpdfstring{Requirements for showing working for
+\(f^{-1}\)}{Requirements for showing working for f\^{}\{-1\}}}\label{requirements-for-showing-working-for-f-1}}
+
+\begin{enumerate}
+\def\labelenumi{\arabic{enumi}.}
+\tightlist
+\item
+ start with \emph{``let \(y=f(x)\)''}
+\item
+ must state \emph{``take inverse''} for line where \(y\) and \(x\) are
+ swapped
+\item
+ do all working in terms of \(y=\dots\)
+\item
+ for square root, state \(\pm\) solutions then show restricted
+\item
+ for inverse \emph{function}, state in function notation
+\end{enumerate}
--- /dev/null
+% Options for packages loaded elsewhere
+\PassOptionsToPackage{unicode}{hyperref}
+\PassOptionsToPackage{hyphens}{url}
+%
+\documentclass[
+]{article}
+\usepackage{lmodern}
+\usepackage{amssymb,amsmath}
+\usepackage{ifxetex,ifluatex}
+\ifnum 0\ifxetex 1\fi\ifluatex 1\fi=0 % if pdftex
+ \usepackage[T1]{fontenc}
+ \usepackage[utf8]{inputenc}
+ \usepackage{textcomp} % provide euro and other symbols
+\else % if luatex or xetex
+ \usepackage{unicode-math}
+ \defaultfontfeatures{Scale=MatchLowercase}
+ \defaultfontfeatures[\rmfamily]{Ligatures=TeX,Scale=1}
+\fi
+% Use upquote if available, for straight quotes in verbatim environments
+\IfFileExists{upquote.sty}{\usepackage{upquote}}{}
+\IfFileExists{microtype.sty}{% use microtype if available
+ \usepackage[]{microtype}
+ \UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts
+}{}
+\makeatletter
+\@ifundefined{KOMAClassName}{% if non-KOMA class
+ \IfFileExists{parskip.sty}{%
+ \usepackage{parskip}
+ }{% else
+ \setlength{\parindent}{0pt}
+ \setlength{\parskip}{6pt plus 2pt minus 1pt}}
+}{% if KOMA class
+ \KOMAoptions{parskip=half}}
+\makeatother
+\usepackage{xcolor}
+\IfFileExists{xurl.sty}{\usepackage{xurl}}{} % add URL line breaks if available
+\IfFileExists{bookmark.sty}{\usepackage{bookmark}}{\usepackage{hyperref}}
+\hypersetup{
+ pdfauthor={Andrew Lorimer},
+ hidelinks,
+ pdfcreator={LaTeX via pandoc}}
+\urlstyle{same} % disable monospaced font for URLs
+\usepackage[a4paper, margin=2cm]{geometry}
+\setlength{\emergencystretch}{3em} % prevent overfull lines
+\providecommand{\tightlist}{%
+ \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
+\setcounter{secnumdepth}{-\maxdimen} % remove section numbering
+\usepackage{setspace}
+\usepackage{fancyhdr}
+\pagestyle{fancy}
+\fancyhead[LO,LE]{Year 12 Methods}
+\fancyhead[CO,CE]{Andrew Lorimer}
+\usepackage{graphicx}
+\usepackage{tabularx}
+\usepackage[dvipsnames]{xcolor}
+
+\author{Andrew Lorimer}
+\date{}
+
+\begin{document}
+
+\hypertarget{polynomials}{%
+\section{Polynomials}\label{polynomials}}
+
+\hypertarget{quadratics}{%
+\subsection{Quadratics}\label{quadratics}}
+
+\newcolumntype{R}{>{\raggedleft\arraybackslash}X}
+\begin{tabularx}{\columnwidth}{Rl}
+ General form& \parbox[t]{5cm}{$x^2 + bx + c = (x+m)(x+n)$\\ where $mn=c, \> m+n=b$} \\
+ \hline
+ Difference of squares & $a^2 - b^2 = (a - b)(a + b)$ \\
+ \hline
+ Perfect squares & \parbox[c]{5cm}{$a^2 \pm 2ab + b^2 = (a \pm b^2)$} \\
+ \hline
+ Completing the square & \parbox[t]{5cm}{$x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}$ \\ $ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}$} \\
+ \hline
+ Quadratic formula & $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$ where $\Delta=b^2-4ac$ \\
+\end{tabularx}
+
+\hypertarget{cubics}{%
+\subsection{Cubics}\label{cubics}}
+
+\textbf{Difference of cubes:} \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)\\
+\textbf{Sum of cubes:} \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)\\
+\textbf{Perfect cubes:} \(a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3\)
+
+\[y=a(bx-h)^3 + c\]
+
+\begin{itemize}
+\tightlist
+\item
+ \(m=0\) at \emph{stationary point of inflection}
+ (i.e.~(\({h \over b}, k)\))
+\item
+ in form \(y=(x-a)^2(x-b)\), local max at \(x=a\), local min at \(x=b\)
+\item
+ in form \(y=a(x-b)(x-c)(x-d)\): \(x\)-intercepts at \(b, c, d\)
+\item
+ in form \(y=a(x-b)^2(x-c)\), touches \(x\)-axis at \(b\), intercept at
+ \(c\)
+\end{itemize}
+
+\hypertarget{linear-and-quadratic-graphs}{%
+\subsection{Linear and quadratic
+graphs}\label{linear-and-quadratic-graphs}}
+
+\hypertarget{forms-of-linear-equations}{%
+\subsubsection{Forms of linear
+equations}\label{forms-of-linear-equations}}
+
+\(y=mx+c\) where \(m\) is gradient and \(c\) is \(y\)-intercept\\
+\({x \over a} + {y \over b}=1\) where \(m\) is gradient and
+\((x_1, y_1)\) lies on the graph\\
+\(y-y_1 = m(x-x_1)\) where \((a,0)\) and \((0,b)\) are \(x\)- and
+\(y\)-intercepts
+
+\hypertarget{line-properties}{%
+\subsection{Line properties}\label{line-properties}}
+
+Parallel lines: \(m_1 = m_2\)\\
+Perpendicular lines: \(m_1 \times m_2 = -1\)\\
+Distance: \(|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
+
+\hypertarget{quartic-graphs}{%
+\subsection{Quartic graphs}\label{quartic-graphs}}
+
+\hypertarget{forms-of-quadratic-equations}{%
+\subsubsection{Forms of quadratic
+equations}\label{forms-of-quadratic-equations}}
+
+\(y=ax^4\)\\
+\(y=a(x-b)(x-c)(x-d)(x-e)\)\\
+\(y=ax^4+cd^2 (c \ge 0)\)\\
+\(y=ax^2(x-b)(x-c)\)\\
+\(y=a(x-b)^2(x-c)^2\)\\
+\(y=a(x-b)(x-c)^3\)
+
+\hypertarget{simultaneous-equations-linear}{%
+\subsection{Simultaneous equations
+(linear)}\label{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}
+
+\hypertarget{solving-protectbegincasespx-qy-a-rx-sy-bprotectendcases-for-01infty-solutions}{%
+\subsubsection{\texorpdfstring{Solving
+\(\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>\)
+for \(\{0,1,\infty\}\)
+solutions}{Solving \textbackslash protect\textbackslash begin\{cases\}px + qy = a \textbackslash\textbackslash{} rx + sy = b\textbackslash protect\textbackslash end\{cases\} \textbackslash\textgreater{} for \textbackslash\{0,1,\textbackslash infty\textbackslash\} solutions}}\label{solving-protectbegincasespx-qy-a-rx-sy-bprotectendcases-for-01infty-solutions}}
+
+where all coefficients are known except for one, and \(a, b\) are known
+
+\begin{enumerate}
+\def\labelenumi{\arabic{enumi}.}
+\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
+
+ \begin{itemize}
+ \tightlist
+ \item
+ \emph{--- for infinite/no solutions: ---}
+ \end{itemize}
+\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}
+
+\hypertarget{solving-protectbegincasesa_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_3protectendcases}{%
+\subsubsection{\texorpdfstring{Solving
+\(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)}{Solving \textbackslash protect\textbackslash begin\{cases\}a\_1 x + b\_1 y + c\_1 z = d\_1 \textbackslash\textbackslash{} a\_2 x + b\_2 y + c\_2 z = d\_2 \textbackslash\textbackslash{} a\_3 x + b\_3 y + c\_3 z = d\_3\textbackslash protect\textbackslash end\{cases\}}}\label{solving-protectbegincasesa_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_3protectendcases}}
+
+\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
+\item
+ etc.
+\end{itemize}
+
+\end{document}
--- /dev/null
+\section{Exponentials \& Logarithms}
+
+\subsubsection*{Logarithmic identities}
+
+\begin{align*}
+ \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 \\
+ \log_a(\frac{m}{n}) &= \log_am - \log_a \\
+ \log_a(m^{-1}) & = -\log_am \\
+ \log_b c &= \frac{\log_a c}{\log_a b}
+\end{align*}
+
+\subsubsection*{Index identities}
+
+\begin{align*}
+ b^{m+n} &= b^m \cdot b^n \\
+ (b^m)^n &= b^{m \cdot n} \\
+ (b \cdot c)^n &= b^n \cdot c^n \\
+ {b^m \div a^n} &= {b^{m-n}}
+\end{align*}
+
+\subsection*{Inverse functions}
+
+For \(f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=a^x\), inverse is:
+
+\[f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=\log_ax\]
+
+\subsection*{Euler's number \(e\)}
+
+\[e= \lim_{n \rightarrow \infty} (1 + {1 \over n})^n\]
+
+\subsection*{Modelling}
+
+\[A = A_0 e^{kt}\]
+
+\begin{itemize}
+\tightlist
+\item
+ \(A_0\) is initial value
+\item
+ \(t\) is time taken
+\item
+ \(k\) is a constant
+\item
+ For continuous growth, \(k > 0\)
+\item
+ For continuous decay, \(k < 0\)
+\end{itemize}
+
+\subsection*{Graphing exponential functions}
+
+\[f(x)=Aa^{k(x-b)} + c, \quad \vert \> a > 1\]
+
+\begin{itemize}
+\tightlist
+\item
+ \textbf{\(y\)-intercept} at \((0, A \cdot a^{-kb}+c)\) as
+ \(x \rightarrow \infty\)
+\item
+ \textbf{horizontal asymptote} at \(y=c\)
+\item
+ \textbf{domain} is \(\mathbb{R}\)
+\item
+ \textbf{range} is \((c, \infty)\)
+\item
+ dilation of factor \(|A|\) from \(x\)-axis
+\item
+ dilation of factor \(1 \over k\) from \(y\)-axis
+\end{itemize}
+
+\begin{tikzpicture}
+ \begin{axis}[restrict x to domain=-0.9:0.9, axis y line = middle, yticklabels={,,}, xticklabels={,,}, enlargelimits, ticks=none]
+ \addplot[red, thick, smooth, samples=100] plot (\x, {pow(2,x)}) node[below, pos=1] {\(2^x\)};
+ \addplot[blue, thick, smooth, samples=100] plot (\x, {pow(3,x)}) node[left, pos=1] {\(3^x\)};
+ \addplot[orange, thick, smooth, samples=100] plot (\x, {pow(e,x)}) node[below, pos=1] {\(e^x\)};
+ \addplot[mark=*] coordinates {(0,1)} node[above left]{\((0,1)\)} ;
+ \addplot[purple, ultra thick, dashed] plot (\x, 0) node[black, below, font=\footnotesize, pos=0.75] {\(y=0\)};
+ \end{axis}
+\end{tikzpicture}
+
+\subsection*{Graphing logarithmic functions}
+
+\(\log_e x\) is the inverse of \(e^x\) (reflection across \(y=x\))
+
+\[f(x)=A \log_a k(x-b) + c\]
+
+where
+
+\begin{itemize}
+\tightlist
+\item
+ \textbf{domain} is \((b, \infty)\)
+\item
+ \textbf{range} is \(\mathbb{R}\)
+\item
+ \textbf{vertical asymptote} at \(x=b\)
+\item
+ \(y\)-intercept exists if \(b<0\)
+\item
+ dilation of factor \(|A|\) from \(x\)-axis
+\item
+ dilation of factor \(1 \over k\) from \(y\)-axis
+\end{itemize}
+\begin{tikzpicture}
+ \begin{axis}[axis lines=middle, xmin=-0.5, xmax=5, ymin=-2, ymax=3, ticks=none]
+ \addplot[purple, ultra thick, dashed] coordinates {(0,-1.8) (0,2.8)} node[black, below right, pos=0.75, font=\footnotesize] {\(x=0\)};
+ \addplot[orange,thick,domain=0.01:4,smooth,samples=100] {ln(x)} node[right, pos=1] {\(\log_e x\)};
+ \addplot[red,thick,domain=0.01:4,smooth,samples=100] {log2(x)} node[right, pos=1] {\(\log_2 x\)};
+ \addplot[blue,thick,domain=0.01:4,smooth,samples=100] {ln(x)/ln(3)} node[below right, pos=1] {\(\log_3 x\)};
+ \addplot[mark=*] coordinates {(1,0)} node[above left]{\((0,1)\)} ;
+ \end{axis}
+\end{tikzpicture}
+
+\subsection*{Finding equations}
+
+\colorbox{cas}{On CAS:}
+\includegraphics[width=0.78125in]{graphics/cas-simultaneous.png}
--- /dev/null
+\definecolor{shade1}{HTML}{ffffff}
+\definecolor{shade2}{HTML}{e6f2ff}
+\definecolor{shade3}{HTML}{cce2ff}
+\section{Transformations}
+
+\textbf{Order of operations:} DRT
+
+\begin{center}dilations --- reflections --- translations\end{center}
+
+\subsection*{Transforming \(x^n\) to \(a(x-h)^n+K\)}
+
+\begin{itemize}
+\tightlist
+\item
+ dilation factor of \(|a|\) units parallel to \(y\)-axis or from
+ \(x\)-axis
+\item
+ if \(a<0\), graph is reflected over \(x\)-axis
+\item
+ translation of \(k\) units parallel to \(y\)-axis or from \(x\)-axis
+\item
+ translation of \(h\) units parallel to \(x\)-axis or from \(y\)-axis
+\item
+ for \((ax)^n\), dilation factor is \(1 \over a\) parallel to
+ \(x\)-axis or from \(y\)-axis
+\item
+ when \(0 < |a| < 1\), graph becomes closer to axis
+\end{itemize}
+
+\subsection*{Transforming \(f(x)\) to \(y=Af[n(x+c)]+b\)}
+
+Applies to exponential, log, trig, \(e^x\), polynomials.\\
+Functions must be written in form \(y=Af[n(x+c)]+b\)
+
+\begin{itemize}
+\tightlist
+\item
+ dilation by factor \(|A|\) from \(x\)-axis (if \(A<0\), reflection
+ across \(y\)-axis)
+\item
+ dilation by factor \(1 \over n\) from \(y\)-axis (if \(n<0\),
+ reflection across \(x\)-axis)
+\item
+ translation of \(c\) units from \(y\)-axis (\(x\)-shift)
+\item
+ translation of \(b\) units from \(x\)-axis (\(y\)-shift)
+\end{itemize}
+
+\subsection*{Dilations}
+
+Two pairs of equivalent processes for \(y=f(x)\):
+
+\begin{enumerate}
+\def\labelenumi{\arabic{enumi}.}
+\item
+ \begin{itemize}
+ \tightlist
+ \item
+ Dilating from \(x\)-axis: \((x, y) \rightarrow (x, by)\)
+ \item
+ Replacing \(y\) with \(y \over b\) to obtain \(y = b f(x)\)
+ \end{itemize}
+\item
+ \begin{itemize}
+ \tightlist
+ \item
+ Dilating from \(y\)-axis: \((x, y) \rightarrow (ax, y)\)
+ \item
+ Replacing \(x\) with \(x \over a\) to obtain \(y = f({x \over a})\)
+ \end{itemize}
+\end{enumerate}
+
+For graph of \(y={1 \over x}\), horizontal \& vertical dilations are
+equivalent (symmetrical). If \(y={a \over x}\), graph is contracted
+rather than dilated.
+
+\subsection*{Matrix transformations}
+
+Find new point \((x^\prime, y^\prime)\). Substitute these into original
+equation to find image with original variables \((x, y)\).
+
+\subsection*{Reflections}
+
+\begin{itemize}
+\tightlist
+\item
+ Reflection \textbf{in} axis = reflection \textbf{over} axis =
+ reflection \textbf{across} axis
+\item
+ Translations do not change
+\end{itemize}
+
+\subsection*{Translations}
+
+For \(y = f(x)\), these processes are equivalent:
+
+\begin{itemize}
+\tightlist
+\item
+ applying the translation \((x, y) \rightarrow (x + h, y + k)\) to the
+ graph of \(y = f(x)\)
+\item
+ replacing \(x\) with \(x-h\) and \(y\) with \(y-k\) to obtain
+ \(y-k = f(x-h)\)
+\end{itemize}
+
+\subsection*{Power functions}
+
+\textbf{Strictly increasing:} \(f(x_2) > f(x_1)\) where \(x_2 > x_1\)
+(including \(x=0\))
+
+\subsubsection*{Odd and even functions}
+
+Even when \(f(x) = -f(x)\)\\
+Odd when \(-f(x) = f(-x)\)
+
+Function is even if it can be reflected across \(y\)-axis
+\(\implies f(x)=f(-x)\)\\
+Function \(x^{\pm {p \over q}}\) is odd if \(q\) is odd
+
+
+\subsubsection*{\(x^{-1 \over n}\) where \(n \in \mathbb{Z}^+\)}
+
+Mostly only on CAS.
+
+We can write
+\(x^{-1 \over n} = {1 \over {x^{1 \over n}}} = {1 \over ^n \sqrt{x}}\)n.\\
+Domain is:
+\(\begin{cases} \mathbb{R} \setminus \{0\}\hspace{0.5em} \text{ if }n\text{ is odd} \\ \mathbb{R}^+ \hspace{2.6em}\text{if }n\text{ is even}\end{cases}\)
+
+If \(n\) is odd, it is an odd function.
+
+\subsubsection*{\(x^{p \over q}\) where \(p, q \in \mathbb{Z}^+\)}
+
+\[x^{p \over q} = \sqrt[q]{x^p}\]
+
+\begin{itemize}
+\tightlist
+\item
+ if \(p > q\), the shape of \(x^p\) is dominant
+\item
+ if \(p < q\), the shape of \(x^{1 \over q}\) is dominant
+\item
+ points \((0, 0)\) and \((1, 1)\) will always lie on graph
+\item
+ Domain is:
+ \(\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}\)
+\end{itemize}
+
+\subsection*{Piecewise functions}
+
+\[\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\]
+
+\textbf{Open circle:} point included\\
+\textbf{Closed circle:} point not included
+
+\subsection*{Operations on functions}
+
+For \(f \pm g\) and \(f \times g\):
+\quad \(\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)\)
+
+Addition of linear piecewise graphs: add \(y\)-values at key points
+
+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}
+
+\subsection*{Composite functions}
+
+\((f \circ g)(x)\) is defined iff
+\(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)
-\PassOptionsToPackage{unicode=true}{hyperref} % options for packages loaded elsewhere
-\PassOptionsToPackage{hyphens}{url}
-%
-\documentclass[]{article}
+\documentclass[standalone]{article}
\usepackage{lmodern}
\usepackage{amssymb,amsmath}
\usepackage{ifxetex,ifluatex}
\KOMAoptions{parskip=half}}
\makeatother
\usepackage{xcolor}
-\IfFileExists{xurl.sty}{\usepackage{xurl}}{} % add URL line breaks if available
-\IfFileExists{bookmark.sty}{\usepackage{bookmark}}{\usepackage{hyperref}}
\urlstyle{same} % don't use monospace font for urls
\usepackage{fullpage}
\usepackage{longtable,booktabs}
\begin{document}
-\hypertarget{transformation}{%
-\section{Transformation}\label{transformation}}
+\section{Transformations}
\textbf{Order of operations:} DRT - Dilations, Reflections, Translations
-\hypertarget{transforming-xn-to-ax-hnk}{%
-\subsection{\texorpdfstring{Transforming \(x^n\) to
-\(a(x-h)^n+K\)}{Transforming x\^{}n to a(x-h)\^{}n+K}}\label{transforming-xn-to-ax-hnk}}
+\subsection{Transforming x\^{}n to a(x-h)\^{}n+K}
\begin{itemize}
\tightlist
when \(0 < |a| < 1\), graph becomes closer to axis
\end{itemize}
-\hypertarget{translations}{%
-\subsection{Translations}\label{translations}}
+\subsection{Translations}
For \(y = f(x)\), these processes are equivalent:
applying the translation \((x, y) \rightarrow (x + h, y + k)\) to the
graph of \(y = f(x)\)
\item
- replacing \(x\) with \(x − h\) and \(y\) with \(y − k\) to obtain
- \(y − k = f (x − h)\)
+ replacing \(x\) with \(x - h\) and \(y\) with \(y - k\) to obtain \(y - k = f (x - h)\)
\end{itemize}
-\hypertarget{dilations}{%
-\subsection{Dilations}\label{dilations}}
+\subsection{Dilations}
For the graph of \(y = f(x)\), there are two pairs of equivalent
processes:
equivalent (symmetrical). If \(y={a \over x}\), graph is contracted
rather than dilated.
-\hypertarget{transforming-fx-to-yafnxcb}{%
-\subsection{\texorpdfstring{Transforming \(f(x)\) to
-\(y=Af[n(x+c)]+b\)}{Transforming f(x) to y=Af{[}n(x+c){]}+b}}\label{transforming-fx-to-yafnxcb}}
+\subsection{Transforming \(f(x)\) to \(y=Af[n(x+c)]+b\)}
Applies to exponential, log, trig, power, polynomial functions.\\
Functions must be written in form \(y=Af[n(x+c)] + b\)
\(c\) - translation from \(y\)-axis (\(x\)-shift)\\
\(b\) - translation from \(x\)-axis (\(y\)-shift)
-\hypertarget{power-functions}{%
-\subsection{Power functions}\label{power-functions}}
+\subsection{Power functions}
\textbf{Strictly increasing:} \(f(x_2) > f(x_1)\) where \(x_2 > x_1\)
(including \(x=0\))
-\hypertarget{odd-and-even-functions}{%
-\subsubsection{Odd and even functions}\label{odd-and-even-functions}}
+\subsubsection{Odd and even functions}
Even when \(f(x) = -f(x)\)\\
Odd when \(-f(x) = f(-x)\)
\(\implies f(x)=f(-x)\)\\
Function \(x^{\pm {p \over q}}\) is odd if \(q\) is odd
-\hypertarget{xn-where-n-in-mathbbz}{%
-\subsubsection{\texorpdfstring{\(x^n\) where
-\(n \in \mathbb{Z}^+\)}{x\^{}n where n \textbackslash{}in \textbackslash{}mathbb\{Z\}\^{}+}}\label{xn-where-n-in-mathbbz}}
+\newcolumntype{C}{>{\centering\arraybackslash} m{3cm} }
+\begin{center}
+\begin{tabular}{m{1.2cm}|C|C}
+ & $n$ is even & $n$ is odd \\
+ \hline
+ \parbox[c]{1.2cm}{$x^n,\\ n \in \mathbb{Z}^+$} & {\includegraphics[height=3cm]{graphics/parabola.png}} & {\includegraphics[height=3cm]{graphics/cubic.png}}\\
+ \parbox[c]{1.2cm}{$x^n$,\\ $n \in \mathbb{Z}^-$} & {\includegraphics[height=3cm]{graphics/truncus.png}} & {\includegraphics[height=3cm]{graphics/hyperbola.png}}\\
+ \parbox[c]{1.2cm}{$x^{1 \over n},\\ n \in \mathbb{Z}^+$} & {\includegraphics[height=3cm]{graphics/square-root-graph.png}} & {\includegraphics[height=3cm]{graphics/cube-root-graph.png}}\\
+\end{tabular}
+\end{center}
+\subsubsection{\(x^n\) where \(n \in \mathbb{Z}^+\)}
-\begin{longtable}[]{@{}ll@{}}
-\toprule
-\(n\) is even: & \(n\) is odd:\tabularnewline
-\midrule
-\endhead
-\includegraphics[width=0.2\textwidth,height=\textheight]{graphics/parabola.png}
-&
-\includegraphics[width=0.2\textwidth,height=\textheight]{graphics/cubic.png}\tabularnewline
-\bottomrule
-\end{longtable}
-
-\hypertarget{xn-where-n-in-mathbbz-}{%
-\subsubsection{\texorpdfstring{\(x^n\) where
-\(n \in \mathbb{Z}^-\)}{x\^{}n where n \textbackslash{}in \textbackslash{}mathbb\{Z\}\^{}-}}\label{xn-where-n-in-mathbbz-}}
-
-\begin{longtable}[]{@{}ll@{}}
-\toprule
-\(n\) is even: & \(n\) is odd:\tabularnewline
-\midrule
-\endhead
-\includegraphics[width=0.2\textwidth,height=\textheight]{graphics/truncus.png}
-&
-\includegraphics[width=0.2\textwidth,height=\textheight]{graphics/hyperbola.png}\tabularnewline
-\bottomrule
-\end{longtable}
-
-\hypertarget{x1-over-n-where-n-in-mathbbz}{%
-\subsubsection{\texorpdfstring{\(x^{1 \over n}\) where
-\(n \in \mathbb{Z}^+\)}{x\^{}\{1 \textbackslash{}over n\} where n \textbackslash{}in \textbackslash{}mathbb\{Z\}\^{}+}}\label{x1-over-n-where-n-in-mathbbz}}
+\subsubsection{\(x^{1 \over n}\) where \(n \in \mathbb{Z}^+\)}
\begin{longtable}[]{@{}ll@{}}
\toprule
\bottomrule
\end{longtable}
-\hypertarget{x-1-over-n-where-n-in-mathbbz}{%
-\subsubsection{\texorpdfstring{\(x^{-1 \over n}\) where
-\(n \in \mathbb{Z}^+\)}{x\^{}\{-1 \textbackslash{}over n\} where n \textbackslash{}in \textbackslash{}mathbb\{Z\}\^{}+}}\label{x-1-over-n-where-n-in-mathbbz}}
+\subsubsection{\(x^{-1 \over n}\) where \(n \in \mathbb{Z}^+\)}
Mostly only on CAS.
If \(n\) is odd, it is an odd function.
-\hypertarget{xp-over-q-where-p-q-in-mathbbz}{%
-\subsubsection{\texorpdfstring{\(x^{p \over q}\) where
-\(p, q \in \mathbb{Z}^+\)}{x\^{}\{p \textbackslash{}over q\} where p, q \textbackslash{}in \textbackslash{}mathbb\{Z\}\^{}+}}\label{xp-over-q-where-p-q-in-mathbbz}}
+\subsubsection{\(x^{p \over q}\) where \(p, q \in \mathbb{Z}^+\)}
\[x^{p \over q} = \sqrt[q]{x^p}\]
\(\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}\)
\end{itemize}
-\hypertarget{combinations-of-functions-piecewisehybrid}{%
-\subsection{Combinations of functions
-(piecewise/hybrid)}\label{combinations-of-functions-piecewisehybrid}}
+\subsection{Combinations of functions (piecewise/hybrid)}
\[\text{e.g.}\quad f(x)=\begin{cases} ^3 \sqrt{x}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\]
Open circle - point included\\
Closed circle - point not included
-\hypertarget{sum-difference-product-of-functions}{%
-\subsubsection{Sum, difference, product of
-functions}\label{sum-difference-product-of-functions}}
+\subsubsection{Sum, difference, product of functions}
\begin{longtable}[]{@{}lll@{}}
\toprule
product
\end{itemize}
-\hypertarget{matrix-transformations}{%
-\subsection{Matrix transformations}\label{matrix-transformations}}
+\subsection{Matrix transformations}
Find new point \((x^\prime, y^\prime)\). Substitute these into original
equation to find image with original variables \((x, y)\).
-\hypertarget{composite-functions}{%
-\subsection{Composite functions}\label{composite-functions}}
+\subsection{Composite functions}
\((f \circ g)(x)\) is defined iff
\(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)
\usepackage{graphicx}
\usepackage{wrapfig}
\usepackage{tikz}
+\usepackage{tkz-fct}
\usepackage{tikz-3dplot}
\usepackage{pgfplots}
\usetikzlibrary{calc}