[methods] collate notes for sac

diff --git a/methods/circ-functions.tex b/methods/circ-functions.tex
index 7870df145d038410cbca68c05664c4ab48de8347..a2d85bfbf0333cb37b647aa76c2b84b468a15261 100644 (file)
--- a/methods/circ-functions.tex
+++ b/methods/circ-functions.tex
@@ -1,197 +1,130 @@
-\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}

 
 \[y=a\tan(nx)\]
 
 
 \[y=a\tan(nx)\]
 

-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}
 
 
 \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}.}
 
 \begin{enumerate}
 \def\labelenumi{\arabic{enumi}.}


@@ -211,5 +144,3 @@ where
 \(2\theta={\pi\over 3}, {2\pi \over 3}, {7\pi \over 3}, {8\pi \over 3}\)
 
 \(\therefore \theta = {\pi \over 6}, {\pi \over 3}, {7 \pi \over 6}, {4\pi \over 3}\)
 \(2\theta={\pi\over 3}, {2\pi \over 3}, {7\pi \over 3}, {8\pi \over 3}\)
 
 \(\therefore \theta = {\pi \over 6}, {\pi \over 3}, {7 \pi \over 6}, {4\pi \over 3}\)

-\columnsend
-\end{document}

diff --git a/methods/methods-collated.pdf b/methods/methods-collated.pdf
new file mode 100644 (file)
index 0000000..808c3b5
Binary files /dev/null and b/methods/methods-collated.pdf differ

diff --git a/methods/methods-collated.poly.gnuplot b/methods/methods-collated.poly.gnuplot
new file mode 100644 (file)
index 0000000..1c600c1
--- /dev/null
+++ b/methods/methods-collated.poly.gnuplot
@@ -0,0 +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)) ;

diff --git a/methods/methods-collated.poly.table b/methods/methods-collated.poly.table
new file mode 100644 (file)
index 0000000..e981d3b
--- /dev/null
+++ b/methods/methods-collated.poly.table
@@ -0,0 +1,1005 @@
+
+# Curve 0 of 1, 1000 points
+# Curve title: "sgn(x)*(abs(x)**(1./3))"
+# x y type
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diff --git a/methods/methods-collated.tex b/methods/methods-collated.tex
new file mode 100644 (file)
index 0000000..b853626
--- /dev/null
+++ b/methods/methods-collated.tex
@@ -0,0 +1,304 @@
+\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}

diff --git a/methods/temp/calculus.tex b/methods/temp/calculus.tex
new file mode 100644 (file)
index 0000000..541a8c0
--- /dev/null
+++ b/methods/temp/calculus.tex
@@ -0,0 +1,169 @@
+\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}
+

diff --git a/methods/temp/inverse.tex b/methods/temp/inverse.tex
new file mode 100644 (file)
index 0000000..f02a4d1
--- /dev/null
+++ b/methods/temp/inverse.tex
@@ -0,0 +1,72 @@
+\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}

diff --git a/methods/temp/polynomials.tex b/methods/temp/polynomials.tex
new file mode 100644 (file)
index 0000000..a24f1c0
--- /dev/null
+++ b/methods/temp/polynomials.tex
@@ -0,0 +1,211 @@
+% 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}

diff --git a/methods/temp/stuff.tex b/methods/temp/stuff.tex
new file mode 100644 (file)
index 0000000..2964bea
--- /dev/null
+++ b/methods/temp/stuff.tex
@@ -0,0 +1,118 @@
+\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}

diff --git a/methods/temp/transformations.tex b/methods/temp/transformations.tex
new file mode 100644 (file)
index 0000000..b688c26
--- /dev/null
+++ b/methods/temp/transformations.tex
@@ -0,0 +1,177 @@
+\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
+\(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)

diff --git a/methods/transformations-ref.tex b/methods/transformations-ref.tex
index 22532c4e956edfc46899efffbe165cacd69aacba..1f0390aac1c92519c5f3a275479e249b855a96d7 100644 (file)
--- a/methods/transformations-ref.tex
+++ b/methods/transformations-ref.tex
@@ -1,7 +1,4 @@
-\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}
 \usepackage{lmodern}
 \usepackage{amssymb,amsmath}
 \usepackage{ifxetex,ifluatex}


@@ -31,8 +28,6 @@
   \KOMAoptions{parskip=half}}
 \makeatother
 \usepackage{xcolor}
   \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}
 \urlstyle{same}  % don't use monospace font for urls
 \usepackage{fullpage}
 \usepackage{longtable,booktabs}


@@ -54,14 +49,11 @@
 
 \begin{document}
 
 
 \begin{document}
 

-\hypertarget{transformation}{%
-\section{Transformation}\label{transformation}}
+\section{Transformations}

 
 \textbf{Order of operations:} DRT - Dilations, Reflections, Translations
 
 
 \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
 
 \begin{itemize}
 \tightlist


@@ -83,8 +75,7 @@
   when \(0 < |a| < 1\), graph becomes closer to axis
 \end{itemize}
 
   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:
 
 
 For \(y = f(x)\), these processes are equivalent:
 


@@ -94,12 +85,10 @@ 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
   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}
 
 \end{itemize}
 

-\hypertarget{dilations}{%
-\subsection{Dilations}\label{dilations}}
+\subsection{Dilations}

 
 For the graph of \(y = f(x)\), there are two pairs of equivalent
 processes:
 
 For the graph of \(y = f(x)\), there are two pairs of equivalent
 processes:


@@ -128,9 +117,7 @@ For graph of \(y={1 \over x}\), horizontal \& vertical dilations are
 equivalent (symmetrical). If \(y={a \over x}\), graph is contracted
 rather than dilated.
 
 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 \(y=Af[n(x+c)] + b\)
 
 Applies to exponential, log, trig, power, polynomial functions.\\
 Functions must be written in form \(y=Af[n(x+c)] + b\)


@@ -142,14 +129,12 @@ reflection across \(x\)-axis)\\
 \(c\) - translation from \(y\)-axis (\(x\)-shift)\\
 \(b\) - translation from \(x\)-axis (\(y\)-shift)
 
 \(c\) - 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 \(x=0\))
 
 
 \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 \(-f(x) = f(-x)\)
 
 Even when \(f(x) = -f(x)\)\\
 Odd when \(-f(x) = f(-x)\)


@@ -158,39 +143,19 @@ 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
 
 \(\implies f(x)=f(-x)\)\\
 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
 
 \begin{longtable}[]{@{}ll@{}}
 \toprule


@@ -203,9 +168,7 @@ Function \(x^{\pm {p \over q}}\) is odd if \(q\) is odd
 \bottomrule
 \end{longtable}
 
 \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.
 
 
 Mostly only on CAS.
 


@@ -216,9 +179,7 @@ Domain is:
 
 If \(n\) is odd, it is an odd function.
 
 
 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 \(p, q \in \mathbb{Z}^+\)}

 
 \[x^{p \over q} = \sqrt[q]{x^p}\]
 
 
 \[x^{p \over q} = \sqrt[q]{x^p}\]
 


@@ -235,18 +196,14 @@ If \(n\) is odd, it is an odd function.
   \(\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}\)
 \end{itemize}
 
   \(\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}\)
 \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
 
 
 \[\text{e.g.}\quad f(x)=\begin{cases} ^3 \sqrt{x}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\]
 
 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
 
 \begin{longtable}[]{@{}lll@{}}
 \toprule


@@ -273,14 +230,12 @@ Product functions:
   product
 \end{itemize}
 
   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 \((x, y)\).
 
 
 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
 \(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)
 
 \((f \circ g)(x)\) is defined iff
 \(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)

diff --git a/spec/spec-collated.tex b/spec/spec-collated.tex
index d75883f103baa7a1614cb5b81448bc2ec37ba7fa..f04ab637a29d3c44f0d6953c8452833b2f0ca23b 100644 (file)
--- a/spec/spec-collated.tex
+++ b/spec/spec-collated.tex
@@ -12,6 +12,7 @@
 \usepackage{graphicx}
 \usepackage{wrapfig}
 \usepackage{tikz}
 \usepackage{graphicx}
 \usepackage{wrapfig}
 \usepackage{tikz}

+\usepackage{tkz-fct}

 \usepackage{tikz-3dplot}
 \usepackage{pgfplots}
 \usetikzlibrary{calc}
 \usepackage{tikz-3dplot}
 \usepackage{pgfplots}
 \usetikzlibrary{calc}