From: Andrew Lorimer Date: Sun, 11 Aug 2019 12:34:19 +0000 (+1000) Subject: [methods] collate notes for sac X-Git-Tag: yr12~67 X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/9822645caf422e51979f03847cba135d266d7fe2?ds=sidebyside [methods] collate notes for sac --- diff --git a/methods/circ-functions.tex b/methods/circ-functions.tex index 7870df1..a2d85bf 100644 --- 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} \[y=a\tan(nx)\] -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}.} @@ -211,5 +144,3 @@ where \(2\theta={\pi\over 3}, {2\pi \over 3}, {7\pi \over 3}, {8\pi \over 3}\) \(\therefore \theta = {\pi \over 6}, {\pi \over 3}, {7 \pi \over 6}, {4\pi \over 3}\) -\columnsend -\end{document} diff --git a/methods/methods-collated.pdf b/methods/methods-collated.pdf new file mode 100644 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 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 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 +-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 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b/methods/methods-collated.tex new file mode 100644 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 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 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 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 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 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 22532c4..1f0390a 100644 --- 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} @@ -31,8 +28,6 @@ \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} @@ -54,14 +49,11 @@ \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 @@ -83,8 +75,7 @@ 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: @@ -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 - 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: @@ -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. -\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 \(y=Af[n(x+c)] + b\) @@ -142,14 +129,12 @@ reflection across \(x\)-axis)\\ \(c\) - 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 \(-f(x) = f(-x)\) @@ -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 -\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 @@ -203,9 +168,7 @@ Function \(x^{\pm {p \over q}}\) is odd if \(q\) is odd \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. @@ -216,9 +179,7 @@ Domain is: 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 \(p, q \in \mathbb{Z}^+\)} \[x^{p \over q} = \sqrt[q]{x^p}\] @@ -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} -\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 @@ -273,14 +230,12 @@ Product functions: 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 \(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\) diff --git a/spec/spec-collated.tex b/spec/spec-collated.tex index d75883f..f04ab63 100644 --- a/spec/spec-collated.tex +++ b/spec/spec-collated.tex @@ -12,6 +12,7 @@ \usepackage{graphicx} \usepackage{wrapfig} \usepackage{tikz} +\usepackage{tkz-fct} \usepackage{tikz-3dplot} \usepackage{pgfplots} \usetikzlibrary{calc}