From: Andrew Lorimer Date: Thu, 23 May 2019 12:16:34 +0000 (+1000) Subject: [spec] add complex numbers notes to SAC file X-Git-Tag: yr12~123 X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/2a6971693dc89bc3a6671a75b7d6939ab575e9ae [spec] add complex numbers notes to SAC file --- diff --git a/spec/spec-collated.pdf b/spec/spec-collated.pdf index 4eafe0c..01682ef 100644 Binary files a/spec/spec-collated.pdf and b/spec/spec-collated.pdf differ diff --git a/spec/spec-collated.tex b/spec/spec-collated.tex index 30f96df..df785b1 100644 --- a/spec/spec-collated.tex +++ b/spec/spec-collated.tex @@ -1,1325 +1,75 @@ -\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} -\else % if luatex or xelatex - \ifxetex - \usepackage{mathspec} - \else - \usepackage{fontspec} - \fi - \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 -}{} -\PassOptionsToPackage{hyphens}{url} % url is loaded by hyperref -\usepackage[unicode=true]{hyperref} -\hypersetup{ - pdfborder={0 0 0}, - breaklinks=true} -\urlstyle{same} % don't use monospace font for urls -\usepackage{longtable,booktabs} -% Fix footnotes in tables (requires footnote package) -\IfFileExists{footnote.sty}{\usepackage{footnote}\makesavenoteenv{long table}}{} -\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} -\IfFileExists{parskip.sty}{% -\usepackage{parskip} -}{% else -\setlength{\parindent}{0pt} -\setlength{\parskip}{6pt plus 2pt minus 1pt} -} -\setlength{\emergencystretch}{3em} % prevent overfull lines -\providecommand{\tightlist}{% - \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} -\setcounter{secnumdepth}{0} -% Redefines (sub)paragraphs to behave more like sections -\ifx\paragraph\undefined\else -\let\oldparagraph\paragraph -\renewcommand{\paragraph}[1]{\oldparagraph{#1}\mbox{}} -\fi -\ifx\subparagraph\undefined\else -\let\oldsubparagraph\subparagraph -\renewcommand{\subparagraph}[1]{\oldsubparagraph{#1}\mbox{}} -\fi - -% set default figure placement to htbp -\makeatletter -\def\fps@figure{htbp} -\makeatother - -\usepackage{harpoon}% -\pagenumbering{gobble} +\documentclass[a4paper]{article} +\usepackage[a4paper,margin=2cm]{geometry} +\usepackage{multicol} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{harpoon} +\usepackage{graphicx} +\usepackage{wrapfig} \usepackage{fancyhdr} - -\title{Year 12 Specialist} -\author{Andrew Lorimer} -\date{2019} - -\begin{document} - \pagestyle{fancy} \fancyhead[LO,LE]{Year 12 Specialist} -\fancyhead[CO,CE]{Andrew Lorimmer} -\maketitle - -\section{Complex \& Imaginary Numbers}\label{complex-imaginary-numbers} - -\subsection{Imaginary numbers}\label{imaginary-numbers} - -\[i^2 = -1 \quad \therefore i = \sqrt {-1}\] - -\subsubsection{Simplifying negative -surds}\label{simplifying-negative-surds} - -\begin{equation}\begin{split}\sqrt{-2} & = \sqrt{-1 \times 2} \\ & = \sqrt{2}i\end{split}\end{equation} - -\subsection{Complex numbers}\label{complex-numbers} - -\[\mathbb{C} = \{a+bi : a, b \in \mathbb{R} \}\] - -General form: \(z=a+bi\)\\ -\(\operatorname{Re}(z) = a, \quad \operatorname{Im}(z) = b\) - -\subsubsection{Addition}\label{addition} - -If \(z_1 = a+bi\) and \(z_2=c+di\), then - -\[z_1+z_2 = (a+c)+(b+d)i\] - -\subsubsection{Subtraction}\label{subtraction} - -If \(z_1=a+bi\) and \(z_2=c+di\), then - -\[z_1 - z_2=(a−c)+(b−d)i\] - -\subsubsection{Multiplication by a real -constant}\label{multiplication-by-a-real-constant} - -If \(z=a+bi\) and \(k \in \mathbb{R}\), then - -\[kz=ka+kbi\] - -\subsubsection{\texorpdfstring{Powers of -\(i\)}{Powers of i}}\label{powers-of-i} - -\begin{itemize} -\tightlist -\item - \(i^{4n} = 1\) -\item - \(i^{4n+1} = i\) -\item - \(i^{4n+2} = -1\) -\item - \(i^{4n+3} = -i\) -\end{itemize} - -For \(i^n\), find remainder \(r\) when \(n \div 4\). Then \(i^n = i^r\). - -\subsubsection{Multiplying complex -expressions}\label{multiplying-complex-expressions} - -If \(z_1 = a+bi\) and \(z_2=c+di\), then - -\[z_1 \times z_2 = (ac-bd)+(ad+bc)i\] - -\subsubsection{Conjugates}\label{conjugates} - -\[\overline{z} = a \mp bi\] - -\subparagraph{Properties}\label{properties} - -\begin{itemize} -\tightlist -\item - \(\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}\) -\item - \(\overline{z_1 z_2} = \overline{z_1} \cdot \overline{z_2}\) -\item - \(\overline{kz} = k \overline{z}, \text{ for } k \in \mathbb{R}\) -\item - \(z \overline{z} = = (a+bi)(a-bi) = a^2+b^2 = |z|^2\) -\item - \(z + \overline{z} = 2 \operatorname{Re}(z)\) -\end{itemize} - -\subsubsection{Modulus}\label{modulus} - -Distance from origin. - -\[|{z}|=\sqrt{a^2+b^2} \quad \therefore z \overline{z} = |z|^2\] - -Properties - -\begin{itemize} -\tightlist -\item - \(|z_1 z_2| = |z_1| |z_2|\) -\item - \(|{z_1 \over z_2}| = {|z_1| \over |z_2|}\) -\item - \(|z_1 + z_2| \le |z_1 + |z_2|\) -\end{itemize} - -\subsubsection{Multiplicative inverse}\label{multiplicative-inverse} - -\begin{equation}\begin{split}z^{-1} & = {1 \over z} \\ & = {{a-bi} \over {a^2+B^2}} \\ & = {\overline{z} \over {|z|^2}}\end{split}\end{equation} - -\subsubsection{Dividing complex numbers}\label{dividing-complex-numbers} - -\[{{z_1} \over {z_2}} = {{z_1\ {z_2}^{-1}}} = {{z_1 \overline{z_2}} \over {{|z_2|}^2}} \quad \text{(multiplicative inverse)}\] - -In practice, rationalise denominator: - -\[{z_1 \over z_2} = {{(a+bi)(c-di)} \over {c^2+d^2}}\] - -\subsection{Argand planes}\label{argand-planes} - -\begin{itemize} -\tightlist -\item - Geometric representation of \(\mathbb{C}\) -\item - horizontal \(= \operatorname{Re}(z)\); vertical - \(= \operatorname{Im}(z)\) -\item - Multiplication by \(i\) results in an anticlockwise rotation of - \(\pi \over 2\) -\end{itemize} - -\vfil \break - -\subsection{Complex polynomials}\label{complex-polynomials} - -\textbf{Include \(\pm\) for all solutions, including imaginary} - -\subsubsection{Sum of two squares -(quadratics)}\label{sum-of-two-squares-quadratics} - -\[z^2+a^2=z^2-(ai)^2=(z+ai)(z-ai)\] - -Complete the square to get to this point. - -\paragraph{Dividing complex -polynomials}\label{dividing-complex-polynomials} - -\(P(z) \div D(z)\) gives quotient \(Q(z)\) and remainder \(R(z)\): - -\[P(z) = D(z)Q(z) + R(z)\] - -\paragraph{Remainder theorem}\label{remainder-theorem} - -Let \(\alpha \in \mathbb{C}\). Remainder of \(P(z) \div (z - \alpha)\) -is \(P(\alpha)\) - -\paragraph{Factor theorem}\label{factor-theorem} - -If \(a+bi\) is a solution to \(P(z)=0\), then: - -\begin{itemize} -\tightlist -\item - \(P(a+bi)=0\) -\item - \(z-(a+bi)\) is a factor of \(P(z)\) -\end{itemize} - -\paragraph{Sum of two cubes}\label{sum-of-two-cubes} - -\[a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\] - -\subsection{Conjugate root theorem}\label{conjugate-root-theorem} - -If \(a+bi\) is a solution to \(P(z)=0\), then the conjugate -\(\overline{z}=a-bi\) is also a solution. - -\subsection{Polar form}\label{polar-form} - -\begin{equation}\begin{split}z & =r \operatorname{cis} \theta \\ & = r(\operatorname{cos}\theta+i \operatorname{sin}\theta) \\ & = a + bi \end{split}\end{equation} - -\begin{itemize} -\tightlist -\item - \(r=|z|=\sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}\) -\item - \(\theta=\operatorname{arg}(z)\) (on CAS: \texttt{arg(a+bi)}) -\item - \textbf{principal argument} is - \(\operatorname{Arg}(z) \in (-\pi, \pi]\) (note capital - \(\operatorname{Arg}\)) -\end{itemize} - -Each complex number has multiple polar representations:\\ -\(z=r \operatorname{cis} \theta = r \operatorname{cis} (\theta+2 n\pi\)) -with \(n \in \mathbb{Z}\) revolutions - -\subsubsection{Conjugate in polar form}\label{conjugate-in-polar-form} - -\[(r \operatorname{cis} \theta)^{-1} = r\operatorname{cis} (- \theta)\] - -Reflection of \(z\) across horizontal axis. - -\subsubsection{Multiplication and division in polar -form}\label{multiplication-and-division-in-polar-form} - -\[z_1z_2=r_1r_2\operatorname{cis}(\theta_1+\theta_2)\] - -\[{z_1 \over z_2} = {r_1 \over r_2} \operatorname{cis}(\theta_1-\theta_2)\] - -\subsection{de Moivres' Theorem}\label{de-moivres-theorem} - -\[(r\operatorname{cis}\theta)^n=r^n\operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}\] - -\subsection{Roots of complex numbers}\label{roots-of-complex-numbers} - -\(n\)th roots of \(z = r \operatorname{cis} \theta\) are - -\[z={r^{1 \over n}} \operatorname{cis}({{\theta + 2 k \pi} \over n})\] - -Same modulus for all solutions. Arguments are separated by -\({2 \pi} \over n\) - -The solutions of \(z^n=a \text{ where } a \in \mathbb{C}\) lie on circle - -\[x^2 + y^2 = (|a|^{1 \over n})^2\] - -\subsection{Sketching complex graphs}\label{sketching-complex-graphs} - -\subsubsection{Straight line}\label{straight-line} - -\begin{itemize} -\tightlist -\item - \(\operatorname{Re}(z) = c\) or \(\operatorname{Im}(z) = c\) - (perpendicular bisector) -\item - \(\operatorname{Arg}(z) = \theta\) -\item - \(|z+a|=|z+bi|\) where \(m={a \over b}\) -\item - \(|z+a|=|z+b| \longrightarrow 2(a-b)x=b^2-a^2\) -\end{itemize} - -\subsubsection{Circle}\label{circle} - -\(|z-z_1|^2 = c^2 |z_2+2|^2\) or \(|z-(a + bi)| = c\) - -\subsubsection{Locus}\label{locus} - -\(\operatorname{Arg}(z) < \theta\) - -\section{Vectors}\label{vectors} - -\begin{itemize} -\tightlist -\item - \textbf{vector:} a directed line segment\\ -\item - arrow indicates direction -\item - length indicates magnitude -\item - column notation: \(\begin{bmatrix} x \\ y \end{bmatrix}\) -\item - vectors with equal magnitude and direction are equivalent -\end{itemize} - -\begin{figure} -\centering -\includegraphics[width=0.20000\textwidth]{graphics/vectors-intro.png} -\caption{}\label{id} -\end{figure} - -\subsection{Vector addition}\label{vector-addition} - -\(\boldsymbol{u} + \boldsymbol{v}\) can be represented by drawing each -vector head to tail then joining the lines.\\ -Addition is commutative (parallelogram) - -\subsection{Scalar multiplication}\label{scalar-multiplication} - -For \(k \in \mathbb{R}^+\), \(k\boldsymbol{u}\) has the same direction -as \(\boldsymbol{u}\) but length is multiplied by a factor of \(k\). - -When multiplied by \(k < 0\), direction is reversed and length is -multplied by \(k\). - -\subsection{Vector subtraction}\label{vector-subtraction} - -To find \(\boldsymbol{u} - \boldsymbol{v}\), add \(\boldsymbol{-v}\) to -\(\boldsymbol{u}\) - -\subsection{Parallel vectors}\label{parallel-vectors} - -Same or opposite direction - -\[\boldsymbol{u} || \boldsymbol{v} \iff \boldsymbol{u} = k \boldsymbol{v} \text{ where } k \in \mathbb{R} \setminus \{0\}\] - -\subsection{Position vectors}\label{position-vectors} - -Vectors may describe a position relative to \(O\). - -For a point \(A\), the position vector is \overrightharp{OA} - -\vfill\eject - -\subsection{Linear combinations of non-parallel -vectors}\label{linear-combinations-of-non-parallel-vectors} - -If two non-zero vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are -not parallel, then: - -\[m\boldsymbol{a} + n\boldsymbol{b} = p \boldsymbol{a} + q \boldsymbol{b}\quad \therefore \quad m = p, \> n = q\] - -\includegraphics[width=0.20000\textwidth]{graphics/parallelogram-vectors.jpg} -\includegraphics[width=0.10000\textwidth]{graphics/vector-subtraction.jpg} - -\subsection{Column vector notation}\label{column-vector-notation} - -A vector between points \(A(x_1,y_1), \> B(x_2,y_2)\) can be represented -as \(\begin{bmatrix}x_2-x_1\\ y_2-y_1 \end{bmatrix}\) - -\subsection{Component notation}\label{component-notation} - -A vector \(\boldsymbol{u} = \begin{bmatrix}x\\ y \end{bmatrix}\) can be -written as \(\boldsymbol{u} = x\boldsymbol{i} + y\boldsymbol{j}\).\\ -\(\boldsymbol{u}\) is the sum of two components \(x\boldsymbol{i}\) and -\(y\boldsymbol{j}\)\\ -Magnitude of vector -\(\boldsymbol{u} = x\boldsymbol{i} + y\boldsymbol{j}\) is denoted by -\(|u|=\sqrt{x^2+y^2}\) - -Basic algebra applies:\\ -\((x\boldsymbol{i} + y\boldsymbol{j}) + (m\boldsymbol{i} + n\boldsymbol{j}) = (x + m)\boldsymbol{i} + (y+n)\boldsymbol{j}\)\\ -Two vectors equal if and only if their components are equal. - -\subsection{\texorpdfstring{Unit vector -\(|\hat{\boldsymbol{a}}|=1\)}{Unit vector \textbar{}\textbackslash{}hat\{\textbackslash{}boldsymbol\{a\}\}\textbar{}=1}}\label{unit-vector-hatboldsymbola1} - -\begin{equation}\begin{split}\hat{\boldsymbol{a}} & = {1 \over {|\boldsymbol{a}|}}\boldsymbol{a} \\ & = \boldsymbol{a} \cdot {|\boldsymbol{a}|}\end{split}\end{equation} - -\subsection{\texorpdfstring{Scalar/dot product -\(\boldsymbol{a} \cdot \boldsymbol{b}\)}{Scalar/dot product \textbackslash{}boldsymbol\{a\} \textbackslash{}cdot \textbackslash{}boldsymbol\{b\}}}\label{scalardot-product-boldsymbola-cdot-boldsymbolb} - -\[\boldsymbol{a} \cdot \boldsymbol{b} = a_1 b_1 + a_2 b_2\] - -\textbf{on CAS:} \texttt{dotP({[}a\ b\ c{]},\ {[}d\ e\ f{]})} - -\subsection{Scalar product properties}\label{scalar-product-properties} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \(k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k{b})\) -\item - \(\boldsymbol{a \cdot 0}=0\) -\item - \(\boldsymbol{a \cdot (b + c)}=\boldsymbol{a \cdot b + a \cdot c}\) -\item - \(\boldsymbol{i \cdot i} = \boldsymbol{j \cdot j} = \boldsymbol{k \cdot k}= 1\) -\item - If \(\boldsymbol{a} \cdot \boldsymbol{b} = 0\), \(\boldsymbol{a}\) and - \(\boldsymbol{b}\) are perpendicular -\item - \(\boldsymbol{a \cdot a} = |\boldsymbol{a}|^2 = a^2\) -\end{enumerate} - -For parallel vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\):\\ -\[\boldsymbol{a \cdot b}=\begin{cases} -|\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction}\\ --|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions} -\end{cases}\] - -\subsection{Geometric scalar products}\label{geometric-scalar-products} - -\[\boldsymbol{a} \cdot \boldsymbol{b} = |\boldsymbol{a}| |\boldsymbol{b}| \cos \theta\] - -where \(0 \le \theta \le \pi\) - -\subsection{Perpendicular vectors}\label{perpendicular-vectors} - -If \(\boldsymbol{a} \cdot \boldsymbol{b} = 0\), then -\(\boldsymbol{a} \perp \boldsymbol{b}\) (since \(\cos 90 = 0\)) - -\subsection{Finding angle between -vectors}\label{finding-angle-between-vectors} - -\textbf{positive direction} - -\[\cos \theta = {{\boldsymbol{a} \cdot \boldsymbol{b}} \over {|\boldsymbol{a}| |\boldsymbol{b}|}} = {{a_1 b_1 + a_2 b_2} \over {|\boldsymbol{a}| |\boldsymbol{b}|}}\] - -\textbf{on CAS:} \texttt{angle({[}a\ b\ c{]},\ {[}a\ b\ c{]})} (Action --\textgreater{} Vector -\textgreater{} Angle) - -\subsection{Angle between vector and -axis}\label{angle-between-vector-and-axis} - -Direction of a vector can be given by the angles it makes with -\(\boldsymbol{i}, \boldsymbol{j}, \boldsymbol{k}\) directions. - -For -\(\boldsymbol{a} = a_1 \boldsymbol{i} + a_2 \boldsymbol{j} + a_3 \boldsymbol{k}\) -which makes angles \(\alpha, \beta, \gamma\) with positive direction of -\(x, y, z\) axes: -\[\cos \alpha = {a_1 \over |\boldsymbol{a}|}, \quad \cos \beta = {a_2 \over |\boldsymbol{a}|}, \quad \cos \gamma = {a_3 \over |\boldsymbol{a}|}\] - -\textbf{on CAS:} \texttt{angle({[}a\ b\ c{]},\ {[}1\ 0\ 0{]})} for angle -between \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) and -\(x\)-axis - -\subsection{Vector projections}\label{vector-projections} - -Vector resolute of \(\boldsymbol{a}\) in direction of \(\boldsymbol{b}\) -is magnitude of \(\boldsymbol{a}\) in direction of \(\boldsymbol{b}\): - -\[\boldsymbol{u}={{\boldsymbol{a}\cdot\boldsymbol{b}}\over |\boldsymbol{b}|^2}\boldsymbol{b}=\left({\boldsymbol{a}\cdot{\boldsymbol{b} \over |\boldsymbol{b}|}}\right)\left({\boldsymbol{b} \over |\boldsymbol{b}|}\right)=(\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}\] - -\subsection{\texorpdfstring{Scalar resolute of \(\boldsymbol{a}\) on -\(\boldsymbol{b}\)}{Scalar resolute of \textbackslash{}boldsymbol\{a\} on \textbackslash{}boldsymbol\{b\}}}\label{scalar-resolute-of-boldsymbola-on-boldsymbolb} - -\[r_s = |\boldsymbol{u}| = \boldsymbol{a} \cdot \hat{\boldsymbol{b}}\] - -\subsection{\texorpdfstring{Vector resolute of -\(\boldsymbol{a} \perp \boldsymbol{b}\)}{Vector resolute of \textbackslash{}boldsymbol\{a\} \textbackslash{}perp \textbackslash{}boldsymbol\{b\}}}\label{vector-resolute-of-boldsymbola-perp-boldsymbolb} - -\[\boldsymbol{w} = \boldsymbol{a} - \boldsymbol{u} \> \text{ where } \boldsymbol{u} \text{ is projection } \boldsymbol{a} \text{ on } \boldsymbol{b}\] - -\subsection{Vector proofs}\label{vector-proofs} - -\subsubsection{Concurrent lines}\label{concurrent-lines} - -\(\ge\) 3 lines intersect at a single point - -\subsubsection{Collinear points}\label{collinear-points} - -\(\ge\) 3 points lie on the same line\\ -\(\implies \vec{OC} = \lambda \vec{OA} + \mu \vec{OB}\) where -\(\lambda + \mu = 1\). If \(C\) is between \(\vec{AB}\), then -\(0 < \mu < 1\)\\ -Points \(A, B, C\) are collinear iff -\(\vec{AC}=m\vec{AB} \text{ where } m \ne 0\) - -\subsubsection{Useful vector properties}\label{useful-vector-properties} - -\begin{itemize} -\tightlist -\item - If \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are parallel, then - \(\boldsymbol{b}=k\boldsymbol{a}\) for some - \(k \in \mathbb{R} \setminus \{0\}\) -\item - If \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are parallel with at - least one point in common, then they lie on the same straight line -\item - Two vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are - perpendicular if \(\boldsymbol{a} \cdot \boldsymbol{b}=0\) -\item - \(\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2\) -\end{itemize} - -\subsection{Linear dependence}\label{linear-dependence} - -Vectors \(\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\) are linearly -dependent if they are non-parallel and: - -\[k\boldsymbol{a}+l\boldsymbol{b}+m\boldsymbol{c} = 0\] -\[\therefore \boldsymbol{c} = m\boldsymbol{a} + n\boldsymbol{b} \quad \text{(simultaneous)}\] - -\(\boldsymbol{a}, \boldsymbol{b},\) and \(\boldsymbol{c}\) are linearly -independent if no vector in the set is expressible as a linear -combination of other vectors in set, or if they are parallel. - -Vector \(\boldsymbol{w}\) is a linear combination of vectors -\(\boldsymbol{v_1}, \boldsymbol{v_2}, \boldsymbol{v_3}\) - -\subsection{Three-dimensional vectors}\label{three-dimensional-vectors} - -Right-hand rule for axes: \(z\) is up or out of page. - -i\includegraphics{graphics/vectors-3d.png} - -\subsection{Parametric vectors}\label{parametric-vectors} - -Parametric equation of line through point \((x_0, y_0, z_0)\) and -parallel to \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) is: - -\begin{equation}\begin{cases}x = x_o + a \cdot t \\ y = y_0 + b \cdot t \\ z = z_0 + c \cdot t\end{cases}\end{equation} - -\section{Circular functions}\label{circular-functions} - -Period of \(a\sin(bx)\) is \({2\pi} \over b\) - -Period of \(a\tan(nx)\) is \(\pi \over n\)\\ -Asymptotes at \(x={2k+1)\pi \over 2n} \> \vert \> k \in \mathbb{Z}\) - -\subsection{Reciprocal functions}\label{reciprocal-functions} - -\subsubsection{Cosecant}\label{cosecant} - -\begin{figure} -\centering -\includegraphics{graphics/csc.png} -\caption{} -\end{figure} - -\[\operatorname{cosec} \theta = {1 \over \sin \theta} \> \vert \> \sin \theta \ne 0\] - -\begin{itemize} -\tightlist -\item - \textbf{Domain} \(= \mathbb{R} \setminus {n\pi : n \in \mathbb{Z}}\) -\item - \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\) -\item - \textbf{Turning points} at - \(\theta = {{(2n + 1)\pi} \over 2} \> \vert \> n \in \mathbb{Z}\) -\item - \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\) -\end{itemize} - -\subsubsection{Secant}\label{secant} - -\begin{figure} -\centering -\includegraphics{graphics/sec.png} -\caption{} -\end{figure} - -\[\operatorname{sec} \theta = {1 \over \cos \theta} \> \vert \> \cos \theta \ne 0\] - -\begin{itemize} -\tightlist -\item - \textbf{Domain} - \(= \mathbb{R} \setminus \{{{(2n + 1) \pi} \over 2 } : n \in \mathbb{Z}\}\) -\item - \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\) -\item - \textbf{Turning points} at - \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\) -\item - \textbf{Asymptotes} at - \(\theta = {{(2n + 1) \pi} \over 2} \> \vert \> n \in \mathbb{Z}\) -\end{itemize} - -\subsubsection{Cotangent}\label{cotangent} - -\begin{figure} -\centering -\includegraphics{graphics/cot.png} -\caption{} -\end{figure} - -\[\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0\] - -\begin{itemize} -\tightlist -\item - \textbf{Domain} \(= \mathbb{R} \setminus \{n \pi: n \in \mathbb{Z}\}\) -\item - \textbf{Range} \(= \mathbb{R}\) -\item - \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\) -\end{itemize} - -\subsubsection{Symmetry properties}\label{symmetry-properties} - -\begin{equation}\begin{split} - \operatorname{sec} (\pi \pm x) & = -\operatorname{sec} x \\ - \operatorname{sec} (-x) & = \operatorname{sec} x \\ - \operatorname{cosec} (\pi \pm x) & = \mp \operatorname{cosec} x \\ - \operatorname{cosec} (-x) & = - \operatorname{cosec} x \\ - \operatorname{cot} (\pi \pm x) & = \pm \operatorname{cot} x \\ - \operatorname{cot} (-x) & = - \operatorname{cot} x -\end{split}\end{equation} - -\subsubsection{Complementary properties}\label{complementary-properties} - -\begin{equation}\begin{split} - \operatorname{sec} \left({\pi \over 2} - x\right) & = \operatorname{cosec} x \\ - \operatorname{cosec} \left({\pi \over 2} - x\right) & = \operatorname{sec} x \\ - \operatorname{cot} \left({\pi \over 2} - x\right) & = \tan x \\ - \tan \left({\pi \over 2} - x\right) & = \operatorname{cot} x -\end{split}\end{equation} - -\subsubsection{Pythagorean identities}\label{pythagorean-identities} - -\begin{equation}\begin{split} - 1 + \operatorname{cot}^2 x & = \operatorname{cosec}^2 x, \quad \text{where } \sin x \ne 0 \\ - 1 + \tan^2 x & = \operatorname{sec}^2 x, \quad \text{where } \cos x \ne 0 -\end{split}\end{equation} - -\subsection{Compound angle formulas}\label{compound-angle-formulas} - -\[\cos(x \pm y) = \cos x + \cos y \mp \sin x \sin y\] -\[\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y\] -\[\tan(x \pm y) = {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}\] - -\subsection{Double angle formulas}\label{double-angle-formulas} - -\begin{equation}\begin{split} - \cos 2x &= \cos^2 x - \sin^2 x \\ - & = 1 - 2\sin^2 x \\ - & = 2 \cos^2 x -1 -\end{split}\end{equation} - -\[\sin 2x = 2 \sin x \cos x\] - -\[\tan 2x = {{2 \tan x} \over {1 - \tan^2 x}}\] - -\subsection{Inverse circular -functions}\label{inverse-circular-functions} - -Inverse functions: \(f(f^{-1}(x)) = x, \quad f(f^{-1}(x)) = x\)\\ -Must be 1:1 to find inverse (reflection in \(y=x\) - -Domain is restricted to make functions 1:1. - -\subsubsection{\texorpdfstring{\(\arcsin\)}{\textbackslash{}arcsin}}\label{arcsin} - -\[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y, \quad \text{where } \sin y = x \text{ and } y \in [{-\pi \over 2}, {\pi \over 2}]\] - -\subsubsection{\texorpdfstring{\(\arcos\)}{\textbackslash{}arcos}}\label{arcos} - -\[\cos^{-1} \rightarrow \mathbb{R}, \quad \cos^{-1} x = y, \quad \text{where } \cos y = x \text{ and } y \in [0, \pi]\] - -\subsubsection{\texorpdfstring{\(\arctan\)}{\textbackslash{}arctan}}\label{arctan} - -\[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y, \quad \text{where } \tan y = x \text{ and } y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\] -\# Differential calculus - -\subsection{Limits}\label{limits} - -\[\lim_{x \rightarrow a}f(x)\] - -\(L^-\) - limit from below - -\(L^+\) - limit from above - -\(\lim_{x \to a} f(x)\) - limit of a point - -\begin{itemize} -\tightlist -\item - Limit exists if \(L^-=L^+\) -\item - If limit exists, point does not. -\end{itemize} - -Limits can be solved using normal techniques (if div 0, factorise) - -\subsection{Limit theorems}\label{limit-theorems} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\) -\item - \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\) -\item - \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\) -\item - \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\) -\end{enumerate} - -Corollary: \(\lim_{x \rightarrow a} c \times f(x)=cF\) where \(c=\) -constant - -\subsection{\texorpdfstring{Solving limits for -\(x\rightarrow\infty\)}{Solving limits for x\textbackslash{}rightarrow\textbackslash{}infty}}\label{solving-limits-for-xrightarrowinfty} - -Factorise so that all values of \(x\) are in denominators. - -e.g. - -\[\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2\] - -\subsection{Continuous functions}\label{continuous-functions} - -A function is continuous if \(L^-=L^+=f(x)\) for all values of \(x\). - -\subsection{Gradients of secants and -tangents}\label{gradients-of-secants-and-tangents} - -Secant (chord) - line joining two points on curve - -Tangent - line that intersects curve at one point - -given \(P(x,y) \quad Q(x+\delta x, y + \delta y)\): gradient of chord -joining \(P\) and \(Q\) is -\({m_{PQ}}={\operatorname{rise} \over \operatorname{run}} = {\delta y \over \delta x}\) - -As \(Q \rightarrow P, \delta x \rightarrow 0\). Chord becomes tangent -(two infinitesimal points are equal). - -Can also be used with functions, where \(h=\delta x\). - -\subsection{First principles -derivative}\label{first-principles-derivative} - -\[f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={dy \over dx}\] - -\[m_{\tan}=\lim_{h \rightarrow 0}f^\prime(x)\] - -\[m_{\vec{PQ}}=f^\prime(x)\] - -first principles derivative: -\[{m_{\text{tangent at }P} =\lim_{h \rightarrow 0}}{{f(x+h)-f(x)}\over h}\] - -\subsection{Gradient at a point}\label{gradient-at-a-point} - -Given point \(P(a, b)\) and function \(f(x)\), the gradient is -\(f^\prime(a)\) - -\subsection{\texorpdfstring{Derivatives of -\(x^n\)}{Derivatives of x\^{}n}}\label{derivatives-of-xn} - -\[{d(ax^n) \over dx}=anx^{n-1}\] - -If \(x=\) constant, derivative is \(0\) - -If \(y=ax^n\), derivative is \(a\times nx^{n-1}\) - -If -\(f(x)={1 \over x}=x^{-1}, \quad f^\prime(x)=-1x^{-2}={-1 \over x^2}\) - -If -\(f(x)=^5\sqrt{x}=x^{1 \over 5}, \quad f^\prime(x)={1 \over 5}x^{-4/5}={1 \over 5 \times ^5\sqrt{x^4}}\) - -If \(f(x)=(x-b)^2, \quad f^\prime(x)=2(x-b)\) - -\[f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}\] - -\subsection{\texorpdfstring{Derivatives of -\(u \pm v\)}{Derivatives of u \textbackslash{}pm v}}\label{derivatives-of-u-pm-v} - -\[{dy \over dx}={du \over dx} \pm {dv \over dx}\] where \(u\) and \(v\) -are functions of \(x\) - -\subsection{Euler's number as a limit}\label{eulers-number-as-a-limit} - -\[\lim_{h \rightarrow 0} {{e^h-1} \over h}=1\] - -\subsection{\texorpdfstring{Chain rule for -\((f\circ g)\)}{Chain rule for (f\textbackslash{}circ g)}}\label{chain-rule-for-fcirc-g} - -If \(f(x) = h(g(x)) = (h \circ g)(x)\): - -\[f^\prime(x) = h^\prime(g(x)) \cdot g^\prime(x)\] - -If \(y=h(u)\) and \(u=g(x)\): - -\[{dy \over dx} = {dy \over du} \cdot {du \over dx}\] -\[{d((ax+b)^n) \over dx} = {d(ax+b) \over dx} \cdot n \cdot (ax+b)^{n-1}\] - -Used with only one expression. - -e.g. \(y=(x^2+5)^7\) - Cannot reasonably expand\\ -Let \(u-x^2+5\) (inner expression)\\ -\({du \over dx} = 2x\)\\ -\(y=u^7\)\\ -\({dy \over du} = 7u^6\) - -\subsection{\texorpdfstring{Product rule for -\(y=uv\)}{Product rule for y=uv}}\label{product-rule-for-yuv} - -\[{dy \over dx} = u{dv \over dx} + v{du \over dx}\] - -\subsection{\texorpdfstring{Quotient rule for -\(y={u \over v}\)}{Quotient rule for y=\{u \textbackslash{}over v\}}}\label{quotient-rule-for-yu-over-v} - -\[{dy \over dx} = {{v{du \over dx} - u{dv \over dx}} \over v^2}\] - -\[f^\prime(x)={{v(x)u^\prime(x)-u(x)v^\prime(x)} \over [v(x)]^2}\] - -\subsection{Logarithms}\label{logarithms} - -\[\log_b (x) = n \quad \operatorname{where} \hspace{0.5em} b^n=x\] - -Wikipedia: - -\begin{quote} -the logarithm of a given number \(x\) is the exponent to which another -fixed number, the base \(b\), must be raised, to produce that number -\(x\) -\end{quote} - -\subsubsection{Logarithmic identities}\label{logarithmic-identities} - -\(\log_b (xy)=\log_b x + \log_b y\)\\ -\(\log_b x^n = n \log_b x\)\\ -\(\log_b y^{x^n} = x^n \log_b y\) - -\subsubsection{Index identities}\label{index-identities} - -\(b^{m+n}=b^m \cdot b^n\)\\ -\((b^m)^n=b^{m \cdot n}\)\\ -\((b \cdot c)^n = b^n \cdot c^n\)\\ -\({a^m \div a^n} = {a^{m-n}}\) - -\subsubsection{\texorpdfstring{\(e\) as a -logarithm}{e as a logarithm}}\label{e-as-a-logarithm} - -\[\operatorname{if} y=e^x, \quad \operatorname{then} x=\log_e y\] -\[\ln x = \log_e x\] - -\subsubsection{Differentiating -logarithms}\label{differentiating-logarithms} - -\[{d(\log_e x)\over dx} = x^{-1} = {1 \over x}\] - -\subsection{Derivative rules}\label{derivative-rules} - -\begin{longtable}[]{@{}ll@{}} -\toprule -\(f(x)\) & \(f^\prime(x)\)\tabularnewline -\midrule -\endhead -\(\sin x\) & \(\cos x\)\tabularnewline -\(\sin ax\) & \(a\cos ax\)\tabularnewline -\(\cos x\) & \(-\sin x\)\tabularnewline -\(\cos ax\) & \(-a \sin ax\)\tabularnewline -\(\tan f(x)\) & \(f^2(x) \sec^2f(x)\)\tabularnewline -\(e^x\) & \(e^x\)\tabularnewline -\(e^{ax}\) & \(ae^{ax}\)\tabularnewline -\(ax^{nx}\) & \(an \cdot e^{nx}\)\tabularnewline -\(\log_e x\) & \(1 \over x\)\tabularnewline -\(\log_e {ax}\) & \(1 \over x\)\tabularnewline -\(\log_e f(x)\) & \(f^\prime (x) \over f(x)\)\tabularnewline -\(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\tabularnewline -\(\sin^{-1} x\) & \(1 \over {\sqrt{1-x^2}}\)\tabularnewline -\(\cos^{-1} x\) & \(-1 \over {sqrt{1-x^2}}\)\tabularnewline -\(\tan^{-1} x\) & \(1 \over {1 + x^2}\)\tabularnewline -\bottomrule -\end{longtable} - -\subsection{Reciprocal derivatives}\label{reciprocal-derivatives} - -\[{1 \over {dy \over dx}} = {dx \over dy}\] - -\subsection{\texorpdfstring{Differentiating -\(x=f(y)\)}{Differentiating x=f(y)}}\label{differentiating-xfy} - -Find \(dx \over dy\). Then -\({dx \over dy} = {1 \over {dy \over dx}} \implies {dy \over dx} = {1 \over {dx \over dy}}\). - -\[{dy \over dx} = {1 \over {dx \over dy}}\] - -\subsection{Second derivative}\label{second-derivative} - -\[f(x) \longrightarrow f^\prime (x) \longrightarrow f^{\prime\prime}(x)\] - -\[\therefore y \longrightarrow {dy \over dx} \longrightarrow {d({dy \over dx}) \over dx} \longrightarrow {d^2 y \over dx^2}\] - -Order of polynomial \(n\)th derivative decrements each time the -derivative is taken - -\subsubsection{Points of Inflection}\label{points-of-inflection} - -\emph{Stationary point} - point of zero gradient (i.e. -\(f^\prime(x)=0\))\\ -\emph{Point of inflection} - point of maximum \(|\)gradient\(|\) (i.e. -\(f^{\prime\prime} = 0\)) - -\begin{itemize} -\tightlist -\item - if \(f^\prime (a) = 0\) and \(f^{\prime\prime}(a) > 0\), then point - \((a, f(a))\) is a local min (curve is concave up) -\item - if \(f^\prime (a) = 0\) and \(f^{\prime\prime} (a) < 0\), then point - \((a, f(a))\) is local max (curve is concave down) -\item - if \(f^{\prime\prime}(a) = 0\), then point \((a, f(a))\) is a point of - inflection -\item - if also \(f^\prime(a)=0\), then it is a stationary point of inflection -\end{itemize} - -\begin{figure} -\centering -\includegraphics{graphics/second-derivatives.png} -\caption{} -\end{figure} - -\subsection{Implicit Differentiation}\label{implicit-differentiation} - -\textbf{On CAS:} Action \(\rightarrow\) Calculation \(\rightarrow\) -\texttt{impDiff(y\^{}2+ax=5,\ x,\ y)}. Returns \(y^\prime= \dots\). - -Used for differentiating circles etc. - -If \(p\) and \(q\) are expressions in \(x\) and \(y\) such that \(p=q\), -for all \(x\) nd \(y\), then: - -\[{dp \over dx} = {dq \over dx} \quad \text{and} \quad {dp \over dy} = {dq \over dy}\] - -\subsection{Integration}\label{integration} - -\[\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)\] - -\[\int x^n \cdot dx = {x^{n+1} \over n+1} + c\] - -\begin{itemize} -\tightlist -\item - area enclosed by curves -\item - \(+c\) should be shown on each step without \(\int\) -\end{itemize} - -\subsubsection{Integral laws}\label{integral-laws} - -\(\int f(x) + g(x) dx = \int f(x) dx + \int g(x) dx\)\\ -\(\int k f(x) dx = k \int f(x) dx\) - -\begin{longtable}[]{@{}ll@{}} -\toprule -\begin{minipage}[b]{0.42\columnwidth}\raggedright\strut -\(f(x)\)\strut -\end{minipage} & \begin{minipage}[b]{0.38\columnwidth}\raggedright\strut -\(\int f(x) \cdot dx\)\strut -\end{minipage}\tabularnewline -\midrule -\endhead -\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut -\(k\) (constant)\strut -\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut -\(kx + c\)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut -\(x^n\)\strut -\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut -\({x^{n+1} \over {n+1}} + c\)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut -\(a x^{-n}\)\strut -\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut -\(a \cdot \log_e x + c\)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut -\({1 \over {ax+b}}\)\strut -\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut -\({1 \over a} \log_e (ax+b) + c\)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut -\((ax+b)^n\)\strut -\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut -\({1 \over {a(n+1)}}(ax+b)^{n-1} + c\)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut -\(e^{kx}\)\strut -\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut -\({1 \over k} e^{kx} + c\)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut -\(e^k\)\strut -\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut -\(e^kx + c\)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut -\(\sin kx\)\strut -\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut -\(-{1 \over k} \cos (kx) + c\)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut -\(\cos kx\)\strut -\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut -\({1 \over k} \sin (kx) + c\)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut -\(\sec^2 kx\)\strut -\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut -\({1 \over k} \tan(kx) + c\)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut -\(1 \over \sqrt{a^2-x^2}\)\strut -\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut -\(\sin^{-1} {x \over a} + c \>\vert\> a>0\)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut -\(-1 \over \sqrt{a^2-x^2}\)\strut -\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut -\(\cos^{-1} {x \over a} + c \>\vert\> a>0\)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut -\(a \over {a^2-x^2}\)\strut -\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut -\(\tan^{-1} {x \over a} + c\)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut -\({f^\prime (x)} \over {f(x)}\)\strut -\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut -\(\log_e f(x) + c\)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut -\(g^\prime(x)\cdot f^\prime(g(x)\)\strut -\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut -\(f(g(x))\) (chain rule)\strut -\end{minipage}\tabularnewline -\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut -\(f(x) \cdot g(x)\)\strut -\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut -\(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\strut -\end{minipage}\tabularnewline -\bottomrule -\end{longtable} - -Note \(\sin^{-1} {x \over a} + \cos^{-1} {x \over a}\) is constant for -all \(x \in (-a, a)\). - -\subsubsection{Definite integrals}\label{definite-integrals} - -\[\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)\] - -\begin{itemize} -\tightlist -\item - Signed area enclosed by: - \(\> y=f(x), \quad y=0, \quad x=a, \quad x=b\). -\item - \emph{Integrand} is \(f\). -\item - \(F(x)\) may be any integral, i.e. \(c\) is inconsequential -\end{itemize} - -\paragraph{Properties}\label{properties-2} - -\[\int^b_a f(x) \> dx = \int^c_a f(x) \> dx + \int^b_c f(x) \> dx\] - -\[\int^a_a f(x) \> dx = 0\] - -\[\int^b_a k \cdot f(x) \> dx = k \int^b_a f(x) \> dx\] - -\[\int^b_a f(x) \pm g(x) \> dx = \int^b_a f(x) \> dx \pm \int^b_a g(x) \> dx\] - -\[\int^b_a f(x) \> dx = - \int^a_b f(x) \> dx\] - -\subsubsection{Integration by -substitution}\label{integration-by-substitution} - -\[\int f(u) {du \over dx} \cdot dx = \int f(u) \cdot du\] - -Note \(f(u)\) must be one-to-one \(\implies\) one \(x\) value for each -\(y\) value - -e.g.~for \(y=\int(2x+1)\sqrt{x+4} \cdot dx\):\\ -let \(u=x+4\)\\ -\(\implies {du \over dx} = 1\)\\ -\(\implies x = u - 4\)\\ -then \(y=\int (2(u-4)+1)u^{1 \over 2} \cdot du\)\\ -Solve as a normal integral - -\paragraph{Definite integrals by -substitution}\label{definite-integrals-by-substitution} - -For \(\int^b_a f(x) {du \over dx} \cdot dx\), evaluate new \(a\) and -\(b\) for \(f(u) \cdot du\). - -\subsubsection{Trigonometric -integration}\label{trigonometric-integration} - -\[\sin^m x \cos^n x \cdot dx\] - -\textbf{\(m\) is odd:}\\ -\(m=2k+1\) where \(k \in \mathbb{Z}\)\\ -\(\implies \sin^{2k+1} x = (\sin^2 z)^k \sin x = (1 - \cos^2 x)^k \sin x\)\\ -Substitute \(u=\cos x\) - -\textbf{\(n\) is odd:}\\ -\(n=2k+1\) where \(k \in \mathbb{Z}\)\\ -\(\implies \cos^{2k+1} x = (\cos^2 x)^k \cos x = (1-\sin^2 x)^k \cos x\)\\ -Subbstitute \(u=\sin x\) - -\textbf{\(m\) and \(n\) are even:}\\ -Use identities: - -\begin{itemize} -\tightlist -\item - \(\sin^2x={1 \over 2}(1-\cos 2x)\) -\item - \(\cos^2x={1 \over 2}(1+\cos 2x)\) -\item - \(\sin 2x = 2 \sin x \cos x\) -\end{itemize} - -\subsection{Partial fractions}\label{partial-fractions} - -On CAS: Action \(\rightarrow\) Transformation \(\rightarrow\) -\texttt{expand/combine}\\ -or Interactive \(\rightarrow\) Transformation \(\rightarrow\) -\texttt{expand} \(\rightarrow\) Partial - -\subsection{Graphing integrals on CAS}\label{graphing-integrals-on-cas} - -In main: Interactive \(\rightarrow\) Calculation \(\rightarrow\) -\(\int\) (\(\rightarrow\) Definite)\\ -Restrictions: \texttt{Define\ f(x)=...} \(\rightarrow\) -\texttt{f(x)\textbar{}x\textgreater{}1} (e.g.) - -\subsection{Applications of -antidifferentiation}\label{applications-of-antidifferentiation} - -\begin{itemize} -\tightlist -\item - \(x\)-intercepts of \(y=f(x)\) identify \(x\)-coordinates of - stationary points on \(y=F(x)\) -\item - nature of stationary points is determined by sign of \(y=f(x)\) on - either side of its \(x\)-intercepts -\item - if \(f(x)\) is a polynomial of degree \(n\), then \(F(x)\) has degree - \(n+1\) -\end{itemize} - -To find stationary points of a function, substitute \(x\) value of given -point into derivative. Solve for \({dy \over dx}=0\). Integrate to find -original function. - -\subsection{Solids of revolution}\label{solids-of-revolution} - -Approximate as sum of infinitesimally-thick cylinders - -\subsubsection{\texorpdfstring{Rotation about -\(x\)-axis}{Rotation about x-axis}}\label{rotation-about-x-axis} - -\begin{align*} - V &= \int^{x=b}_{x-a} \pi y^2 \> dx \\ - &= \pi \int^b_a (f(x))^2 \> dx -\end{align*} - -\subsubsection{\texorpdfstring{Rotation about -\(y\)-axis}{Rotation about y-axis}}\label{rotation-about-y-axis} - -\begin{align*} - V &= \int^{y=b}_{y=a} \pi x^2 \> dy \\ - &= \pi \int^b_a (f(y))^2 \> dy -\end{align*} - -\subsubsection{\texorpdfstring{Regions not bound by -\(y=0\)}{Regions not bound by y=0}}\label{regions-not-bound-by-y0} - -\[V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx\]\\ -where \(f(x) > g(x)\) - -\subsection{Length of a curve}\label{length-of-a-curve} - -\[L = \int^b_a \sqrt{1 + ({dy \over dx})^2} \> dx \quad \text{(Cartesian)}\] - -\[L = \int^b_a \sqrt{{dx \over dt} + ({dy \over dt})^2} \> dt \quad \text{(parametric)}\] - -Evaluate on CAS. Or use Interactive \(\rightarrow\) Calculation -\(\rightarrow\) Line \(\rightarrow\) \texttt{arcLen}. - -\subsection{Rates}\label{rates} - -\subsubsection{Related rates}\label{related-rates} - -\[{da \over db} \quad \text{(change in } a \text{ with respect to } b)\] - -\subsubsection{Gradient at a point on parametric -curve}\label{gradient-at-a-point-on-parametric-curve} - -\[{dy \over dx} = {{dy \over dt} \div {dx \over dt}} \> \vert \> {dx \over dt} \ne 0\] - -\[{d^2 \over dx^2} = {d(y^\prime) \over dx} = {{dy^\prime \over dt} \div {dx \over dt}} \> \vert \> y^\prime = {dy \over dx}\] - -\subsection{Rational functions}\label{rational-functions} - -\[f(x) = {P(x) \over Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}\] - -\subsubsection{Addition of ordinates}\label{addition-of-ordinates} - -\begin{itemize} -\tightlist -\item - when two graphs have the same ordinate, \(y\)-coordinate is double the - ordinate -\item - when two graphs have opposite ordinates, \(y\)-coordinate is 0 i.e. - (\(x\)-intercept) -\item - when one of the ordinates is 0, the resulting ordinate is equal to the - other ordinate -\end{itemize} - -\subsection{Fundamental theorem of -calculus}\label{fundamental-theorem-of-calculus} - -If \(f\) is continuous on \([a, b]\), then - -\[\int^b_a f(x) \> dx = F(b) - F(a)\] - -where \(F\) is any antiderivative of \(f\) - -\subsection{Differential equations}\label{differential-equations} - -One or more derivatives +\fancyhead[CO,CE]{Andrew Lorimer} +\begin{document} -\textbf{Order} - highest power inside derivative\\ -\textbf{Degree} - highest power of highest derivative\\ -e.g. \({\left(dy^2 \over d^2 x\right)}^3\): order 2, degree 3 +\begin{multicols}{2} -\subsubsection{Verifying solutions}\label{verifying-solutions} + \section{Complex numbers} -Start with \(y=\dots\), and differentiate. Substitute into original -equation. + \[\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}\] -\subsubsection{Function of the dependent -variable}\label{function-of-the-dependent-variable} + \subsection*{Operations} -If \({dy \over dx}=g(y)\), then -\({dx \over dy} = 1 \div {dy \over dx} = {1 \over g(y)}\). Integrate -both sides to solve equation. Only add \(c\) on one side. Express -\(e^c\) as \(A\). + \begin{align*} + z_1 \pm z_2&=(a \pm c)(b \pm d)i\\ + k \times z &= ka + kbi\\ + z_1 \cdot z_2 &= ac-bd+(ad+bc)i\\ + z_1 \div z_2 &= (z_1 \overline{z_2}) \div |z_2|^2 + \end{align*} -\subsubsection{Mixing problems}\label{mixing-problems} + \subsection*{Conjugate} -\[\left({dm \over dt}\right)_\Sigma = \left({dm \over dt}\right)_{\text{in}} - \left({dm \over dt}\right)_{\text{out}}\] + \[\overline{z} = a \pm bi\] -\subsubsection{Separation of variables}\label{separation-of-variables} + \subsubsection*{Properties} -If \({dy \over dx}=f(x)g(y)\), then: + \begin{align*} + \overline{z_1 \pm z_2} &= \overline{z_1}\pm\overline{z_2}\\ + \overline{z_1 \cdot z_2} &= \overline{z_1}\cdot\overline{z_2}\\ + \overline{kz} &= k\overline{z} \quad | \quad k \in \mathbb{R}\\ + z\overline{z} &= (a+bi)(a-bi)\\ + &= a^2 + b^2\\ + &= |z|^2 + \end{align*} -\[\int f(x) \> dx = \int {1 \over g(y)} \> dy\] + \subsection*{Modulus} -\subsubsection{Using definite integrals to solve -DEs}\label{using-definite-integrals-to-solve-des} + \[|z|=|\vec{Oz}|=\sqrt{a^2 + b^2}\] -Used for situations where solutions to \({dy \over dx} = f(x)\) is not -required. + \subsubsection*{Properties} -In some cases, it may not be possible to obtain an exact solution. + \begin{align*} + |z_1z_2|&=|z_1||z_2|\\ + \left|\frac{z_1}{z_2}\right|&=\frac{|z_1|}{|z_2|}\\ + |z_1+z_2|&\le|z_1|+|z_2| + \end{align*} -Approximate solutions can be found by numerically evaluating a definite -integral. + \subsection*{Multiplicative inverse} -\subsubsection{Using Euler's method to solve a differential -equation}\label{using-eulers-method-to-solve-a-differential-equation} + \begin{align*} + z^{-1}&=\frac{a-bi}{a^2+b^2}\\ + &=\frac{\overline{z}}{|z|^2} + a + \end{align*} -\[{{f(x+h) - f(x)} \over h } \approx f^\prime (x) \quad \text{for small } h\] + \subsection*{Dividing over \(\mathbb{C}\)} -\[\implies f(x+h) \approx f(x) + hf^\prime(x)\] + \begin{align*} + \frac{z_1}{z_2}&=z_1z_2^{-1}\\ + &=\frac{z_1\overline{z_2}}{|z_2|^2}\\ + &=\frac{(a+bi)(c-di)}{c^2+d^2}\\ + & \qquad \text{(rationalise denominator)} + \end{align*} +\end{multicols} \end{document}