b669b55bad715280a6045bc24a11d05ec2486039
   1\documentclass[a4paper]{article}
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   4\usepackage{multirow}
   5\usepackage{amsmath}
   6\usepackage{amssymb}
   7\usepackage{harpoon}
   8\usepackage{tabularx}
   9\usepackage[dvipsnames, table]{xcolor}
  10\usepackage{blindtext}
  11\usepackage{graphicx}
  12\usepackage{wrapfig}
  13\usepackage{tikz}
  14\usepackage{tikz-3dplot}
  15\usepackage{pgfplots}
  16\usetikzlibrary{calc}
  17\usetikzlibrary{angles}
  18\usetikzlibrary{datavisualization.formats.functions}
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  21\usepackage{fancyhdr}
  22\pagestyle{fancy}
  23\fancyhead[LO,LE]{Year 12 Specialist}
  24\fancyhead[CO,CE]{Andrew Lorimer}
  25
  26\usepackage{mathtools}
  27\usepackage{xcolor} % used only to show the phantomed stuff
  28\renewcommand\hphantom[1]{{\color[gray]{.6}#1}} % comment out!
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  34\definecolor{cas}{HTML}{e6f0fe}
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  36\newcommand{\midarrow}{\tikz \draw[-triangle 90] (0,0) -- +(.1,0);}
  37
  38\begin{document}
  39
  40\begin{multicols}{2}
  41
  42  \section{Complex numbers}
  43
  44    \[\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}\]
  45
  46    \begin{align*}
  47      \text{Cartesian form: } & a+bi\\
  48      \text{Polar form: } & r\operatorname{cis}\theta
  49    \end{align*}
  50
  51    \subsection*{Operations}
  52
  53\definecolor{shade1}{HTML}{ffffff}
  54\definecolor{shade2}{HTML}{e6f2ff}
  55  \definecolor{shade3}{HTML}{cce2ff}
  56      \begin{tabularx}{\columnwidth}{r|X|X}
  57        & \textbf{Cartesian} & \textbf{Polar} \\
  58        \hline
  59        \(z_1 \pm z_2\) & \((a \pm c)(b \pm d)i\) & convert to \(a+bi\)\\
  60        \hline
  61        \(+k \times z\) & \multirow{2}{*}{\(ka \pm kbi\)} & \(kr\operatorname{cis} \theta\)\\
  62        \cline{1-1}\cline{3-3}
  63        \(-k \times z\) & & \(kr \operatorname{cis}(\theta\pm \pi)\)\\
  64        \hline
  65        \(z_1 \cdot z_2\) & \(ac-bd+(ad+bc)i\) & \(r_1r_2 \operatorname{cis}(\theta_1 + \theta_2)\)\\
  66        \hline
  67        \(z_1 \div z_2\) & \((z_1 \overline{z_2}) \div |z_2|^2\) & \(\left(\frac{r_1}{r_2}\right) \operatorname{cis}(\theta_1 - \theta_2)\)
  68      \end{tabularx}
  69
  70      \subsubsection*{Scalar multiplication in polar form}
  71      
  72        For \(k \in \mathbb{R}^+\):
  73        \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\theta\]
  74
  75        \noindent For \(k \in \mathbb{R}^-\):
  76        \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\left(\begin{cases}\theta - \pi & |0<\operatorname{Arg}(z)\le \pi \\ \theta + \pi & |-\pi<\operatorname{Arg}(z)\le 0\end{cases}\right)\]
  77
  78    \subsection*{Conjugate}
  79
  80      \begin{align*}
  81        \overline{z} &= a \mp bi\\
  82        &= r \operatorname{cis}(-\theta)
  83      \end{align*}
  84
  85      \noindent \colorbox{cas}{On CAS: \texttt{conjg(a+bi)}}
  86
  87      \subsubsection*{Properties}
  88
  89        \begin{align*}
  90          \overline{z_1 \pm z_2} &= \overline{z_1}\pm\overline{z_2}\\
  91          \overline{z_1 \cdot z_2} &= \overline{z_1}\cdot\overline{z_2}\\
  92          \overline{kz} &= k\overline{z} \quad | \quad k \in \mathbb{R}\\
  93          z\overline{z} &= (a+bi)(a-bi)\\
  94          &= a^2 + b^2\\
  95          &= |z|^2
  96        \end{align*}
  97
  98    \subsection*{Modulus}
  99
 100      \[|z|=|\vec{Oz}|=\sqrt{a^2 + b^2}\]
 101
 102      \subsubsection*{Properties}
 103
 104        \begin{align*}
 105          |z_1z_2|&=|z_1||z_2|\\
 106          \left|\frac{z_1}{z_2}\right|&=\frac{|z_1|}{|z_2|}\\
 107          |z_1+z_2|&\le|z_1|+|z_2|
 108        \end{align*}
 109
 110    \subsection*{Multiplicative inverse}
 111
 112      \begin{align*}
 113        z^{-1}&=\frac{a-bi}{a^2+b^2}\\
 114        &=\frac{\overline{z}}{|z|^2}a\\
 115        &=r \operatorname{cis}(-\theta)
 116      \end{align*}
 117
 118    \subsection*{Dividing over \(\mathbb{C}\)}
 119
 120      \begin{align*}
 121        \frac{z_1}{z_2}&=z_1z_2^{-1}\\
 122        &=\frac{z_1\overline{z_2}}{|z_2|^2}\\
 123        &=\frac{(a+bi)(c-di)}{c^2+d^2}\\
 124        & \qquad \text{(rationalise denominator)}
 125      \end{align*}
 126
 127    \subsection*{Polar form}
 128
 129      \begin{align*}
 130        z&=r\operatorname{cis}\theta\\
 131        &=r(\cos \theta + i \sin \theta)
 132      \end{align*}
 133
 134      \begin{itemize}
 135        \item{\(r=|z|=\sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}\)}
 136        \item{\(\theta = \operatorname{arg}(z)\) \quad \colorbox{cas}{On CAS: \texttt{arg(a+bi)}}}
 137        \item{\(\operatorname{Arg}(z) \in (-\pi,\pi)\) \quad \bf{(principal argument)}}
 138        \item{\colorbox{cas}{Convert on CAS:}\\ \verb|compToTrig(a+bi)| \(\iff\) \verb|cExpand{r·cisX}|}
 139        \item{Multiple representations:\\\(r\operatorname{cis}\theta=r\operatorname{cis}(\theta+2n\pi)\) with \(n \in \mathbb{Z}\) revolutions}
 140        \item{\(\operatorname{cis}\pi=-1,\qquad \operatorname{cis}0=1\)}
 141      \end{itemize}
 142
 143    \subsection*{de Moivres' theorem}
 144
 145    \[(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}\]
 146
 147    \subsection*{Complex polynomials}
 148    
 149      Include \(\pm\) for all solutions, incl. imaginary
 150
 151      \begin{tabularx}{\columnwidth}{ R{0.55} X  }
 152        \hline
 153        Sum of squares & \(\begin{aligned} 
 154        z^2 + a^2 &= z^2-(ai)^2\\
 155        &= (z+ai)(z-ai) \end{aligned}\) \\
 156        \hline
 157        Sum of cubes & \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)\\
 158        \hline
 159        Division & \(P(z)=D(z)Q(z)+R(z)\) \\
 160        \hline
 161        Remainder theorem & Let \(\alpha \in \mathbb{C}\). Remainder of \(P(z) \div (z-\alpha)\) is \(P(\alpha)\)\\
 162        \hline
 163        Factor theorem & \(z-\alpha\) is a factor of \(P(z) \iff P(\alpha)=0\) for \(\alpha \in \mathbb{C}\)\\
 164        \hline
 165        Conjugate root theorem & \(P(z)=0 \text{ at } z=a\pm bi\) (\(\implies\) both \(z_1\) and \(\overline{z_1}\) are solutions)
 166      \end{tabularx}
 167
 168    \subsection*{Roots}
 169
 170      \(n\)th roots of \(z=r\operatorname{cis}\theta\) are:
 171
 172      \[z = r^{\frac{1}{n}} \operatorname{cis}\left(\frac{\theta+2k\pi}{n}\right)\]
 173
 174      \begin{itemize}
 175
 176        \item{Same modulus for all solutions}
 177        \item{Arguments are separated by \(\frac{2\pi}{n}\)}
 178        \item{Solutions of \(z^n=a\) where \(a \in \mathbb{C}\) lie on the circle \(x^2+y^2=\left(|a|^{\frac{1}{n}}\right)^2\) \quad (intervals of \(\frac{2\pi}{n}\))}
 179      \end{itemize}
 180
 181      \noindent For \(0=az^2+bz+c\), use quadratic formula:
 182
 183      \[z=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]
 184
 185    \subsection*{Fundamental theorem of algebra}
 186
 187      A polynomial of degree \(n\) can be factorised into \(n\) linear factors in \(\mathbb{C}\):
 188
 189        \[\implies P(z)=a_n(z-\alpha_1)(z-\alpha_2)(z-\alpha_3)\dots(z-\alpha_n)\]
 190        \[\text{ where } \alpha_1,\alpha_2,\alpha_3,\dots,\alpha_n \in \mathbb{C}\]
 191
 192    \subsection*{Argand planes}
 193    
 194      \begin{center}\begin{tikzpicture}[scale=2]
 195        \draw [->] (-0.2,0) -- (1.5,0) node [right]  {$\operatorname{Re}(z)$};
 196        \draw [->] (0,-0.2) -- (0,1.5) node [above] {$\operatorname{Im}(z)$};
 197        \coordinate (P) at (1,1);
 198        \coordinate (a) at (1,0);
 199        \coordinate (b) at (0,1);
 200        \coordinate (O) at (0,0);
 201        \draw (0,0) -- (P) node[pos=0.5, above left]{\(r\)} node[pos=1, right]{\(\begin{aligned}z&=a+bi\\&=r\operatorname{cis}\theta\end{aligned}\)};
 202        \draw [gray, dashed] (1,1) -- (1,0) node[black, pos=1, below]{\(a\)};
 203        \draw [gray, dashed] (1,1) -- (0,1) node[black, pos=1, left]{\(b\)};
 204        \begin{scope}
 205          \path[clip] (O) -- (P) -- (a);
 206          \fill[red, opacity=0.5, draw=black] (O) circle (2mm);
 207          \node at ($(O)+(20:3mm)$) {$\theta$};
 208        \end{scope}
 209        \filldraw (P) circle (0.5pt);
 210      \end{tikzpicture}\end{center}
 211
 212      \begin{itemize}
 213        \item{Multiplication by \(i \implies\) CCW rotation of \(\frac{\pi}{2}\)}
 214        \item{Addition: \(z_1 + z_2 \equiv\) \overrightharp{\(Oz_1\)} + \overrightharp{\(Oz_2\)}}
 215      \end{itemize}
 216
 217    \subsection*{Sketching complex graphs}
 218      
 219      \subsubsection*{Linear}
 220
 221        \begin{itemize}
 222          \item{\(\operatorname{Re}(z)=c\) or \(\operatorname{Im}(z)=c\) (perpendicular bisector)}
 223          \item{\(\operatorname{Im}(z)=m\operatorname{Re}(z)\)}
 224          \item{\(|z+a|=|z+b| \implies 2(a-b)x=b^2-a^2\)\\Geometric: equidistant from \(a,b\)}
 225        \end{itemize}
 226
 227      \subsubsection*{Circles}
 228
 229        \begin{itemize}
 230          \item \(|z-z_1|^2=c^2|z_2+2|^2\)
 231          \item \(|z-(a+bi)|=c \implies (x-a)^2+_(y-b)^2=c^2\)
 232        \end{itemize}
 233
 234      \noindent \textbf{Loci} \qquad \(\operatorname{Arg}(z)<\theta\)
 235
 236        \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
 237          \draw [->] (0,0) -- (1,0) node [right]  {$\operatorname{Re}(z)$};
 238          \draw [->] (0,-0.5) -- (0,1) node [above] {$\operatorname{Im}(z)$};
 239          \draw [<-, dashed, thick, blue] (-1,0) -- (0,0);
 240          \draw [->, thick, blue] (0,0) -- (1,1);
 241          \fill [gray, opacity=0.2, domain=-1:1, variable=\x] (-1,-0.5) -- (-1,0) -- (0, 0) -- (1,1) -- (1,-0.5) -- cycle;
 242          \begin{scope}
 243            \path[clip] (0,0) -- (1,1) -- (1,0);
 244            \fill[red, opacity=0.5, draw=black] (0,0) circle (2mm);
 245            \node at ($(0,0)+(20:3mm)$) {$\frac{\pi}{4}$};
 246          \end{scope}
 247          \node [font=\footnotesize] at (0.5,-0.25) {\(\operatorname{Arg}(z)\le\frac{\pi}{4}\)};
 248          \node [blue, mydot] {};
 249        \end{tikzpicture}\end{center}
 250
 251      \noindent \textbf{Rays} \qquad \(\operatorname{Arg}(z-b)=\theta\)
 252
 253        \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
 254          \draw [->] (-0.75,0) -- (1.5,0) node [right]  {$\operatorname{Re}(z)$};
 255          \draw [->] (0,-1) -- (0,1) node [above] {$\operatorname{Im}(z)$};
 256          \draw [->, thick, brown] (-0.25,0) -- (-0.75,-1);
 257          \node [above, font=\footnotesize] at (-0.25,0) {\(\frac{1}{4}\)};
 258          \begin{scope}
 259            \path[clip] (-0.25,0) -- (-0.75,-1) -- (0,0);
 260            \fill[orange, opacity=0.5, draw=black] (-0.25,0) circle (2mm);
 261          \end{scope}
 262          \node at (-0.08,-0.3) {\(\frac{\pi}{8}\)};
 263          \node [font=\footnotesize, left] at (-0.75,-1) {\(\operatorname{Arg}(z+\frac{1}{4})=\frac{\pi}{8}\)};
 264          \node [brown, mydot] at (-0.25,0) {};
 265          \draw [<->, thick, green] (0,-1) -- (1.5,0.5) node [pos=0.25, black, font=\footnotesize, right] {\(|z-2|=|z-(1+i)|\)};
 266          \node [left, font=\footnotesize] at (0,-1) {\(-1\)};
 267          \node [below, font=\footnotesize] at (1,0) {\(1\)};
 268        \end{tikzpicture}\end{center}
 269
 270    \section{Vectors}
 271\begin{center}\begin{tikzpicture}
 272  \draw [->] (-0.5,0) -- (3,0) node [right]  {\(x\)};
 273          \draw [->] (0,-0.5) -- (0,3) node [above] {\(y\)};
 274          \draw [orange, ->, thick] (0.5,0.5) -- (2.5,2.5) node [pos=0.5, above] {\(\vec{u}\)};
 275         \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
 276        \draw [gray, dashed, thick] (0.5,0.5) -- (2.5,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, below]{\(x\vec{i}\)};
 277        \draw [gray, dashed, thick] (2.5,0.5) -- (2.5,2.5) node [pos=0.5] {\midarrow};
 278         \end{scope}
 279        \node[black, right] at (2.5,1.5) {\(y\vec{j}\)};
 280\end{tikzpicture}\end{center}
 281\subsection*{Column notation}
 282
 283\[\begin{bmatrix}x\\ y \end{bmatrix} \iff x\boldsymbol{i} + y\boldsymbol{j}\]
 284\(\begin{bmatrix}x_2-x_1\\ y_2-y_1 \end{bmatrix}\) \quad between \(A(x_1,y_1), \> B(x_2,y_2)\)
 285
 286\subsection*{Scalar multiplication}
 287
 288\[k\cdot (x\boldsymbol{i}+y\boldsymbol{j})=kx\boldsymbol{i}+ky\boldsymbol{j}\]
 289
 290\noindent For \(k \in \mathbb{R}^-\), direction is reversed
 291
 292\subsection*{Vector addition}
 293\begin{center}\begin{tikzpicture}[scale=1]
 294          \coordinate (A) at (0,0);
 295          \coordinate (B) at (2,2);
 296          \draw [->, thick, red] (0,0) -- (2,2) node [pos=0.5, below right] {\(\vec{u}=2\vec{i}+2\vec{j}\)};
 297          \draw [->, thick, blue] (2,2) -- (1,4) node [pos=0.5, above right] {\(\vec{v}=-\vec{i}+2\vec{j}\)};
 298          \draw [->, thick, orange] (0,0) -- (1,4) node [pos=0.5, left] {\(\vec{u}+\vec{v}=\vec{i}+4\vec{j}\)};
 299\end{tikzpicture}\end{center}
 300
 301\[(x\boldsymbol{i}+y\boldsymbol{j}) \pm (a\boldsymbol{i}+b\boldsymbol{j})=(x \pm a)\boldsymbol{i}+(y \pm b)\boldsymbol{j}\]
 302
 303\begin{itemize}
 304  \item Draw each vector head to tail then join lines
 305  \item Addition is commutative (parallelogram)
 306  \item \(\boldsymbol{u}-\boldsymbol{v}=\boldsymbol{u}+(-\boldsymbol{v}) \implies \overrightharp{AB}=\boldsymbol{b}-\boldsymbol{a}\)
 307\end{itemize}
 308
 309\subsection*{Magnitude}
 310
 311\[|(x\boldsymbol{i} + y\boldsymbol{j})|=\sqrt{x^2+y^2}\]
 312
 313\subsection*{Parallel vectors}
 314
 315\[\boldsymbol{u} || \boldsymbol{v} \iff \boldsymbol{u} = k \boldsymbol{v} \text{ where } k \in \mathbb{R} \setminus \{0\}\]
 316
 317For parallel vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\):\\
 318\[\boldsymbol{a \cdot b}=\begin{cases}
 319|\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction}\\
 320-|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions}
 321\end{cases}\]
 322%\includegraphics[width=0.2,height=\textheight]{graphics/parallelogram-vectors.jpg}
 323%\includegraphics[width=1]{graphics/vector-subtraction.jpg}
 324
 325\subsection*{Perpendicular vectors}
 326
 327\[\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b} = 0\ \quad \text{(since \(\cos 90 = 0\))}\]
 328
 329\subsection*{Unit vector \(|\hat{\boldsymbol{a}}|=1\)}
 330\[\begin{split}\hat{\boldsymbol{a}} & = {\frac{1}{|\boldsymbol{a}|}}\boldsymbol{a} \\ & = \boldsymbol{a} \cdot {|\boldsymbol{a}|}\end{split}\]
 331
 332  \subsection*{Scalar product \(\boldsymbol{a} \cdot \boldsymbol{b}\)}
 333
 334
 335\begin{center}\begin{tikzpicture}[scale=2]
 336  \draw [->] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{b}\)};
 337  \draw [->] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{a}\)};
 338          \begin{scope}
 339            \path[clip] (1,0.5) -- (1,0) -- (0,0);
 340            \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
 341            \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
 342          \end{scope}
 343\end{tikzpicture}\end{center}
 344\begin{align*}\boldsymbol{a} \cdot \boldsymbol{b} &= a_1 b_1 + a_2 b_2 \\  &= |\boldsymbol{a}| |\boldsymbol{b}| \cos \theta \\ &\quad (\> 0 \le \theta \le \pi) \text{ - from cosine rule}\end{align*}
 345\noindent\colorbox{cas}{On CAS: \texttt{dotP({[}a\ b\ c{]},\ {[}d\ e\ f{]})}}
 346
 347\subsubsection*{Properties}
 348
 349\begin{enumerate}
 350\item
 351  \(k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k\boldsymbol{b})\)
 352\item
 353  \(\boldsymbol{a \cdot 0}=0\)
 354\item
 355  \(\boldsymbol{a} \cdot (\boldsymbol{b} + \boldsymbol{c})=\boldsymbol{a} \cdot \boldsymbol{b} + \boldsymbol{a} \cdot \boldsymbol{c}\)
 356\item
 357  \(\boldsymbol{i \cdot i} = \boldsymbol{j \cdot j} = \boldsymbol{k \cdot k}= 1\)
 358\item
 359  \(\boldsymbol{a} \cdot \boldsymbol{b} = 0 \quad \implies \quad \boldsymbol{a} \perp \boldsymbol{b}\)
 360\item
 361  \(\boldsymbol{a \cdot a} = |\boldsymbol{a}|^2 = a^2\)
 362\end{enumerate}
 363
 364\subsection*{Angle between vectors}
 365
 366\[\cos \theta = \frac{\boldsymbol{a} \cdot \boldsymbol{b}}{|\boldsymbol{a}| |\boldsymbol{b}|} = \frac{a_1 b_1 + a_2 b_2}{|\boldsymbol{a}| |\boldsymbol{b}|}\]
 367
 368\noindent \colorbox{cas}{On CAS:} \texttt{angle([a b c], [a b c])}
 369
 370(Action \(\rightarrow\) Vector \(\rightarrow\)Angle)
 371
 372\subsection*{Angle between vector and axis}
 373
 374\noindent For\(\boldsymbol{a} = a_1 \boldsymbol{i} + a_2 \boldsymbol{j} + a_3 \boldsymbol{k}\)
 375which makes angles \(\alpha, \beta, \gamma\) with positive side of
 376\(x, y, z\) axes:
 377\[\cos \alpha = \frac{a_1}{|\boldsymbol{a}|}, \quad \cos \beta = \frac{a_2}{|\boldsymbol{a}|}, \quad \cos \gamma = \frac{a_3}{|\boldsymbol{a}|}\]
 378
 379\noindent \colorbox{cas}{On CAS:} \texttt{angle({[}a\ b\ c{]},\ {[}1\ 0\ 0{]})}\\for angle
 380between \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) and
 381\(x\)-axis
 382
 383\subsection*{Projections \& resolutes}
 384
 385\begin{tikzpicture}[scale=3]
 386  \draw [->, purple] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{a}\)};
 387  \draw [->, orange] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{u}\)};
 388  \draw [->, blue] (1,0) -- (2,0) node [pos=0.5, below] {\(\boldsymbol{b}\)};
 389          \begin{scope}
 390            \path[clip] (1,0.5) -- (1,0) -- (0,0);
 391            \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
 392            \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
 393          \end{scope}
 394         \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
 395        \draw [gray, dashed, thick] (1,0) -- (1,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, right, rotate=-90]{\(\boldsymbol{w}\)};
 396          \end{scope}
 397\draw (0,0) coordinate (O)
 398  (1,0) coordinate (A)
 399  (1,0.5) coordinate (B)
 400  pic [draw,red,angle radius=2mm] {right angle = O--A--B};
 401\end{tikzpicture}
 402
 403\subsubsection*{\(\parallel\boldsymbol{b}\) (vector projection/resolute)}
 404
 405\begin{align*}
 406  \boldsymbol{u} & = \frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|^2}\boldsymbol{b} \\
 407  & = \left(\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\right)\left(\frac{\boldsymbol{b}}{|\boldsymbol{b}|}\right) \\
 408  & = (\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}
 409\end{align*}
 410
 411\subsubsection*{\(\perp\boldsymbol{b}\) (perpendicular projection)}
 412\[\boldsymbol{w} = \boldsymbol{a} - \boldsymbol{u}\]
 413
 414\subsubsection*{\(|\boldsymbol{u}|\) (scalar resolute)}
 415\begin{align*}
 416  r_s &= |\boldsymbol{u}|\\
 417  &= \boldsymbol{a} \cdot \hat{\boldsymbol{b}}\\
 418  &=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}
 419\end{align*}
 420
 421\subsubsection*{Rectangular (\(\parallel,\perp\)) components}
 422
 423\[\boldsymbol{a}=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}+\left(\boldsymbol{a}-\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}\right)\]
 424
 425
 426\subsection*{Vector proofs}
 427
 428\textbf{Concurrent:} intersection of \(\ge\) 3 lines
 429
 430\begin{tikzpicture}
 431  \draw [blue] (0,0) -- (1,1);
 432  \draw [red] (1,0) -- (0,1);
 433  \draw [brown] (0.4,0) -- (0.6,1);
 434        \filldraw (0.5,0.5) circle (2pt);
 435\end{tikzpicture}
 436
 437\subsubsection*{Collinear points}
 438
 439\(\ge\) 3 points lie on the same line
 440
 441\begin{tikzpicture}
 442  \draw [purple] (0,0) -- (4,1);
 443  \filldraw (2,0.5) circle (2pt) node [above] {\(C\)};
 444  \filldraw (1,0.25) circle (2pt) node [above] {\(A\)};
 445  \filldraw (3,0.75) circle (2pt) node [above] {\(B\)};
 446  \coordinate (O) at (2.8,-0.2);
 447  \node at (O) [below] {\(O\)}; 
 448         \begin{scope}[->, orange, thick] 
 449           \draw (O) -- (2,0.5) node [pos=0.5, above, font=\footnotesize, black] {\(\boldsymbol{c}\)};
 450           \draw (O) -- (1,0.25) node [pos=0.5, below, font=\footnotesize, black] {\(\boldsymbol{a}\)};
 451           \draw (O) -- (3,0.75) node [pos=0.5, right, font=\footnotesize, black] {\(\boldsymbol{b}\)};
 452         \end{scope}
 453\end{tikzpicture}
 454
 455\begin{align*}
 456  \text{e.g. Prove that}\\
 457  \overrightharp{AC}=m\overrightharp{AB} \iff \boldsymbol{c}&=(1-m)\boldsymbol{a}+m\boldsymbol{b}\\
 458  \implies \boldsymbol{c} &= \overrightharp{OA} + \overrightharp{AC}\\
 459  &= \overrightharp{OA} + m\overrightharp{AB}\\
 460  &=\boldsymbol{a}+m(\boldsymbol{b}-\boldsymbol{a})\\
 461  &=\boldsymbol{a}+m\boldsymbol{b}-m\boldsymbol{a}\\
 462  &=(1-m)\boldsymbol{a}+m{b}
 463\end{align*}
 464\begin{align*}
 465  \text{Also, } \implies \overrightharp{OC} &= \lambda \vec{OA} + \mu \overrightharp{OB} \\
 466  \text{where } \lambda + \mu &= 1\\
 467  \text{If } C \text{ lies along } \overrightharp{AB}, & \implies 0 < \mu < 1
 468\end{align*}
 469
 470
 471\subsubsection*{Parallelograms}
 472
 473\begin{center}\begin{tikzpicture}
 474  \coordinate (O) at (0,0) node [below left] {\(O\)};
 475  \coordinate (A) at (4,0);
 476  \coordinate (B) at (6,2);
 477  \coordinate (C) at (2,2);
 478  \coordinate (D) at (6,0);
 479
 480    \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (O)--(A) node [below left] {\(A\)};
 481    \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (A)--(B) node [above right] {\(B\)};
 482    \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (B)--(C) node [above left] {\(C\)};
 483    \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (C)--(O);
 484
 485    \draw [gray, dashed] (O) -- (B) node [pos=0.75] {\(\diagdown\diagdown\)} node [pos=0.25] {\(\diagdown\diagdown\)};
 486    \draw [gray, dashed] (A) -- (C) node [pos=0.75] {\(\diagup\)} node [pos=0.25] {\(\diagup\)};
 487          \begin{scope}
 488            \path[clip] (C) -- (A) -- (O);
 489            \fill[orange, opacity=0.5, draw=black] (0,0) circle (4mm);
 490            \node at ($(0,0)+(20:8mm)$) {\(\theta\)};
 491          \end{scope}
 492          \draw [gray, thick, dotted] (B) -- (D) node [pos=0.5, right, black, font=\footnotesize] {\(|\boldsymbol{c}|\sin\theta\)} (A) -- (D) node [pos=0.5, below, black, font=\footnotesize] {\(|\boldsymbol{c}|\cos\theta\)};
 493  \draw pic [draw,thick,red,angle radius=2mm] {right angle=O--D--B};
 494\end{tikzpicture}\end{center}
 495
 496\begin{itemize}
 497  \item
 498    Diagonals \(\overrightharp{OB}, \overrightharp{AC}\) bisect each other
 499  \item
 500    If diagonals are equal length, it is a rectangle
 501  \item
 502    \(|\overrightharp{OB}|^2 + |\overrightharp{CA}|^2 = |\overrightharp{OA}|^2 + |\overrightharp{AB}|^2 + |\overrightharp{CB}|^2 + |\overrightharp{OC}|^2\)
 503  \item
 504    Area \(=\boldsymbol{c} \cdot \boldsymbol{a}\)
 505\end{itemize}
 506
 507  \subsubsection*{Useful vector properties}
 508
 509\begin{itemize}
 510\item
 511  \(\boldsymbol{a} \parallel \boldsymbol{b} \implies \boldsymbol{b}=k\boldsymbol{a}\) for some
 512  \(k \in \mathbb{R} \setminus \{0\}\)
 513\item
 514  If \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are parallel with at
 515  least one point in common, then they lie on the same straight line
 516\item
 517  \(\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b}=0\)
 518\item
 519  \(\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2\)
 520\end{itemize}
 521
 522\subsection*{Linear dependence}
 523
 524\(\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\) are linearly dependent if they are \(\nparallel\) and:
 525\begin{align*}
 526  0&=k\boldsymbol{a}+l\boldsymbol{b}+m\boldsymbol{c}\\
 527  \therefore \boldsymbol{c} &= m\boldsymbol{a} + n\boldsymbol{b} \quad \text{(simultaneous)}
 528\end{align*}
 529
 530\noindent \(\boldsymbol{a}, \boldsymbol{b},\) and \(\boldsymbol{c}\) are linearly
 531independent if no vector in the set is expressible as a linear
 532combination of other vectors in set, or if they are parallel.
 533
 534\subsection*{Three-dimensional vectors}
 535
 536Right-hand rule for axes: \(z\) is up or out of page.
 537
 538\tdplotsetmaincoords{60}{120} 
 539\begin{center}\begin{tikzpicture} [scale=3, tdplot_main_coords, axis/.style={->,thick}, 
 540vector/.style={-stealth,red,very thick}, 
 541vector guide/.style={dashed,gray,thick}]
 542
 543%standard tikz coordinate definition using x, y, z coords
 544\coordinate (O) at (0,0,0);
 545
 546%tikz-3dplot coordinate definition using x, y, z coords
 547
 548\pgfmathsetmacro{\ax}{1}
 549\pgfmathsetmacro{\ay}{1}
 550\pgfmathsetmacro{\az}{1}
 551
 552\coordinate (P) at (\ax,\ay,\az);
 553
 554%draw axes
 555\draw[axis] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$};
 556\draw[axis] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$};
 557\draw[axis] (0,0,0) -- (0,0,1) node[anchor=south]{$z$};
 558
 559%draw a vector from O to P
 560\draw[vector] (O) -- (P);
 561
 562%draw guide lines to components
 563\draw[vector guide]         (O) -- (\ax,\ay,0);
 564\draw[vector guide] (\ax,\ay,0) -- (P);
 565\draw[vector guide]         (P) -- (0,0,\az);
 566\draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
 567\draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
 568\draw[vector guide] (\ax,\ay,0) -- (\ax,0,0);
 569\node[tdplot_main_coords,above right]
 570at (\ax,\ay,\az){(\ax, \ay, \az)};
 571\end{tikzpicture}\end{center}
 572
 573\subsection*{Parametric vectors}
 574
 575Parametric equation of line through point \((x_0, y_0, z_0)\) and
 576parallel to \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) is:
 577
 578\[\begin{cases}x = x_o + a \cdot t \\ y = y_0 + b \cdot t \\ z = z_0 + c \cdot t\end{cases}\]
 579
 580\section{Circular functions}
 581
 582\(\sin(bx)\) or \(\cos(bx)\): period \(=\frac{2\pi}{b}\)
 583
 584\noindent \(\tan(nx)\): period \(=\frac{\pi}{n}\)\\
 585\indent\indent\indent asymptotes at \(x=\frac{(2k+1)\pi}{2n} \> \vert \> k \in \mathbb{Z}\)
 586
 587\subsection*{Reciprocal functions}
 588
 589\subsubsection*{Cosecant}
 590
 591\[\operatorname{cosec} \theta = \frac{1}{\sin \theta} \> \vert \> \sin \theta \ne 0\]
 592
 593\begin{itemize}
 594\item
 595  \textbf{Domain} \(= \mathbb{R} \setminus {n\pi : n \in \mathbb{Z}}\)
 596\item
 597  \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
 598\item
 599  \textbf{Turning points} at
 600    \(\theta = \frac{(2n + 1)\pi}{2} \> \vert \> n \in \mathbb{Z}\)
 601\item
 602  \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
 603\end{itemize}
 604
 605\subsubsection*{Secant}
 606
 607
 608\begin{center}\includegraphics[width=0.7\columnwidth]{graphics/sec.png}\end{center}
 609
 610\[\operatorname{sec} \theta = \frac{1}{\cos \theta} \> \vert \> \cos \theta \ne 0\]
 611
 612\begin{itemize}
 613
 614\item
 615  \textbf{Domain}
 616    \(= \mathbb{R} \setminus \frac{(2n + 1) \pi}{2} : n \in \mathbb{Z}\}\)
 617\item
 618  \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
 619\item
 620  \textbf{Turning points} at
 621  \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
 622\item
 623  \textbf{Asymptotes} at
 624    \(\theta = \frac{(2n + 1) \pi}{2} \> \vert \> n \in \mathbb{Z}\)
 625\end{itemize}
 626
 627\subsubsection*{Cotangent}
 628
 629\begin{center}\includegraphics[width=0.7\columnwidth]{graphics/cot.png}\end{center}
 630
 631\[\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0\]
 632
 633\begin{itemize}
 634
 635\item
 636  \textbf{Domain} \(= \mathbb{R} \setminus \{n \pi: n \in \mathbb{Z}\}\)
 637\item
 638  \textbf{Range} \(= \mathbb{R}\)
 639\item
 640  \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
 641\end{itemize}
 642
 643\subsubsection*{Symmetry properties}
 644
 645\[\begin{split}
 646  \operatorname{sec} (\pi \pm x) & = -\operatorname{sec} x \\
 647  \operatorname{sec} (-x) & = \operatorname{sec} x \\
 648  \operatorname{cosec} (\pi \pm x) & = \mp \operatorname{cosec} x \\
 649  \operatorname{cosec} (-x) & = - \operatorname{cosec} x \\
 650  \operatorname{cot} (\pi \pm x) & = \pm \operatorname{cot} x \\
 651  \operatorname{cot} (-x) & = - \operatorname{cot} x
 652\end{split}\]
 653
 654\subsubsection*{Complementary properties}
 655
 656\[\begin{split}
 657  \operatorname{sec} \left({\pi \over 2} - x\right) & = \operatorname{cosec} x \\
 658  \operatorname{cosec} \left({\pi \over 2} - x\right) & = \operatorname{sec} x \\
 659  \operatorname{cot} \left({\pi \over 2} - x\right) & = \tan x \\
 660  \tan \left({\pi \over 2} - x\right) & = \operatorname{cot} x
 661\end{split}\]
 662
 663\subsubsection*{Pythagorean identities}
 664
 665\[\begin{split}
 666  1 + \operatorname{cot}^2 x & = \operatorname{cosec}^2 x, \quad \text{where } \sin x \ne 0 \\
 667  1 + \tan^2 x & = \operatorname{sec}^2 x, \quad \text{where } \cos x \ne 0
 668\end{split}\]
 669
 670\subsection*{Compound angle formulas}
 671
 672\[\cos(x \pm y) = \cos x + \cos y \mp \sin x \sin y\]
 673\[\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y\]
 674\[\tan(x \pm y) = {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}\]
 675
 676\subsection*{Double angle formulas}
 677
 678\[\begin{split}
 679  \cos 2x &= \cos^2 x - \sin^2 x \\
 680  & = 1 - 2\sin^2 x \\
 681  & = 2 \cos^2 x -1
 682\end{split}\]
 683
 684\[\sin 2x = 2 \sin x \cos x\]
 685
 686\[\tan 2x = {{2 \tan x} \over {1 - \tan^2 x}}\]
 687
 688\subsection*{Inverse circular functions}
 689
 690Inverse functions: \(f(f^{-1}(x)) = x, \quad f(f^{-1}(x)) = x\)\\
 691Must be 1:1 to find inverse (reflection in \(y=x\)).\\
 692Domain is restricted to make functions 1:1.
 693
 694\[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y\]
 695\hfill where \(\sin y = x, \> y \in [{-\pi \over 2}, {\pi \over 2}]\)
 696
 697\[\cos^{-1}: [-1,1] \rightarrow \mathbb{R}, \quad \cos^{-1} x = y\]
 698\hfill where \(\cos y = x, \> y \in [0, \pi]\)
 699
 700\[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\]
 701\hfill where \(\tan y = x, \> y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\)
 702
 703
 704\section{Differential calculus}
 705
 706\subsection*{Limits}
 707
 708\[\lim_{x \rightarrow a}f(x)\]
 709\(L^-,\quad L^+\) \qquad limit from below/above\\
 710\(\lim_{x \to a} f(x)\) \quad limit of a point\\
 711
 712\noindent For solving \(x\rightarrow\infty\), put all \(x\) terms in denominators\\
 713    e.g. \[\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2\]
 714
 715\subsubsection*{Limit theorems}
 716
 717\begin{enumerate}
 718\item
 719  For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\)
 720\item
 721  \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\)
 722\item
 723  \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\)
 724    \item
 725\(\therefore \lim_{x \rightarrow a} c \times f(x)=cF\) where \(c=\) constant
 726\item
 727  \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\)
 728\item
 729  \(f(x)\) is continuous \(\iff L^-=L^+=f(x) \> \forall x\)
 730\end{enumerate}
 731
 732\subsection*{Gradients of secants and tangents}
 733
 734\textbf{Secant (chord)} - line joining two points on curve\\
 735\textbf{Tangent} - line that intersects curve at one point
 736
 737\subsection*{First principles derivative}
 738
 739\[f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={\frac{dy}{dx}}\]
 740
 741\subsubsection*{Logarithmic identities}
 742
 743\(\log_b (xy)=\log_b x + \log_b y\)\\
 744\(\log_b x^n = n \log_b x\)\\
 745\(\log_b y^{x^n} = x^n \log_b y\)
 746
 747\subsubsection*{Index identities}
 748
 749\(b^{m+n}=b^m \cdot b^n\)\\
 750\((b^m)^n=b^{m \cdot n}\)\\
 751\((b \cdot c)^n = b^n \cdot c^n\)\\
 752\({a^m \div a^n} = {a^{m-n}}\)
 753
 754\subsection*{Derivative rules}
 755
 756\renewcommand{\arraystretch}{1.4}
 757\begin{tabularx}{\columnwidth}{rX}
 758  \hline
 759\(f(x)\) & \(f^\prime(x)\)\\
 760\hline
 761\(\sin x\) & \(\cos x\)\\
 762\(\sin ax\) & \(a\cos ax\)\\
 763\(\cos x\) & \(-\sin x\)\\
 764\(\cos ax\) & \(-a \sin ax\)\\
 765\(\tan f(x)\) & \(f^2(x) \sec^2f(x)\)\\
 766\(e^x\) & \(e^x\)\\
 767\(e^{ax}\) & \(ae^{ax}\)\\
 768\(ax^{nx}\) & \(an \cdot e^{nx}\)\\
 769  \(\log_e x\) & \(\dfrac{1}{x}\)\\
 770  \(\log_e {ax}\) & \(\dfrac{1}{x}\)\\
 771  \(\log_e f(x)\) & \(\dfrac{f^\prime (x)}{f(x)}\)\\
 772\(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\\
 773  \(\sin^{-1} x\) & \(\dfrac{1}{\sqrt{1-x^2}}\)\\
 774  \(\cos^{-1} x\) & \(\dfrac{-1}{sqrt{1-x^2}}\)\\
 775  \(\tan^{-1} x\) & \(\dfrac{1}{1 + x^2}\)\\
 776  \hline
 777\end{tabularx}
 778
 779\subsection*{Reciprocal derivatives}
 780
 781\[\frac{1}{\frac{dy}{dx}} = \frac{dx}{dy}\]
 782
 783\subsection*{Differentiating \(x=f(y)\)}
 784\begin{align*}
 785  \text{Find }& \frac{dx}{dy}\\
 786  \text{Then, }\frac{dx}{dy} &= \frac{1}{\frac{dy}{dx}} \\
 787  \implies {\frac{dy}{dx}} &= \frac{1}{\frac{dx}{dy}}\\
 788  \therefore {\frac{dy}{dx}} &= \frac{1}{\frac{dx}{dy}}
 789\end{align*}
 790
 791\subsection*{Second derivative}
 792\begin{align*}f(x) \longrightarrow &f^\prime (x) \longrightarrow f^{\prime\prime}(x)\\
 793\implies y \longrightarrow &\frac{dy}{dx} \longrightarrow \frac{d^2 y}{dx^2}\end{align*}
 794
 795\noindent Order of polynomial \(n\)th derivative decrements each time the derivative is taken
 796
 797\subsubsection*{Points of Inflection}
 798
 799\emph{Stationary point} - i.e.
 800\(f^\prime(x)=0\)\\
 801\emph{Point of inflection} - max \(|\)gradient\(|\) (i.e.
 802\(f^{\prime\prime} = 0\))
 803%\begin{table*}[ht]
 804%\centering
 805%  \begin{tabularx}{\textwidth}{XXXX}
 806%\hline
 807%    \rowcolor{shade2}
 808%    & \(\dfrac{d^2 y}{dx^2} > 0\)  & \(\dfrac{d^2y}{dx^2}<0\) & \(\dfrac{d^2y}{dx^2}=0\) (inflection) \\
 809%\hline
 810%    \(\frac{dy}{dx}>0\) & \begin{tikzpicture} \draw[domain=1:2,smooth,variable=\x,blue] plot ({\x},{(1/10)*\x*\x*\x}) plot ({\x},{0.675*\x-0.677}); \end{tikzpicture} & cell 3\\
 811%cell 1 & cell 2 & cell 3\\
 812%\hline
 813%\end{tabularx}
 814%\end{table*}
 815\begin{itemize}
 816
 817\item
 818  if \(f^\prime (a) = 0\) and \(f^{\prime\prime}(a) > 0\), then point
 819  \((a, f(a))\) is a local min (curve is concave up)
 820\item
 821  if \(f^\prime (a) = 0\) and \(f^{\prime\prime} (a) < 0\), then point
 822  \((a, f(a))\) is local max (curve is concave down)
 823\item
 824  if \(f^{\prime\prime}(a) = 0\), then point \((a, f(a))\) is a point of
 825  inflection
 826\item
 827  if also \(f^\prime(a)=0\), then it is a stationary point of inflection
 828\end{itemize}
 829
 830\begin{table*}[ht]
 831  \centering
 832  \includegraphics[width=0.7\textwidth]{graphics/second-derivatives.png}
 833\end{table*}
 834
 835\subsection*{Implicit Differentiation}
 836
 837\noindent Used for differentiating circles etc.
 838
 839If \(p\) and \(q\) are expressions in \(x\) and \(y\) such that \(p=q\),
 840for all \(x\) and \(y\), then:
 841
 842\[{\frac{dp}{dx}} = {\frac{dq}{dx}} \quad \text{and} \quad {\frac{dp}{dy}} = {\frac{dq}{dy}}\]
 843
 844\noindent \colorbox{cas}{\textbf{On CAS:}}\\
 845Action \(\rightarrow\) Calculation \(\rightarrow\) \texttt{impDiff(y\^{}2+ax=5,\ x,\ y)}\\
 846Returns \(y^\prime= \dots\).
 847
 848\subsection*{Integration}
 849
 850\[\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)\]
 851
 852\subsection*{Integral laws}
 853
 854\renewcommand{\arraystretch}{1.4}
 855\begin{tabularx}{\columnwidth}{rX}
 856\hline
 857  \(f(x)\) & \(\int f(x) \cdot dx\) \\
 858  \hline
 859  \(k\) (constant) & \(kx + c\)\\
 860  \(x^n\) & \(\dfrac{1}{n+1} x^{n+1}\) \\
 861  \(a x^{-n}\) &\(a \cdot \log_e x + c\)\\
 862  \(\dfrac{1}{ax+b}\) &\(\dfrac{1}{a} \log_e (ax+b) + c\)\\
 863  \((ax+b)^n\) & \(\dfrac{1}{a(n+1)}(ax+b)^{n-1} + c\)\\
 864  \(e^{kx}\) & \(\dfrac{1}{k} e^{kx} + c\)\\
 865  \(e^k\) & \(e^kx + c\)\\
 866  \(\sin kx\) & \(\dfrac{-1}{k} \cos (kx) + c\)\\
 867  \(\cos kx\) & \(\frac{1}{k} \sin (kx) + c\)\\
 868  \(\sec^2 kx\) & \(\frac{1}{k} \tan(kx) + c\)\\
 869  \(\dfrac{1}{\sqrt{a^2-x^2}}\) & \(\sin^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
 870  \(\dfrac{-1}{\sqrt{a^2-x^2}}\) & \(\cos^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
 871  \(\frac{a}{a^2-x^2}\) & \(\tan^{-1} \frac{x}{a} + c\)\\
 872  \(\frac{f^\prime (x)}{f(x)}\) & \(\log_e f(x) + c\)\\
 873  \(g^\prime(x)\cdot f^\prime(g(x)\) & \(f(g(x))\) (chain rule)\\
 874  \(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\
 875  \hline
 876\end{tabularx}
 877
 878Note \(\sin^{-1} {x \over a} + \cos^{-1} {x \over a}\) is constant \(\forall x \in (-a, a)\)
 879
 880\subsection*{Definite integrals}
 881
 882\[\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)\]
 883
 884\begin{itemize}
 885
 886\item
 887  Signed area enclosed by\\
 888  \(\> y=f(x), \quad y=0, \quad x=a, \quad x=b\).
 889\item
 890  \emph{Integrand} is \(f\).
 891\end{itemize}
 892
 893\subsubsection*{Properties}
 894
 895\[\int^b_a f(x) \> dx = \int^c_a f(x) \> dx + \int^b_c f(x) \> dx\]
 896
 897\[\int^a_a f(x) \> dx = 0\]
 898
 899\[\int^b_a k \cdot f(x) \> dx = k \int^b_a f(x) \> dx\]
 900
 901\[\int^b_a f(x) \pm g(x) \> dx = \int^b_a f(x) \> dx \pm \int^b_a g(x) \> dx\]
 902
 903\[\int^b_a f(x) \> dx = - \int^a_b f(x) \> dx\]
 904
 905\subsection*{Integration by substitution}
 906
 907\[\int f(u) {\frac{du}{dx}} \cdot dx = \int f(u) \cdot du\]
 908
 909\noindent Note \(f(u)\) must be 1:1 \(\implies\) one \(x\) for each \(y\)
 910\begin{align*}\text{e.g. for } y&=\int(2x+1)\sqrt{x+4} \cdot dx\\
 911  \text{let } u&=x+4\\
 912  \implies& {\frac{du}{dx}} = 1\\
 913  \implies& x = u - 4\\
 914  \text{then } &y=\int (2(u-4)+1)u^{\frac{1}{2}} \cdot du\\
 915  &\text{(solve as  normal integral)}
 916\end{align*}
 917
 918\subsubsection*{Definite integrals by substitution}
 919
 920For \(\int^b_a f(x) {\frac{du}{dx}} \cdot dx\), evaluate new \(a\) and
 921\(b\) for \(f(u) \cdot du\).
 922
 923\subsubsection*{Trigonometric integration}
 924
 925\[\sin^m x \cos^n x \cdot dx\]
 926
 927\paragraph{\textbf{\(m\) is odd:}}
 928\(m=2k+1\) where \(k \in \mathbb{Z}\)\\
 929\(\implies \sin^{2k+1} x = (\sin^2 z)^k \sin x = (1 - \cos^2 x)^k \sin x\)\\
 930Substitute \(u=\cos x\)
 931
 932\paragraph{\textbf{\(n\) is odd:}}
 933\(n=2k+1\) where \(k \in \mathbb{Z}\)\\
 934\(\implies \cos^{2k+1} x = (\cos^2 x)^k \cos x = (1-\sin^2 x)^k \cos x\)\\
 935Substitute \(u=\sin x\)
 936
 937\paragraph{\textbf{\(m\) and \(n\) are even:}}
 938use identities...
 939
 940\begin{itemize}
 941
 942\item
 943  \(\sin^2x={1 \over 2}(1-\cos 2x)\)
 944\item
 945  \(\cos^2x={1 \over 2}(1+\cos 2x)\)
 946\item
 947  \(\sin 2x = 2 \sin x \cos x\)
 948\end{itemize}
 949
 950\subsection*{Partial fractions}
 951
 952\colorbox{cas}{On CAS:}\\
 953\indent Action \(\rightarrow\) Transformation \(\rightarrow\)
 954\texttt{expand/combine}\\
 955\indent Interactive \(\rightarrow\) Transformation \(\rightarrow\)
 956Expand \(\rightarrow\) Partial
 957
 958\subsection*{Graphing integrals on CAS}
 959
 960\colorbox{cas}{In main:} Interactive \(\rightarrow\) Calculation \(\rightarrow\)
 961\(\int\) (\(\rightarrow\) Definite)\\
 962Restrictions: \texttt{Define\ f(x)=..} then \texttt{f(x)\textbar{}x\textgreater{}..}
 963
 964\subsection*{Applications of antidifferentiation}
 965
 966\begin{itemize}
 967
 968\item
 969  \(x\)-intercepts of \(y=f(x)\) identify \(x\)-coordinates of
 970  stationary points on \(y=F(x)\)
 971\item
 972  nature of stationary points is determined by sign of \(y=f(x)\) on
 973  either side of its \(x\)-intercepts
 974\item
 975  if \(f(x)\) is a polynomial of degree \(n\), then \(F(x)\) has degree
 976  \(n+1\)
 977\end{itemize}
 978
 979To find stationary points of a function, substitute \(x\) value of given
 980point into derivative. Solve for \({\frac{dy}{dx}}=0\). Integrate to find
 981original function.
 982
 983\subsection*{Solids of revolution}
 984
 985Approximate as sum of infinitesimally-thick cylinders
 986
 987\subsubsection*{Rotation about \(x\)-axis}
 988
 989\begin{align*}
 990  V &= \int^{x=b}_{x-a} \pi y^2 \> dx \\
 991    &= \pi \int^b_a (f(x))^2 \> dx
 992\end{align*}
 993
 994\subsubsection*{Rotation about \(y\)-axis}
 995
 996\begin{align*}
 997  V &= \int^{y=b}_{y=a} \pi x^2 \> dy \\
 998    &= \pi \int^b_a (f(y))^2 \> dy
 999\end{align*}
1000
1001\subsubsection*{Regions not bound by \(y=0\)}
1002
1003\[V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx\]
1004\hfill where \(f(x) > g(x)\)
1005
1006\subsection*{Length of a curve}
1007
1008\[L = \int^b_a \sqrt{1 + ({\frac{dy}{dx}})^2} \> dx \quad \text{(Cartesian)}\]
1009
1010\[L = \int^b_a \sqrt{{\frac{dx}{dt}} + ({\frac{dy}{dt}})^2} \> dt \quad \text{(parametric)}\]
1011
1012\noindent \colorbox{cas}{On CAS:}\\
1013\indent Evaluate formula,\\
1014\indent or Interactive \(\rightarrow\) Calculation
1015\(\rightarrow\) Line \(\rightarrow\) \texttt{arcLen}
1016
1017\subsection*{Rates}
1018
1019\subsubsection*{Gradient at a point on parametric curve}
1020
1021\[{\frac{dy}{dx}} = {{\frac{dy}{dt}} \div {\frac{dx}{dt}}} \> \vert \> {\frac{dx}{dt}} \ne 0\]
1022
1023\[\frac{d^2}{dx^2} = \frac{d(y^\prime)}{dx} = {\frac{dy^\prime}{dt} \div {\frac{dx}{dt}}} \> \vert \> y^\prime = {\frac{dy}{dx}}\]
1024
1025\subsection*{Rational functions}
1026
1027\[f(x) = \frac{P(x)}{Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}\]
1028
1029\subsubsection*{Addition of ordinates}
1030
1031\begin{itemize}
1032
1033\item
1034  when two graphs have the same ordinate, \(y\)-coordinate is double the
1035  ordinate
1036\item
1037  when two graphs have opposite ordinates, \(y\)-coordinate is 0 i.e.
1038  (\(x\)-intercept)
1039\item
1040  when one of the ordinates is 0, the resulting ordinate is equal to the
1041  other ordinate
1042\end{itemize}
1043
1044\subsection*{Fundamental theorem of calculus}
1045
1046If \(f\) is continuous on \([a, b]\), then
1047
1048\[\int^b_a f(x) \> dx = F(b) - F(a)\]
1049\hfill where \(F = \int f \> dx\)
1050
1051\subsection*{Differential equations}
1052
1053\noindent\textbf{Order} - highest power inside derivative\\
1054\textbf{Degree} - highest power of highest derivative\\
1055e.g. \({\left(\dfrac{dy^2}{d^2} x\right)}^3\) \qquad order 2, degree 3
1056
1057\subsubsection*{Verifying solutions}
1058
1059Start with \(y=\dots\), and differentiate. Substitute into original
1060equation.
1061
1062\subsubsection*{Function of the dependent
1063variable}
1064
1065If \({\frac{dy}{dx}}=g(y)\), then
1066\(\frac{dx}{dy} = 1 \div \frac{dy}{dx} = \frac{1}{g(y)}\). Integrate both sides to solve equation. Only add \(c\) on one side. Express
1067\(e^c\) as \(A\).
1068
1069\subsubsection*{Mixing problems}
1070
1071\[\left(\frac{dm}{dt}\right)_\Sigma = \left(\frac{dm}{dt}\right)_{\text{in}} - \left(\frac{dm}{dt}_{\text{out}}\right)\]
1072
1073\subsubsection*{Separation of variables}
1074
1075If \({\frac{dy}{dx}}=f(x)g(y)\), then:
1076
1077\[\int f(x) \> dx = \int \frac{1}{g(y)} \> dy\]
1078
1079\subsubsection*{Euler's method for solving DEs}
1080
1081\[\frac{f(x+h) - f(x)}{h} \approx f^\prime (x) \quad \text{for small } h\]
1082
1083\[\implies f(x+h) \approx f(x) + hf^\prime(x)\]
1084
1085  \end{multicols}
1086\end{document}