39ab74ecb586908cb18c2851ac08382cc756a136
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   5\usepackage{amsmath}
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   7\usepackage{harpoon}
   8\usepackage{tabularx}
   9\usepackage{makecell}
  10\usepackage[dvipsnames, table]{xcolor}
  11\usepackage{blindtext}
  12\usepackage{graphicx}
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  16\usepackage{pgfplots}
  17\usetikzlibrary{calc}
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  19\usetikzlibrary{datavisualization.formats.functions}
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  22\usepackage{fancyhdr}
  23\pagestyle{fancy}
  24\fancyhead[LO,LE]{Year 12 Specialist}
  25\fancyhead[CO,CE]{Andrew Lorimer}
  26
  27\usepackage{mathtools}
  28\usepackage{xcolor} % used only to show the phantomed stuff
  29\renewcommand\hphantom[1]{{\color[gray]{.6}#1}} % comment out!
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  44                  \pgfplotsset{every axis/.append style={
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  50                  }}
  51\begin{document}
  52
  53\begin{multicols}{2}
  54
  55  \section{Complex numbers}
  56
  57  \[\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}\]
  58
  59  \begin{align*}
  60    \text{Cartesian form: } & a+bi\\
  61    \text{Polar form: } & r\operatorname{cis}\theta
  62  \end{align*}
  63
  64  \subsection*{Operations}
  65
  66  \definecolor{shade1}{HTML}{ffffff}
  67  \definecolor{shade2}{HTML}{e6f2ff}
  68  \definecolor{shade3}{HTML}{cce2ff}
  69  \begin{tabularx}{\columnwidth}{r|X|X}
  70    & \textbf{Cartesian} & \textbf{Polar} \\
  71    \hline
  72    \(z_1 \pm z_2\) & \((a \pm c)(b \pm d)i\) & convert to \(a+bi\)\\
  73    \hline
  74    \(+k \times z\) & \multirow{2}{*}{\(ka \pm kbi\)} & \(kr\operatorname{cis} \theta\)\\
  75    \cline{1-1}\cline{3-3}
  76    \(-k \times z\) & & \(kr \operatorname{cis}(\theta\pm \pi)\)\\
  77    \hline
  78    \(z_1 \cdot z_2\) & \(ac-bd+(ad+bc)i\) & \(r_1r_2 \operatorname{cis}(\theta_1 + \theta_2)\)\\
  79    \hline
  80    \(z_1 \div z_2\) & \((z_1 \overline{z_2}) \div |z_2|^2\) & \(\left(\frac{r_1}{r_2}\right) \operatorname{cis}(\theta_1 - \theta_2)\)
  81  \end{tabularx}
  82
  83  \subsubsection*{Scalar multiplication in polar form}
  84
  85  For \(k \in \mathbb{R}^+\):
  86  \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\theta\]
  87
  88  \noindent For \(k \in \mathbb{R}^-\):
  89  \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\left(\begin{cases}\theta - \pi & |0<\operatorname{Arg}(z)\le \pi \\ \theta + \pi & |-\pi<\operatorname{Arg}(z)\le 0\end{cases}\right)\]
  90
  91    \subsection*{Conjugate}
  92
  93    \begin{align*}
  94      \overline{z} &= a \mp bi\\
  95      &= r \operatorname{cis}(-\theta)
  96    \end{align*}
  97
  98    \noindent \colorbox{cas}{On CAS: \texttt{conjg(a+bi)}}
  99
 100    \subsubsection*{Properties}
 101
 102    \begin{align*}
 103      \overline{z_1 \pm z_2} &= \overline{z_1}\pm\overline{z_2}\\
 104      \overline{z_1 \cdot z_2} &= \overline{z_1}\cdot\overline{z_2}\\
 105      \overline{kz} &= k\overline{z} \quad | \quad k \in \mathbb{R}\\
 106      z\overline{z} &= (a+bi)(a-bi)\\
 107      &= a^2 + b^2\\
 108      &= |z|^2
 109    \end{align*}
 110
 111    \subsection*{Modulus}
 112
 113    \[|z|=|\vec{Oz}|=\sqrt{a^2 + b^2}\]
 114
 115    \subsubsection*{Properties}
 116
 117    \begin{align*}
 118      |z_1z_2|&=|z_1||z_2|\\
 119      \left|\frac{z_1}{z_2}\right|&=\frac{|z_1|}{|z_2|}\\
 120      |z_1+z_2|&\le|z_1|+|z_2|
 121    \end{align*}
 122
 123    \subsection*{Multiplicative inverse}
 124
 125    \begin{align*}
 126      z^{-1}&=\frac{a-bi}{a^2+b^2}\\
 127      &=\frac{\overline{z}}{|z|^2}a\\
 128      &=r \operatorname{cis}(-\theta)
 129    \end{align*}
 130
 131    \subsection*{Dividing over \(\mathbb{C}\)}
 132
 133    \begin{align*}
 134      \frac{z_1}{z_2}&=z_1z_2^{-1}\\
 135      &=\frac{z_1\overline{z_2}}{|z_2|^2}\\
 136      &=\frac{(a+bi)(c-di)}{c^2+d^2}\\
 137      & \qquad \text{(rationalise denominator)}
 138    \end{align*}
 139
 140    \subsection*{Polar form}
 141
 142    \begin{align*}
 143      z&=r\operatorname{cis}\theta\\
 144      &=r(\cos \theta + i \sin \theta)
 145    \end{align*}
 146
 147    \begin{itemize}
 148      \item{\(r=|z|=\sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}\)}
 149      \item{\(\theta = \operatorname{arg}(z)\) \quad \colorbox{cas}{On CAS: \texttt{arg(a+bi)}}}
 150      \item{\(\operatorname{Arg}(z) \in (-\pi,\pi)\) \quad \bf{(principal argument)}}
 151      \item{\colorbox{cas}{Convert on CAS:}\\ \verb|compToTrig(a+bi)| \(\iff\) \verb|cExpand{r·cisX}|}
 152      \item{Multiple representations:\\\(r\operatorname{cis}\theta=r\operatorname{cis}(\theta+2n\pi)\) with \(n \in \mathbb{Z}\) revolutions}
 153      \item{\(\operatorname{cis}\pi=-1,\qquad \operatorname{cis}0=1\)}
 154    \end{itemize}
 155
 156    \subsection*{de Moivres' theorem}
 157
 158    \[(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}\]
 159
 160    \subsection*{Complex polynomials}
 161
 162    Include \(\pm\) for all solutions, incl. imaginary
 163
 164    \begin{tabularx}{\columnwidth}{ R{0.55} X  }
 165      \hline
 166      Sum of squares & \(\begin{aligned} 
 167        z^2 + a^2 &= z^2-(ai)^2\\
 168      &= (z+ai)(z-ai) \end{aligned}\) \\
 169      \hline
 170      Sum of cubes & \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)\\
 171      \hline
 172      Division & \(P(z)=D(z)Q(z)+R(z)\) \\
 173      \hline
 174      Remainder theorem & Let \(\alpha \in \mathbb{C}\). Remainder of \(P(z) \div (z-\alpha)\) is \(P(\alpha)\)\\
 175      \hline
 176      Factor theorem & \(z-\alpha\) is a factor of \(P(z) \iff P(\alpha)=0\) for \(\alpha \in \mathbb{C}\)\\
 177      \hline
 178      Conjugate root theorem & \(P(z)=0 \text{ at } z=a\pm bi\) (\(\implies\) both \(z_1\) and \(\overline{z_1}\) are solutions)\\
 179      \hline
 180    \end{tabularx}
 181
 182    \subsection*{\(n\)th roots}
 183
 184    \(n\)th roots of \(z=r\operatorname{cis}\theta\) are:
 185
 186    \[z = r^{\frac{1}{n}} \operatorname{cis}\left(\frac{\theta+2k\pi}{n}\right)\]
 187
 188    \begin{itemize}
 189
 190      \item{Same modulus for all solutions}
 191      \item{Arguments separated by \(\frac{2\pi}{n} \therefore\) there are \(n\) roots}
 192      \item{If one square root is \(a+bi\), the other is \(-a-bi\)}
 193      \item{Give one implicit \(n\)th root \(z_1\), function is \(z=z_1^n\)}
 194      \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 \(\frac{2\pi}{n}\))}
 195    \end{itemize}
 196
 197    \noindent For \(0=az^2+bz+c\), use quadratic formula:
 198
 199    \[z=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]
 200
 201    \subsection*{Fundamental theorem of algebra}
 202
 203    A polynomial of degree \(n\) can be factorised into \(n\) linear factors in \(\mathbb{C}\):
 204
 205    \[\implies P(z)=a_n(z-\alpha_1)(z-\alpha_2)(z-\alpha_3)\dots(z-\alpha_n)\]
 206    \[\text{ where } \alpha_1,\alpha_2,\alpha_3,\dots,\alpha_n \in \mathbb{C}\]
 207
 208    \subsection*{Argand planes}
 209
 210    \begin{center}\begin{tikzpicture}[scale=2]
 211      \draw [->] (-0.2,0) -- (1.5,0) node [right]  {$\operatorname{Re}(z)$};
 212      \draw [->] (0,-0.2) -- (0,1.5) node [above] {$\operatorname{Im}(z)$};
 213      \coordinate (P) at (1,1);
 214      \coordinate (a) at (1,0);
 215      \coordinate (b) at (0,1);
 216      \coordinate (O) at (0,0);
 217      \draw (0,0) -- (P) node[pos=0.5, above left]{\(r\)} node[pos=1, right]{\(\begin{aligned}z&=a+bi\\&=r\operatorname{cis}\theta\end{aligned}\)};
 218        \draw [gray, dashed] (1,1) -- (1,0) node[black, pos=1, below]{\(a\)};
 219        \draw [gray, dashed] (1,1) -- (0,1) node[black, pos=1, left]{\(b\)};
 220        \begin{scope}
 221          \path[clip] (O) -- (P) -- (a);
 222          \fill[red, opacity=0.5, draw=black] (O) circle (2mm);
 223          \node at ($(O)+(20:3mm)$) {$\theta$};
 224        \end{scope}
 225        \filldraw (P) circle (0.5pt);
 226    \end{tikzpicture}\end{center}
 227
 228    \begin{itemize}
 229      \item{Multiplication by \(i \implies\) CCW rotation of \(\frac{\pi}{2}\)}
 230      \item{Addition: \(z_1 + z_2 \equiv\) \overrightharp{\(Oz_1\)} + \overrightharp{\(Oz_2\)}}
 231    \end{itemize}
 232
 233    \subsection*{Sketching complex graphs}
 234
 235    \subsubsection*{Linear}
 236
 237    \begin{itemize}
 238      \item{\(\operatorname{Re}(z)=c\) or \(\operatorname{Im}(z)=c\) (perpendicular bisector)}
 239      \item{\(\operatorname{Im}(z)=m\operatorname{Re}(z)\)}
 240      \item{\(|z+a|=|z+b| \implies 2(a-b)x=b^2-a^2\)\\Geometric: equidistant from \(a,b\)}
 241    \end{itemize}
 242
 243    \subsubsection*{Circles}
 244
 245    \begin{itemize}
 246      \item \(|z-z_1|^2=c^2|z_2+2|^2\)
 247      \item \(|z-(a+bi)|=c \implies (x-a)^2+_(y-b)^2=c^2\)
 248    \end{itemize}
 249
 250    \noindent \textbf{Loci} \qquad \(\operatorname{Arg}(z)<\theta\)
 251
 252    \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
 253      \draw [->] (0,0) -- (1,0) node [right]  {$\operatorname{Re}(z)$};
 254      \draw [->] (0,-0.5) -- (0,1) node [above] {$\operatorname{Im}(z)$};
 255      \draw [<-, dashed, thick, blue] (-1,0) -- (0,0);
 256      \draw [->, thick, blue] (0,0) -- (1,1);
 257      \fill [gray, opacity=0.2, domain=-1:1, variable=\x] (-1,-0.5) -- (-1,0) -- (0, 0) -- (1,1) -- (1,-0.5) -- cycle;
 258      \begin{scope}
 259        \path[clip] (0,0) -- (1,1) -- (1,0);
 260        \fill[red, opacity=0.5, draw=black] (0,0) circle (2mm);
 261        \node at ($(0,0)+(20:3mm)$) {$\frac{\pi}{4}$};
 262      \end{scope}
 263      \node [font=\footnotesize] at (0.5,-0.25) {\(\operatorname{Arg}(z)\le\frac{\pi}{4}\)};
 264      \node [blue, mydot] {};
 265    \end{tikzpicture}\end{center}
 266
 267    \noindent \textbf{Rays} \qquad \(\operatorname{Arg}(z-b)=\theta\)
 268
 269    \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
 270      \draw [->] (-0.75,0) -- (1.5,0) node [right]  {$\operatorname{Re}(z)$};
 271      \draw [->] (0,-1) -- (0,1) node [above] {$\operatorname{Im}(z)$};
 272      \draw [->, thick, brown] (-0.25,0) -- (-0.75,-1);
 273      \node [above, font=\footnotesize] at (-0.25,0) {\(\frac{1}{4}\)};
 274      \begin{scope}
 275        \path[clip] (-0.25,0) -- (-0.75,-1) -- (0,0);
 276        \fill[orange, opacity=0.5, draw=black] (-0.25,0) circle (2mm);
 277      \end{scope}
 278      \node at (-0.08,-0.3) {\(\frac{\pi}{8}\)};
 279      \node [font=\footnotesize, left] at (-0.75,-1) {\(\operatorname{Arg}(z+\frac{1}{4})=\frac{\pi}{8}\)};
 280      \node [brown, mydot] at (-0.25,0) {};
 281      \draw [<->, thick, green] (0,-1) -- (1.5,0.5) node [pos=0.25, black, font=\footnotesize, right] {\(|z-2|=|z-(1+i)|\)};
 282      \node [left, font=\footnotesize] at (0,-1) {\(-1\)};
 283      \node [below, font=\footnotesize] at (1,0) {\(1\)};
 284    \end{tikzpicture}\end{center}
 285
 286    \section{Vectors}
 287    \begin{center}\begin{tikzpicture}
 288      \draw [->] (-0.5,0) -- (3,0) node [right]  {\(x\)};
 289      \draw [->] (0,-0.5) -- (0,3) node [above] {\(y\)};
 290      \draw [orange, ->, thick] (0.5,0.5) -- (2.5,2.5) node [pos=0.5, above] {\(\vec{u}\)};
 291      \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
 292        \draw [gray, dashed, thick] (0.5,0.5) -- (2.5,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, below]{\(x\vec{i}\)};
 293        \draw [gray, dashed, thick] (2.5,0.5) -- (2.5,2.5) node [pos=0.5] {\midarrow};
 294      \end{scope}
 295      \node[black, right] at (2.5,1.5) {\(y\vec{j}\)};
 296    \end{tikzpicture}\end{center}
 297    \subsection*{Column notation}
 298
 299    \[\begin{bmatrix}x\\ y \end{bmatrix} \iff x\boldsymbol{i} + y\boldsymbol{j}\]
 300      \(\begin{bmatrix}x_2-x_1\\ y_2-y_1 \end{bmatrix}\) \quad between \(A(x_1,y_1), \> B(x_2,y_2)\)
 301
 302        \subsection*{Scalar multiplication}
 303
 304        \[k\cdot (x\boldsymbol{i}+y\boldsymbol{j})=kx\boldsymbol{i}+ky\boldsymbol{j}\]
 305
 306        \noindent For \(k \in \mathbb{R}^-\), direction is reversed
 307
 308        \subsection*{Vector addition}
 309        \begin{center}\begin{tikzpicture}[scale=1]
 310          \coordinate (A) at (0,0);
 311          \coordinate (B) at (2,2);
 312          \draw [->, thick, red] (0,0) -- (2,2) node [pos=0.5, below right] {\(\vec{u}=2\vec{i}+2\vec{j}\)};
 313          \draw [->, thick, blue] (2,2) -- (1,4) node [pos=0.5, above right] {\(\vec{v}=-\vec{i}+2\vec{j}\)};
 314          \draw [->, thick, orange] (0,0) -- (1,4) node [pos=0.5, left] {\(\vec{u}+\vec{v}=\vec{i}+4\vec{j}\)};
 315        \end{tikzpicture}\end{center}
 316
 317        \[(x\boldsymbol{i}+y\boldsymbol{j}) \pm (a\boldsymbol{i}+b\boldsymbol{j})=(x \pm a)\boldsymbol{i}+(y \pm b)\boldsymbol{j}\]
 318
 319        \begin{itemize}
 320          \item Draw each vector head to tail then join lines
 321          \item Addition is commutative (parallelogram)
 322          \item \(\boldsymbol{u}-\boldsymbol{v}=\boldsymbol{u}+(-\boldsymbol{v}) \implies \overrightharp{AB}=\boldsymbol{b}-\boldsymbol{a}\)
 323        \end{itemize}
 324
 325        \subsection*{Magnitude}
 326
 327        \[|(x\boldsymbol{i} + y\boldsymbol{j})|=\sqrt{x^2+y^2}\]
 328
 329        \subsection*{Parallel vectors}
 330
 331        \[\boldsymbol{u} || \boldsymbol{v} \iff \boldsymbol{u} = k \boldsymbol{v} \text{ where } k \in \mathbb{R} \setminus \{0\}\]
 332
 333        For parallel vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\):\\
 334        \[\boldsymbol{a \cdot b}=\begin{cases}
 335          |\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction}\\
 336          -|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions}
 337        \end{cases}\]
 338        %\includegraphics[width=0.2,height=\textheight]{graphics/parallelogram-vectors.jpg}
 339        %\includegraphics[width=1]{graphics/vector-subtraction.jpg}
 340
 341        \subsection*{Perpendicular vectors}
 342
 343        \[\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b} = 0\ \quad \text{(since \(\cos 90 = 0\))}\]
 344
 345        \subsection*{Unit vector \(|\hat{\boldsymbol{a}}|=1\)}
 346        \[\begin{split}\hat{\boldsymbol{a}} & = {\frac{1}{|\boldsymbol{a}|}}\boldsymbol{a} \\ & = \boldsymbol{a} \cdot {|\boldsymbol{a}|}\end{split}\]
 347
 348          \subsection*{Scalar product \(\boldsymbol{a} \cdot \boldsymbol{b}\)}
 349
 350
 351          \begin{center}\begin{tikzpicture}[scale=2]
 352            \draw [->] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{b}\)};
 353            \draw [->] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{a}\)};
 354            \begin{scope}
 355              \path[clip] (1,0.5) -- (1,0) -- (0,0);
 356              \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
 357              \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
 358            \end{scope}
 359          \end{tikzpicture}\end{center}
 360          \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*}
 361            \noindent\colorbox{cas}{On CAS: \texttt{dotP({[}a\ b\ c{]},\ {[}d\ e\ f{]})}}
 362
 363            \subsubsection*{Properties}
 364
 365            \begin{enumerate}
 366              \item
 367                \(k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k\boldsymbol{b})\)
 368              \item
 369                \(\boldsymbol{a \cdot 0}=0\)
 370              \item
 371                \(\boldsymbol{a} \cdot (\boldsymbol{b} + \boldsymbol{c})=\boldsymbol{a} \cdot \boldsymbol{b} + \boldsymbol{a} \cdot \boldsymbol{c}\)
 372              \item
 373                \(\boldsymbol{i \cdot i} = \boldsymbol{j \cdot j} = \boldsymbol{k \cdot k}= 1\)
 374              \item
 375                \(\boldsymbol{a} \cdot \boldsymbol{b} = 0 \quad \implies \quad \boldsymbol{a} \perp \boldsymbol{b}\)
 376              \item
 377                \(\boldsymbol{a \cdot a} = |\boldsymbol{a}|^2 = a^2\)
 378            \end{enumerate}
 379
 380            \subsection*{Angle between vectors}
 381
 382            \[\cos \theta = \frac{\boldsymbol{a} \cdot \boldsymbol{b}}{|\boldsymbol{a}| |\boldsymbol{b}|} = \frac{a_1 b_1 + a_2 b_2}{|\boldsymbol{a}| |\boldsymbol{b}|}\]
 383
 384            \noindent \colorbox{cas}{On CAS:} \texttt{angle([a b c], [a b c])}
 385
 386            (Action \(\rightarrow\) Vector \(\rightarrow\)Angle)
 387
 388            \subsection*{Angle between vector and axis}
 389
 390            \noindent For\(\boldsymbol{a} = a_1 \boldsymbol{i} + a_2 \boldsymbol{j} + a_3 \boldsymbol{k}\)
 391            which makes angles \(\alpha, \beta, \gamma\) with positive side of
 392            \(x, y, z\) axes:
 393            \[\cos \alpha = \frac{a_1}{|\boldsymbol{a}|}, \quad \cos \beta = \frac{a_2}{|\boldsymbol{a}|}, \quad \cos \gamma = \frac{a_3}{|\boldsymbol{a}|}\]
 394
 395            \noindent \colorbox{cas}{On CAS:} \texttt{angle({[}a\ b\ c{]},\ {[}1\ 0\ 0{]})}\\for angle
 396            between \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) and
 397            \(x\)-axis
 398
 399            \subsection*{Projections \& resolutes}
 400
 401            \begin{tikzpicture}[scale=3]
 402              \draw [->, purple] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{a}\)};
 403              \draw [->, orange] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{u}\)};
 404              \draw [->, blue] (1,0) -- (2,0) node [pos=0.5, below] {\(\boldsymbol{b}\)};
 405              \begin{scope}
 406                \path[clip] (1,0.5) -- (1,0) -- (0,0);
 407                \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
 408                \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
 409              \end{scope}
 410              \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
 411                \draw [gray, dashed, thick] (1,0) -- (1,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, right, rotate=-90]{\(\boldsymbol{w}\)};
 412              \end{scope}
 413              \draw (0,0) coordinate (O)
 414              (1,0) coordinate (A)
 415              (1,0.5) coordinate (B)
 416              pic [draw,red,angle radius=2mm] {right angle = O--A--B};
 417            \end{tikzpicture}
 418
 419            \subsubsection*{\(\parallel\boldsymbol{b}\) (vector projection/resolute)}
 420
 421            \begin{align*}
 422              \boldsymbol{u} & = \frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|^2}\boldsymbol{b} \\
 423              & = \left(\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\right)\left(\frac{\boldsymbol{b}}{|\boldsymbol{b}|}\right) \\
 424              & = (\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}
 425            \end{align*}
 426
 427            \subsubsection*{\(\perp\boldsymbol{b}\) (perpendicular projection)}
 428            \[\boldsymbol{w} = \boldsymbol{a} - \boldsymbol{u}\]
 429
 430            \subsubsection*{\(|\boldsymbol{u}|\) (scalar projection/resolute)}
 431            \begin{align*}
 432              s &= |\boldsymbol{u}|\\
 433              &= \boldsymbol{a} \cdot \hat{\boldsymbol{b}}\\
 434              &=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\\
 435              &= |\boldsymbol{a}| \cos \theta
 436            \end{align*}
 437
 438            \subsubsection*{Rectangular (\(\parallel,\perp\)) components}
 439
 440            \[\boldsymbol{a}=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}+\left(\boldsymbol{a}-\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}\right)\]
 441
 442
 443            \subsection*{Vector proofs}
 444
 445            \textbf{Concurrent:} intersection of \(\ge\) 3 lines
 446
 447            \begin{tikzpicture}
 448              \draw [blue] (0,0) -- (1,1);
 449              \draw [red] (1,0) -- (0,1);
 450              \draw [brown] (0.4,0) -- (0.6,1);
 451              \filldraw (0.5,0.5) circle (2pt);
 452            \end{tikzpicture}
 453
 454            \subsubsection*{Collinear points}
 455
 456            \(\ge\) 3 points lie on the same line
 457
 458            \begin{tikzpicture}
 459              \draw [purple] (0,0) -- (4,1);
 460              \filldraw (2,0.5) circle (2pt) node [above] {\(C\)};
 461              \filldraw (1,0.25) circle (2pt) node [above] {\(A\)};
 462              \filldraw (3,0.75) circle (2pt) node [above] {\(B\)};
 463              \coordinate (O) at (2.8,-0.2);
 464              \node at (O) [below] {\(O\)}; 
 465              \begin{scope}[->, orange, thick] 
 466                \draw (O) -- (2,0.5) node [pos=0.5, above, font=\footnotesize, black] {\(\boldsymbol{c}\)};
 467                \draw (O) -- (1,0.25) node [pos=0.5, below, font=\footnotesize, black] {\(\boldsymbol{a}\)};
 468                \draw (O) -- (3,0.75) node [pos=0.5, right, font=\footnotesize, black] {\(\boldsymbol{b}\)};
 469              \end{scope}
 470            \end{tikzpicture}
 471
 472            \begin{align*}
 473              \text{e.g. Prove that}\\
 474              \overrightharp{AC}=m\overrightharp{AB} \iff \boldsymbol{c}&=(1-m)\boldsymbol{a}+m\boldsymbol{b}\\
 475              \implies \boldsymbol{c} &= \overrightharp{OA} + \overrightharp{AC}\\
 476              &= \overrightharp{OA} + m\overrightharp{AB}\\
 477              &=\boldsymbol{a}+m(\boldsymbol{b}-\boldsymbol{a})\\
 478              &=\boldsymbol{a}+m\boldsymbol{b}-m\boldsymbol{a}\\
 479              &=(1-m)\boldsymbol{a}+m{b}
 480            \end{align*}
 481            \begin{align*}
 482              \text{Also, } \implies \overrightharp{OC} &= \lambda \vec{OA} + \mu \overrightharp{OB} \\
 483              \text{where } \lambda + \mu &= 1\\
 484              \text{If } C \text{ lies along } \overrightharp{AB}, & \implies 0 < \mu < 1
 485            \end{align*}
 486
 487
 488            \subsubsection*{Parallelograms}
 489
 490            \begin{center}\begin{tikzpicture}
 491              \coordinate (O) at (0,0) node [below left] {\(O\)};
 492              \coordinate (A) at (4,0);
 493              \coordinate (B) at (6,2);
 494              \coordinate (C) at (2,2);
 495              \coordinate (D) at (6,0);
 496
 497              \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (O)--(A) node [below left] {\(A\)};
 498              \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (A)--(B) node [above right] {\(B\)};
 499              \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (B)--(C) node [above left] {\(C\)};
 500              \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (C)--(O);
 501
 502              \draw [gray, dashed] (O) -- (B) node [pos=0.75] {\(\diagdown\diagdown\)} node [pos=0.25] {\(\diagdown\diagdown\)};
 503              \draw [gray, dashed] (A) -- (C) node [pos=0.75] {\(\diagup\)} node [pos=0.25] {\(\diagup\)};
 504              \begin{scope}
 505                \path[clip] (C) -- (A) -- (O);
 506                \fill[orange, opacity=0.5, draw=black] (0,0) circle (4mm);
 507                \node at ($(0,0)+(20:8mm)$) {\(\theta\)};
 508              \end{scope}
 509              \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] {\(|\boldsymbol{c}|\cos\theta\)};
 510              \draw pic [draw,thick,red,angle radius=2mm] {right angle=O--D--B};
 511            \end{tikzpicture}\end{center}
 512
 513            \begin{itemize}
 514              \item
 515                Diagonals \(\overrightharp{OB}, \overrightharp{AC}\) bisect each other
 516              \item
 517                If diagonals are equal length, it is a rectangle
 518              \item
 519                \(|\overrightharp{OB}|^2 + |\overrightharp{CA}|^2 = |\overrightharp{OA}|^2 + |\overrightharp{AB}|^2 + |\overrightharp{CB}|^2 + |\overrightharp{OC}|^2\)
 520              \item
 521                Area \(=\boldsymbol{c} \cdot \boldsymbol{a}\)
 522            \end{itemize}
 523
 524            \subsubsection*{Useful vector properties}
 525
 526            \begin{itemize}
 527              \item
 528                \(\boldsymbol{a} \parallel \boldsymbol{b} \implies \boldsymbol{b}=k\boldsymbol{a}\) for some
 529                \(k \in \mathbb{R} \setminus \{0\}\)
 530              \item
 531                If \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are parallel with at
 532                least one point in common, then they lie on the same straight line
 533              \item
 534                \(\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b}=0\)
 535              \item
 536                \(\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2\)
 537            \end{itemize}
 538
 539            \subsection*{Linear dependence}
 540
 541            \(\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\) are linearly dependent if they are \(\nparallel\) and:
 542            \begin{align*}
 543              0&=k\boldsymbol{a}+l\boldsymbol{b}+m\boldsymbol{c}\\
 544              \therefore \boldsymbol{c} &= m\boldsymbol{a} + n\boldsymbol{b} \quad \text{(simultaneous)}
 545            \end{align*}
 546
 547            \noindent \(\boldsymbol{a}, \boldsymbol{b},\) and \(\boldsymbol{c}\) are linearly
 548            independent if no vector in the set is expressible as a linear
 549            combination of other vectors in set, or if they are parallel.
 550
 551            \subsection*{Three-dimensional vectors}
 552
 553            Right-hand rule for axes: \(z\) is up or out of page.
 554
 555            \tdplotsetmaincoords{60}{120} 
 556            \begin{center}\begin{tikzpicture} [scale=3, tdplot_main_coords, axis/.style={->,thick}, 
 557              vector/.style={-stealth,red,very thick}, 
 558              vector guide/.style={dashed,gray,thick}]
 559
 560              %standard tikz coordinate definition using x, y, z coords
 561              \coordinate (O) at (0,0,0);
 562
 563              %tikz-3dplot coordinate definition using x, y, z coords
 564
 565              \pgfmathsetmacro{\ax}{1}
 566              \pgfmathsetmacro{\ay}{1}
 567              \pgfmathsetmacro{\az}{1}
 568
 569              \coordinate (P) at (\ax,\ay,\az);
 570
 571              %draw axes
 572              \draw[axis] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$};
 573              \draw[axis] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$};
 574              \draw[axis] (0,0,0) -- (0,0,1) node[anchor=south]{$z$};
 575
 576              %draw a vector from O to P
 577              \draw[vector] (O) -- (P);
 578
 579              %draw guide lines to components
 580              \draw[vector guide]         (O) -- (\ax,\ay,0);
 581              \draw[vector guide] (\ax,\ay,0) -- (P);
 582              \draw[vector guide]         (P) -- (0,0,\az);
 583              \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
 584              \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
 585              \draw[vector guide] (\ax,\ay,0) -- (\ax,0,0);
 586              \node[tdplot_main_coords,above right]
 587              at (\ax,\ay,\az){(\ax, \ay, \az)};
 588            \end{tikzpicture}\end{center}
 589
 590            \subsection*{Parametric vectors}
 591
 592            Parametric equation of line through point \((x_0, y_0, z_0)\) and
 593            parallel to \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) is:
 594
 595            \[\begin{cases}x = x_o + a \cdot t \\ y = y_0 + b \cdot t \\ z = z_0 + c \cdot t\end{cases}\]
 596
 597              \section{Circular functions}
 598
 599              \(\sin(bx)\) or \(\cos(bx)\): period \(=\frac{2\pi}{b}\)
 600
 601              \noindent \(\tan(nx)\): period \(=\frac{\pi}{n}\)\\
 602              \indent\indent\indent asymptotes at \(x=\frac{(2k+1)\pi}{2n} \> \vert \> k \in \mathbb{Z}\)
 603
 604              \subsection*{Reciprocal functions}
 605
 606              \subsubsection*{Cosecant}
 607
 608              \[\operatorname{cosec} \theta = \frac{1}{\sin \theta} \> \vert \> \sin \theta \ne 0\]
 609
 610              \begin{itemize}
 611                \item
 612                  \textbf{Domain} \(= \mathbb{R} \setminus {n\pi : n \in \mathbb{Z}}\)
 613                \item
 614                  \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
 615                \item
 616                  \textbf{Turning points} at
 617                  \(\theta = \frac{(2n + 1)\pi}{2} \> \vert \> n \in \mathbb{Z}\)
 618                \item
 619                  \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
 620              \end{itemize}
 621
 622              \subsubsection*{Secant}
 623
 624\begin{tikzpicture}
 625  \begin{axis}[ytick={-1,1}, yticklabels={\(-1\), \(1\)}, xmin=-7,xmax=7,ymin=-3,ymax=3,enlargelimits=true, xtick={-6.2830, -3.1415, 3.1415, 6.2830},xticklabels={\(-2\pi\), \(-\pi\), \(\pi\), \(2\pi\)}]
 626%    \addplot[blue, domain=-6.2830:6.2830,unbounded coords=jump,samples=80] {sec(deg(x))};
 627    \addplot[blue, restrict y to domain=-10:10, domain=-7:7,samples=100] {sec(deg(x))} node [pos=0.93, black, right] {\(\operatorname{sec} x\)};
 628    \addplot[red, dashed, domain=-7:7,samples=100] {cos(deg(x))};
 629    \draw [gray, dotted, thick] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{rel axis cs:0,1});
 630    \draw [gray, dotted, thick] ({axis cs:4.71239,0}|-{rel axis cs:0,0}) -- ({axis cs:4.71239,0}|-{rel axis cs:0,1});
 631    \draw [gray, dotted, thick] ({axis cs:-4.71239,0}|-{rel axis cs:0,0}) -- ({axis cs:-4.71239,0}|-{rel axis cs:0,1});
 632    \draw [gray, dotted, thick] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1});
 633\end{axis}
 634    \node [black] at (7,3.5) {\(\cos x\)};
 635\end{tikzpicture}
 636
 637                \[\operatorname{sec} \theta = \frac{1}{\cos \theta} \> \vert \> \cos \theta \ne 0\]
 638
 639                \begin{itemize}
 640
 641                  \item
 642                    \textbf{Domain}
 643                    \(= \mathbb{R} \setminus \frac{(2n + 1) \pi}{2} : n \in \mathbb{Z}\}\)
 644                  \item
 645                    \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
 646                  \item
 647                    \textbf{Turning points} at
 648                    \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
 649                  \item
 650                    \textbf{Asymptotes} at
 651                    \(\theta = \frac{(2n + 1) \pi}{2} \> \vert \> n \in \mathbb{Z}\)
 652                \end{itemize}
 653
 654                \subsubsection*{Cotangent}
 655
 656\begin{tikzpicture}
 657  \begin{axis}[xmin=-3,xmax=3,ymin=-1.5,ymax=1.5,enlargelimits=true, xtick={-3.1415, -1.5708, 1.5708, 3.1415},xticklabels={\(-\pi\), \(-\frac{\pi}{2}\), \(\frac{\pi}{2}\), \(\pi\)}]
 658    \addplot[blue, smooth, domain=-3:-0.1,unbounded coords=jump,samples=105] {cot(deg(x))} node [pos=0.3, left] {\(\operatorname{cot} x\)};
 659\addplot[blue, smooth, domain=0.1:3,unbounded coords=jump,samples=105] {cot(deg(x))};
 660\addplot[red, smooth, dashed] gnuplot [domain=-1.5:1.5,unbounded coords=jump,samples=105] {tan(x)};
 661\addplot[red, smooth, dashed] gnuplot [domain=-3.5:-1.8,unbounded coords=jump,samples=105] {tan(x)} node [pos=0.5, right] {\(\tan x\)};
 662\addplot[red, smooth, dashed] gnuplot [domain=1.8:3.5,unbounded coords=jump,samples=105] {tan(x)};
 663    \draw [thick, red, dotted] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1});
 664    \draw [thick, blue, dotted] ({axis cs:-3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:-3.1415,0}|-{rel axis cs:0,1});
 665    \draw [thick, blue, dotted] ({axis cs:0,0}|-{rel axis cs:0,0}) -- ({axis cs:0,0}|-{rel axis cs:0,1});
 666    \draw [thick, blue, dotted] ({axis cs:3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:3.1415,0}|-{rel axis cs:0,1});
 667    \draw [thick, red, dotted] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{rel axis cs:0,1});
 668\end{axis}
 669\end{tikzpicture}
 670
 671                  \[\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0\]
 672
 673                  \begin{itemize}
 674
 675                    \item
 676                      \textbf{Domain} \(= \mathbb{R} \setminus \{n \pi: n \in \mathbb{Z}\}\)
 677                    \item
 678                      \textbf{Range} \(= \mathbb{R}\)
 679                    \item
 680                      \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
 681                  \end{itemize}
 682
 683                  \subsubsection*{Symmetry properties}
 684
 685                  \[\begin{split}
 686                    \operatorname{sec} (\pi \pm x) & = -\operatorname{sec} x \\
 687                    \operatorname{sec} (-x) & = \operatorname{sec} x \\
 688                    \operatorname{cosec} (\pi \pm x) & = \mp \operatorname{cosec} x \\
 689                    \operatorname{cosec} (-x) & = - \operatorname{cosec} x \\
 690                    \operatorname{cot} (\pi \pm x) & = \pm \operatorname{cot} x \\
 691                    \operatorname{cot} (-x) & = - \operatorname{cot} x
 692                  \end{split}\]
 693
 694                  \subsubsection*{Complementary properties}
 695
 696                  \[\begin{split}
 697                    \operatorname{sec} \left({\pi \over 2} - x\right) & = \operatorname{cosec} x \\
 698                    \operatorname{cosec} \left({\pi \over 2} - x\right) & = \operatorname{sec} x \\
 699                    \operatorname{cot} \left({\pi \over 2} - x\right) & = \tan x \\
 700                    \tan \left({\pi \over 2} - x\right) & = \operatorname{cot} x
 701                  \end{split}\]
 702
 703                  \subsubsection*{Pythagorean identities}
 704
 705                  \[\begin{split}
 706                    1 + \operatorname{cot}^2 x & = \operatorname{cosec}^2 x, \quad \text{where } \sin x \ne 0 \\
 707                    1 + \tan^2 x & = \operatorname{sec}^2 x, \quad \text{where } \cos x \ne 0
 708                  \end{split}\]
 709
 710                  \subsection*{Compound angle formulas}
 711
 712                  \[\cos(x \pm y) = \cos x + \cos y \mp \sin x \sin y\]
 713                  \[\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y\]
 714                  \[\tan(x \pm y) = {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}\]
 715
 716                  \subsection*{Double angle formulas}
 717
 718                  \[\begin{split}
 719                    \cos 2x &= \cos^2 x - \sin^2 x \\
 720                    & = 1 - 2\sin^2 x \\
 721                    & = 2 \cos^2 x -1
 722                  \end{split}\]
 723
 724                  \[\sin 2x = 2 \sin x \cos x\]
 725
 726                  \[\tan 2x = {{2 \tan x} \over {1 - \tan^2 x}}\]
 727
 728                  \subsection*{Inverse circular functions}
 729
 730                  \begin{tikzpicture}
 731                    \begin{axis}[ymin=-2, ymax=4, xmin=-1.1, xmax=1.1, ytick={-1.5708, 1.5708, 3.14159},yticklabels={$-\frac{\pi}{2}$, $\frac{\pi}{2}$, $\pi$}]
 732                      \addplot[color=red, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {asin(x)} node [pos=0.25, below right] {\(\sin^{-1}x\)};
 733                      \addplot[color=blue, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {acos(x)} node [pos=0.25, below left] {\(\cos^{-1}x\)};
 734                      \addplot[mark=*, red] coordinates {(-1,-1.5708)} node[right, font=\footnotesize]{\((-1,-\frac{\pi}{2})\)} ;
 735                      \addplot[mark=*, red] coordinates {(1,1.5708)} node[left, font=\footnotesize]{\((1,\frac{\pi}{2})\)} ;
 736                      \addplot[mark=*, blue] coordinates {(1,0)};
 737                      \addplot[mark=*, blue] coordinates {(-1,3.1415)} node[right, font=\footnotesize]{\((-1,\pi)\)} ;
 738                    \end{axis}
 739                  \end{tikzpicture}\\
 740
 741                  Inverse functions: \(f(f^{-1}(x)) = x\) (restrict domain)
 742
 743                  \[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y\]
 744                  \hfill where \(\sin y = x, \> y \in [{-\pi \over 2}, {\pi \over 2}]\)
 745
 746                  \[\cos^{-1}: [-1,1] \rightarrow \mathbb{R}, \quad \cos^{-1} x = y\]
 747                  \hfill where \(\cos y = x, \> y \in [0, \pi]\)
 748
 749                  \[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\]
 750                  \hfill where \(\tan y = x, \> y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\)
 751
 752                  \begin{tikzpicture}
 753                    \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={\(-\frac{\pi}{2}\),\(\frac{\pi}{2}\)}]
 754                      \addplot[color=orange, smooth] gnuplot [domain=-35:35, unbounded coords=jump,samples=350] {atan(x)} node [pos=0.5, above left] {\(\tan^{-1}x\)};
 755                      \addplot[->, gray, dotted, thick, domain=-35:35] {1.5708};
 756                      \addplot[->, gray, dotted, thick, domain=-35:35] {-1.5708};
 757                    \end{axis}
 758                  \end{tikzpicture}
 759\columnbreak
 760                  \section{Differential calculus}
 761
 762                  \subsection*{Limits}
 763
 764                  \[\lim_{x \rightarrow a}f(x)\]
 765                  \(L^-,\quad L^+\) \qquad limit from below/above\\
 766                  \(\lim_{x \to a} f(x)\) \quad limit of a point\\
 767
 768                  \noindent For solving \(x\rightarrow\infty\), put all \(x\) terms in denominators\\
 769                  e.g. \[\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2\]
 770
 771                  \subsubsection*{Limit theorems}
 772
 773                  \begin{enumerate}
 774                    \item
 775                      For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\)
 776                    \item
 777                      \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\)
 778                    \item
 779                      \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\)
 780                    \item
 781                      \(\therefore \lim_{x \rightarrow a} c \times f(x)=cF\) where \(c=\) constant
 782                    \item
 783                      \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\)
 784                    \item
 785                      \(f(x)\) is continuous \(\iff L^-=L^+=f(x) \> \forall x\)
 786                  \end{enumerate}
 787
 788                  \subsection*{Gradients of secants and tangents}
 789
 790                  \textbf{Secant (chord)} - line joining two points on curve\\
 791                  \textbf{Tangent} - line that intersects curve at one point
 792
 793                  \subsection*{First principles derivative}
 794
 795                  \[f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={\frac{dy}{dx}}\]
 796
 797                  \subsubsection*{Logarithmic identities}
 798
 799                  \(\log_b (xy)=\log_b x + \log_b y\)\\
 800                  \(\log_b x^n = n \log_b x\)\\
 801                  \(\log_b y^{x^n} = x^n \log_b y\)
 802
 803                  \subsubsection*{Index identities}
 804
 805                  \(b^{m+n}=b^m \cdot b^n\)\\
 806                  \((b^m)^n=b^{m \cdot n}\)\\
 807                  \((b \cdot c)^n = b^n \cdot c^n\)\\
 808                  \({a^m \div a^n} = {a^{m-n}}\)
 809
 810                  \subsection*{Derivative rules}
 811
 812                  \renewcommand{\arraystretch}{1.4}
 813                  \begin{tabularx}{\columnwidth}{rX}
 814                    \hline
 815                    \(f(x)\) & \(f^\prime(x)\)\\
 816                    \hline
 817                    \(\sin x\) & \(\cos x\)\\
 818                    \(\sin ax\) & \(a\cos ax\)\\
 819                    \(\cos x\) & \(-\sin x\)\\
 820                    \(\cos ax\) & \(-a \sin ax\)\\
 821                    \(\tan f(x)\) & \(f^2(x) \sec^2f(x)\)\\
 822                    \(e^x\) & \(e^x\)\\
 823                    \(e^{ax}\) & \(ae^{ax}\)\\
 824                    \(ax^{nx}\) & \(an \cdot e^{nx}\)\\
 825                    \(\log_e x\) & \(\dfrac{1}{x}\)\\
 826                    \(\log_e {ax}\) & \(\dfrac{1}{x}\)\\
 827                    \(\log_e f(x)\) & \(\dfrac{f^\prime (x)}{f(x)}\)\\
 828                    \(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\\
 829                    \(\sin^{-1} x\) & \(\dfrac{1}{\sqrt{1-x^2}}\)\\
 830                    \(\cos^{-1} x\) & \(\dfrac{-1}{sqrt{1-x^2}}\)\\
 831                    \(\tan^{-1} x\) & \(\dfrac{1}{1 + x^2}\)\\
 832                    \(\frac{d}{dy}f(y)\) & \(\dfrac{1}{\frac{dx}{dy}}\) (reciprocal)\\
 833                    \(uv\) & \(u \frac{dv}{dx}+v\frac{du}{dx} (product rule)\)\\
 834                    \(\dfrac{u}{v}\) & \(\dfrac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\) (quotient rule)\\
 835                    \(f(g(x))\) & \(f^\prime(g(x))\cdot g^\prime(x)\)\\
 836                    \hline
 837                  \end{tabularx}
 838
 839                  \subsection*{Reciprocal derivatives}
 840
 841                  \[\frac{1}{\frac{dy}{dx}} = \frac{dx}{dy}\]
 842
 843                  \subsection*{Differentiating \(x=f(y)\)}
 844                  \begin{align*}
 845                    \text{Find }& \frac{dx}{dy}\\
 846                    \text{Then, }\frac{dx}{dy} &= \frac{1}{\frac{dy}{dx}} \\
 847                    \implies {\frac{dy}{dx}} &= \frac{1}{\frac{dx}{dy}}\\
 848                    \therefore {\frac{dy}{dx}} &= \frac{1}{\frac{dx}{dy}}
 849                  \end{align*}
 850
 851                  \subsection*{Second derivative}
 852                  \begin{align*}f(x) \longrightarrow &f^\prime (x) \longrightarrow f^{\prime\prime}(x)\\
 853                  \implies y \longrightarrow &\frac{dy}{dx} \longrightarrow \frac{d^2 y}{dx^2}\end{align*}
 854
 855                  \noindent Order of polynomial \(n\)th derivative decrements each time the derivative is taken
 856
 857                  \subsubsection*{Points of Inflection}
 858
 859                  \emph{Stationary point} - i.e.
 860                  \(f^\prime(x)=0\)\\
 861                  \emph{Point of inflection} - max \(|\)gradient\(|\) (i.e.
 862                  \(f^{\prime\prime} = 0\))
 863
 864
 865                  \pgfplotsset{every axis/.append style={
 866                    axis x line=none,    % put the x axis in the middle
 867                    axis y line=none,    % put the y axis in the middle
 868                  }}
 869                  \begin{table*}[ht]
 870                    \centering
 871                    \begin{tabularx}{\textwidth}{rXXX}
 872                      \hline
 873                      \rowcolor{shade2}
 874                      & \centering\(\dfrac{d^2 y}{dx^2} > 0\)  & \centering \(\dfrac{d^2y}{dx^2}<0\) & \(\dfrac{d^2y}{dx^2}=0\) (inflection) \\
 875                      \hline
 876                      \(\frac{dy}{dx}>0\) &
 877                      \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=-3,  xmax=0.8, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(e^(x))};  \addplot[red] {x/2.5+0.75}; \end{axis}\end{tikzpicture} \\Rising (concave up)}&
 878                        \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=0.1, xmax=4,   scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(ln(x))};  \addplot[red] {x/1.5-0.56}; \end{axis}\end{tikzpicture} \\Rising (concave down)}&
 879                          \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=-1.5,  xmax=1.5,   scale=0.2, samples=100] \addplot[blue] {(sin((deg x)))}; \addplot[red] {x}; \end{axis}\end{tikzpicture} \\Rising inflection point}\\
 880                            \hline
 881                            \(\dfrac{dy}{dx}<0\) &
 882                            \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=-.5, xmax=1, ymin=-.5, ymax=.5, scale=0.2, samples=100] \addplot[blue] {(1/(x+1)-1}; \addplot[red] {-x}; \end{axis}\end{tikzpicture} \\Falling (concave up)}&
 883                              \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=0,  xmax=1.5, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(2-x*x)^(1/2)};  \addplot[red] {-x+2}; \end{axis}\end{tikzpicture} \\Falling (concave down)}&
 884                                \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=1.5,  xmax=4.5,   scale=0.2, samples=100] \addplot[blue] {(sin((deg x)))}; \addplot[red] {-x+3.1415}; \end{axis}\end{tikzpicture} \\Falling inflection point}\\
 885                                  \hline
 886                                  \(\dfrac{dy}{dx}=0\)&
 887                                  \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=-1,  xmax=1,   scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(x*x))}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \\Local minimum}&                       \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=-1,  xmax=1,   scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(-x*x))}; \addplot[red, very thick] {0}; \end{axis}\end{tikzpicture} \\Local maximum}&
 888                                    \makecell{\\\begin{tikzpicture}\begin{axis}[xmin=-1,  xmax=1,   scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(x*x*x))}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \(\>\) \begin{tikzpicture}\begin{axis}[xmin=-1,  xmax=1,   scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(-x*x*x))}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture}  \\Stationary inflection point}\\
 889                                      \hline
 890                    \end{tabularx}
 891                  \end{table*}
 892                  \begin{itemize}
 893                    \item
 894                      if \(f^\prime (a) = 0\) and \(f^{\prime\prime}(a) > 0\), then point
 895                      \((a, f(a))\) is a local min (curve is concave up)
 896                    \item
 897                      if \(f^\prime (a) = 0\) and \(f^{\prime\prime} (a) < 0\), then point
 898                      \((a, f(a))\) is local max (curve is concave down)
 899                    \item
 900                      if \(f^{\prime\prime}(a) = 0\), then point \((a, f(a))\) is a point of
 901                      inflection
 902                    \item
 903                      if also \(f^\prime(a)=0\), then it is a stationary point of inflection
 904                  \end{itemize}
 905
 906                  \subsection*{Implicit Differentiation}
 907
 908                  \noindent Used for differentiating circles etc.
 909
 910                  If \(p\) and \(q\) are expressions in \(x\) and \(y\) such that \(p=q\),
 911                  for all \(x\) and \(y\), then:
 912
 913                  \[{\frac{dp}{dx}} = {\frac{dq}{dx}} \quad \text{and} \quad {\frac{dp}{dy}} = {\frac{dq}{dy}}\]
 914
 915                  \noindent \colorbox{cas}{\textbf{On CAS:}}\\
 916                  Action \(\rightarrow\) Calculation \(\rightarrow\) \texttt{impDiff(y\^{}2+ax=5,\ x,\ y)}\\
 917                  Returns \(y^\prime= \dots\).
 918
 919                  \subsection*{Integration}
 920
 921                  \[\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)\]
 922
 923                  \subsection*{Integral laws}
 924
 925                  \renewcommand{\arraystretch}{1.4}
 926                  \begin{tabularx}{\columnwidth}{rX}
 927                    \hline
 928                    \(f(x)\) & \(\int f(x) \cdot dx\) \\
 929                    \hline
 930                    \(k\) (constant) & \(kx + c\)\\
 931                    \(x^n\) & \(\dfrac{1}{n+1} x^{n+1}\) \\
 932                    \(a x^{-n}\) &\(a \cdot \log_e |x| + c\)\\
 933                    \(\dfrac{1}{ax+b}\) &\(\dfrac{1}{a} \log_e (ax+b) + c\)\\
 934                    \((ax+b)^n\) & \(\dfrac{1}{a(n+1)}(ax+b)^{n-1} + c\>|\>n\ne 1\)\\
 935                    \((ax+b)^{-1}\) & \(\dfrac{1}{a}\log_e |ax+b|+c\)\\
 936                    \(e^{kx}\) & \(\dfrac{1}{k} e^{kx} + c\)\\
 937                    \(e^k\) & \(e^kx + c\)\\
 938                    \(\sin kx\) & \(\dfrac{-1}{k} \cos (kx) + c\)\\
 939                    \(\cos kx\) & \(\dfrac{1}{k} \sin (kx) + c\)\\
 940                    \(\sec^2 kx\) & \(\dfrac{1}{k} \tan(kx) + c\)\\
 941                    \(\dfrac{1}{\sqrt{a^2-x^2}}\) & \(\sin^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
 942                    \(\dfrac{-1}{\sqrt{a^2-x^2}}\) & \(\cos^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
 943                    \(\frac{a}{a^2-x^2}\) & \(\tan^{-1} \frac{x}{a} + c\)\\
 944                    \(\frac{f^\prime (x)}{f(x)}\) & \(\log_e f(x) + c\)\\
 945                    \(\int f(u) \cdot \frac{du}{dx} \cdot dx\) & \(\int f(u) \cdot du\) (substitution)\\
 946                    \(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\
 947                    \hline
 948                  \end{tabularx}
 949
 950                  Note \(\sin^{-1} {x \over a} + \cos^{-1} {x \over a}\) is constant \(\forall x \in (-a, a)\)
 951
 952                  \subsection*{Definite integrals}
 953
 954                  \[\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)\]
 955
 956                  \begin{itemize}
 957
 958                    \item
 959                      Signed area enclosed by\\
 960                      \(\> y=f(x), \quad y=0, \quad x=a, \quad x=b\).
 961                    \item
 962                      \emph{Integrand} is \(f\).
 963                  \end{itemize}
 964
 965                  \subsubsection*{Properties}
 966
 967                  \[\int^b_a f(x) \> dx = \int^c_a f(x) \> dx + \int^b_c f(x) \> dx\]
 968
 969                  \[\int^a_a f(x) \> dx = 0\]
 970
 971                  \[\int^b_a k \cdot f(x) \> dx = k \int^b_a f(x) \> dx\]
 972
 973                  \[\int^b_a f(x) \pm g(x) \> dx = \int^b_a f(x) \> dx \pm \int^b_a g(x) \> dx\]
 974
 975                  \[\int^b_a f(x) \> dx = - \int^a_b f(x) \> dx\]
 976
 977                  \subsection*{Integration by substitution}
 978
 979                  \[\int f(u) {\frac{du}{dx}} \cdot dx = \int f(u) \cdot du\]
 980
 981                  \noindent Note \(f(u)\) must be 1:1 \(\implies\) one \(x\) for each \(y\)
 982                  \begin{align*}\text{e.g. for } y&=\int(2x+1)\sqrt{x+4} \cdot dx\\
 983                    \text{let } u&=x+4\\
 984                    \implies& {\frac{du}{dx}} = 1\\
 985                    \implies& x = u - 4\\
 986                    \text{then } &y=\int (2(u-4)+1)u^{\frac{1}{2}} \cdot du\\
 987                    &\text{(solve as  normal integral)}
 988                  \end{align*}
 989
 990                  \subsubsection*{Definite integrals by substitution}
 991
 992                  For \(\int^b_a f(x) {\frac{du}{dx}} \cdot dx\), evaluate new \(a\) and
 993                  \(b\) for \(f(u) \cdot du\).
 994
 995                  \subsubsection*{Trigonometric integration}
 996
 997                  \[\sin^m x \cos^n x \cdot dx\]
 998
 999                  \paragraph{\textbf{\(m\) is odd:}}
1000                  \(m=2k+1\) where \(k \in \mathbb{Z}\)\\
1001                  \(\implies \sin^{2k+1} x = (\sin^2 z)^k \sin x = (1 - \cos^2 x)^k \sin x\)\\
1002                  Substitute \(u=\cos x\)
1003
1004                  \paragraph{\textbf{\(n\) is odd:}}
1005                  \(n=2k+1\) where \(k \in \mathbb{Z}\)\\
1006                  \(\implies \cos^{2k+1} x = (\cos^2 x)^k \cos x = (1-\sin^2 x)^k \cos x\)\\
1007                  Substitute \(u=\sin x\)
1008
1009                  \paragraph{\textbf{\(m\) and \(n\) are even:}}
1010                  use identities...
1011
1012                  \begin{itemize}
1013
1014                    \item
1015                      \(\sin^2x={1 \over 2}(1-\cos 2x)\)
1016                    \item
1017                      \(\cos^2x={1 \over 2}(1+\cos 2x)\)
1018                    \item
1019                      \(\sin 2x = 2 \sin x \cos x\)
1020                  \end{itemize}
1021
1022                  \subsection*{Partial fractions}
1023
1024                  \colorbox{cas}{On CAS:}\\
1025                  \indent Action \(\rightarrow\) Transformation \(\rightarrow\)
1026                  \texttt{expand/combine}\\
1027                  \indent Interactive \(\rightarrow\) Transformation \(\rightarrow\)
1028                  Expand \(\rightarrow\) Partial
1029
1030                  \subsection*{Graphing integrals on CAS}
1031
1032                  \colorbox{cas}{In main:} Interactive \(\rightarrow\) Calculation \(\rightarrow\)
1033                  \(\int\) (\(\rightarrow\) Definite)\\
1034                  Restrictions: \texttt{Define\ f(x)=..} then \texttt{f(x)\textbar{}x\textgreater{}..}
1035
1036                  \subsection*{Applications of antidifferentiation}
1037
1038                  \begin{itemize}
1039
1040                    \item
1041                      \(x\)-intercepts of \(y=f(x)\) identify \(x\)-coordinates of
1042                      stationary points on \(y=F(x)\)
1043                    \item
1044                      nature of stationary points is determined by sign of \(y=f(x)\) on
1045                      either side of its \(x\)-intercepts
1046                    \item
1047                      if \(f(x)\) is a polynomial of degree \(n\), then \(F(x)\) has degree
1048                      \(n+1\)
1049                  \end{itemize}
1050
1051                  To find stationary points of a function, substitute \(x\) value of given
1052                  point into derivative. Solve for \({\frac{dy}{dx}}=0\). Integrate to find
1053                  original function.
1054
1055                  \subsection*{Solids of revolution}
1056
1057                  Approximate as sum of infinitesimally-thick cylinders
1058
1059                  \subsubsection*{Rotation about \(x\)-axis}
1060
1061                  \begin{align*}
1062                    V &= \int^{x=b}_{x-a} \pi y^2 \> dx \\
1063                    &= \pi \int^b_a (f(x))^2 \> dx
1064                  \end{align*}
1065
1066                  \subsubsection*{Rotation about \(y\)-axis}
1067
1068                  \begin{align*}
1069                    V &= \int^{y=b}_{y=a} \pi x^2 \> dy \\
1070                    &= \pi \int^b_a (f(y))^2 \> dy
1071                  \end{align*}
1072
1073                  \subsubsection*{Regions not bound by \(y=0\)}
1074
1075                  \[V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx\]
1076                  \hfill where \(f(x) > g(x)\)
1077
1078                  \subsection*{Length of a curve}
1079
1080                  \[L = \int^b_a \sqrt{1 + ({\frac{dy}{dx}})^2} \> dx \quad \text{(Cartesian)}\]
1081
1082                  \[L = \int^b_a \sqrt{{\frac{dx}{dt}} + ({\frac{dy}{dt}})^2} \> dt \quad \text{(parametric)}\]
1083
1084                  \noindent \colorbox{cas}{On CAS:}\\
1085                  \indent Evaluate formula,\\
1086                  \indent or Interactive \(\rightarrow\) Calculation
1087                  \(\rightarrow\) Line \(\rightarrow\) \texttt{arcLen}
1088
1089                  \subsection*{Rates}
1090
1091                  \subsubsection*{Gradient at a point on parametric curve}
1092
1093                  \[{\frac{dy}{dx}} = {{\frac{dy}{dt}} \div {\frac{dx}{dt}}} \> \vert \> {\frac{dx}{dt}} \ne 0\]
1094
1095                  \[\frac{d^2}{dx^2} = \frac{d(y^\prime)}{dx} = {\frac{dy^\prime}{dt} \div {\frac{dx}{dt}}} \> \vert \> y^\prime = {\frac{dy}{dx}}\]
1096
1097                  \subsection*{Rational functions}
1098
1099                  \[f(x) = \frac{P(x)}{Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}\]
1100
1101                  \subsubsection*{Addition of ordinates}
1102
1103                  \begin{itemize}
1104
1105                    \item
1106                      when two graphs have the same ordinate, \(y\)-coordinate is double the
1107                      ordinate
1108                    \item
1109                      when two graphs have opposite ordinates, \(y\)-coordinate is 0 i.e.
1110                      (\(x\)-intercept)
1111                    \item
1112                      when one of the ordinates is 0, the resulting ordinate is equal to the
1113                      other ordinate
1114                  \end{itemize}
1115
1116                  \subsection*{Fundamental theorem of calculus}
1117
1118                  If \(f\) is continuous on \([a, b]\), then
1119
1120                  \[\int^b_a f(x) \> dx = F(b) - F(a)\]
1121                  \hfill where \(F = \int f \> dx\)
1122                  
1123                  \subsection*{Differential equations}
1124
1125                  \noindent\textbf{Order} - highest power inside derivative\\
1126                  \textbf{Degree} - highest power of highest derivative\\
1127                  e.g. \({\left(\dfrac{dy^2}{d^2} x\right)}^3\) \qquad order 2, degree 3
1128
1129                  \subsubsection*{Verifying solutions}
1130
1131                  Start with \(y=\dots\), and differentiate. Substitute into original
1132                  equation.
1133
1134                  \subsubsection*{Function of the dependent
1135                  variable}
1136
1137                  If \({\frac{dy}{dx}}=g(y)\), then
1138                  \(\frac{dx}{dy} = 1 \div \frac{dy}{dx} = \frac{1}{g(y)}\). Integrate both sides to solve equation. Only add \(c\) on one side. Express
1139                  \(e^c\) as \(A\).
1140
1141
1142
1143                  \subsubsection*{Mixing problems}
1144
1145                  \[\left(\frac{dm}{dt}\right)_\Sigma = \left(\frac{dm}{dt}\right)_{\text{in}} - \left(\frac{dm}{dt}_{\text{out}}\right)\]
1146
1147                  \subsubsection*{Separation of variables}
1148
1149                  If \({\frac{dy}{dx}}=f(x)g(y)\), then:
1150
1151                  \[\int f(x) \> dx = \int \frac{1}{g(y)} \> dy\]
1152
1153                  \subsubsection*{Euler's method for solving DEs}
1154
1155                  \[\frac{f(x+h) - f(x)}{h} \approx f^\prime (x) \quad \text{for small } h\]
1156
1157                  \[\implies f(x+h) \approx f(x) + hf^\prime(x)\]
1158
1159                \end{multicols}
1160              \end{document}