f04ab637a29d3c44f0d6953c8452833b2f0ca23b
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   7\usepackage{harpoon}
   8\usepackage{tabularx}
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  18\usetikzlibrary{calc}
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  20\usetikzlibrary{datavisualization.formats.functions}
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  23\usepackage{fancyhdr}
  24\pagestyle{fancy}
  25\fancyhead[LO,LE]{Year 12 Specialist}
  26\fancyhead[CO,CE]{Andrew Lorimer}
  27
  28\usepackage{mathtools}
  29\usepackage{xcolor} % used only to show the phantomed stuff
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  51                  }}
  52\begin{document}
  53
  54\begin{multicols}{2}
  55
  56  \section{Complex numbers}
  57
  58  \[\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}\]
  59
  60  \begin{align*}
  61    \text{Cartesian form: } & a+bi\\
  62    \text{Polar form: } & r\operatorname{cis}\theta
  63  \end{align*}
  64
  65  \subsection*{Operations}
  66
  67  \definecolor{shade1}{HTML}{ffffff}
  68  \definecolor{shade2}{HTML}{e6f2ff}
  69  \definecolor{shade3}{HTML}{cce2ff}
  70  \begin{tabularx}{\columnwidth}{r|X|X}
  71    & \textbf{Cartesian} & \textbf{Polar} \\
  72    \hline
  73    \(z_1 \pm z_2\) & \((a \pm c)(b \pm d)i\) & convert to \(a+bi\)\\
  74    \hline
  75    \(+k \times z\) & \multirow{2}{*}{\(ka \pm kbi\)} & \(kr\operatorname{cis} \theta\)\\
  76    \cline{1-1}\cline{3-3}
  77    \(-k \times z\) & & \(kr \operatorname{cis}(\theta\pm \pi)\)\\
  78    \hline
  79    \(z_1 \cdot z_2\) & \(ac-bd+(ad+bc)i\) & \(r_1r_2 \operatorname{cis}(\theta_1 + \theta_2)\)\\
  80    \hline
  81    \(z_1 \div z_2\) & \((z_1 \overline{z_2}) \div |z_2|^2\) & \(\left(\frac{r_1}{r_2}\right) \operatorname{cis}(\theta_1 - \theta_2)\)
  82  \end{tabularx}
  83
  84  \subsubsection*{Scalar multiplication in polar form}
  85
  86  For \(k \in \mathbb{R}^+\):
  87  \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\theta\]
  88
  89  \noindent For \(k \in \mathbb{R}^-\):
  90  \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\left(\begin{cases}\theta - \pi & |0<\operatorname{Arg}(z)\le \pi \\ \theta + \pi & |-\pi<\operatorname{Arg}(z)\le 0\end{cases}\right)\]
  91
  92    \subsection*{Conjugate}
  93
  94    \begin{align*}
  95      \overline{z} &= a \mp bi\\
  96      &= r \operatorname{cis}(-\theta)
  97    \end{align*}
  98
  99    \noindent \colorbox{cas}{On CAS: \texttt{conjg(a+bi)}}
 100
 101    \subsubsection*{Properties}
 102
 103    \begin{align*}
 104      \overline{z_1 \pm z_2} &= \overline{z_1}\pm\overline{z_2}\\
 105      \overline{z_1 \cdot z_2} &= \overline{z_1}\cdot\overline{z_2}\\
 106      \overline{kz} &= k\overline{z} \quad | \quad k \in \mathbb{R}\\
 107      z\overline{z} &= (a+bi)(a-bi)\\
 108      &= a^2 + b^2\\
 109      &= |z|^2
 110    \end{align*}
 111
 112    \subsection*{Modulus}
 113
 114    \[|z|=|\vec{Oz}|=\sqrt{a^2 + b^2}\]
 115
 116    \subsubsection*{Properties}
 117
 118    \begin{align*}
 119      |z_1z_2|&=|z_1||z_2|\\
 120      \left|\frac{z_1}{z_2}\right|&=\frac{|z_1|}{|z_2|}\\
 121      |z_1+z_2|&\le|z_1|+|z_2|
 122    \end{align*}
 123
 124    \subsection*{Multiplicative inverse}
 125
 126    \begin{align*}
 127      z^{-1}&=\frac{a-bi}{a^2+b^2}\\
 128      &=\frac{\overline{z}}{|z|^2}a\\
 129      &=r \operatorname{cis}(-\theta)
 130    \end{align*}
 131
 132    \subsection*{Dividing over \(\mathbb{C}\)}
 133
 134    \begin{align*}
 135      \frac{z_1}{z_2}&=z_1z_2^{-1}\\
 136      &=\frac{z_1\overline{z_2}}{|z_2|^2}\\
 137      &=\frac{(a+bi)(c-di)}{c^2+d^2}\\
 138      & \qquad \text{(rationalise denominator)}
 139    \end{align*}
 140
 141    \subsection*{Polar form}
 142
 143    \begin{align*}
 144      z&=r\operatorname{cis}\theta\\
 145      &=r(\cos \theta + i \sin \theta)
 146    \end{align*}
 147
 148    \begin{itemize}
 149      \item{\(r=|z|=\sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}\)}
 150      \item{\(\theta = \operatorname{arg}(z)\) \quad \colorbox{cas}{On CAS: \texttt{arg(a+bi)}}}
 151      \item{\(\operatorname{Arg}(z) \in (-\pi,\pi)\) \quad \bf{(principal argument)}}
 152      \item{\colorbox{cas}{Convert on CAS:}\\ \verb|compToTrig(a+bi)| \(\iff\) \verb|cExpand{r·cisX}|}
 153      \item{Multiple representations:\\\(r\operatorname{cis}\theta=r\operatorname{cis}(\theta+2n\pi)\) with \(n \in \mathbb{Z}\) revolutions}
 154      \item{\(\operatorname{cis}\pi=-1,\qquad \operatorname{cis}0=1\)}
 155    \end{itemize}
 156
 157    \subsection*{de Moivres' theorem}
 158
 159    \[(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}\]
 160
 161    \subsection*{Complex polynomials}
 162
 163    Include \(\pm\) for all solutions, incl. imaginary
 164
 165    \begin{tabularx}{\columnwidth}{ R{0.55} X  }
 166      \hline
 167      Sum of squares & \(\begin{aligned} 
 168        z^2 + a^2 &= z^2-(ai)^2\\
 169      &= (z+ai)(z-ai) \end{aligned}\) \\
 170      \hline
 171      Sum of cubes & \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)\\
 172      \hline
 173      Division & \(P(z)=D(z)Q(z)+R(z)\) \\
 174      \hline
 175      Remainder theorem & Let \(\alpha \in \mathbb{C}\). Remainder of \(P(z) \div (z-\alpha)\) is \(P(\alpha)\)\\
 176      \hline
 177      Factor theorem & \(z-\alpha\) is a factor of \(P(z) \iff P(\alpha)=0\) for \(\alpha \in \mathbb{C}\)\\
 178      \hline
 179      Conjugate root theorem & \(P(z)=0 \text{ at } z=a\pm bi\) (\(\implies\) both \(z_1\) and \(\overline{z_1}\) are solutions)\\
 180      \hline
 181    \end{tabularx}
 182
 183    \subsection*{\(n\)th roots}
 184
 185    \(n\)th roots of \(z=r\operatorname{cis}\theta\) are:
 186
 187    \[z = r^{\frac{1}{n}} \operatorname{cis}\left(\frac{\theta+2k\pi}{n}\right)\]
 188
 189    \begin{itemize}
 190
 191      \item{Same modulus for all solutions}
 192      \item{Arguments separated by \(\frac{2\pi}{n} \therefore\) there are \(n\) roots}
 193      \item{If one square root is \(a+bi\), the other is \(-a-bi\)}
 194      \item{Give one implicit \(n\)th root \(z_1\), function is \(z=z_1^n\)}
 195      \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 \(\frac{2\pi}{n}\))}
 196    \end{itemize}
 197
 198    \noindent For \(0=az^2+bz+c\), use quadratic formula:
 199
 200    \[z=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]
 201
 202    \subsection*{Fundamental theorem of algebra}
 203
 204    A polynomial of degree \(n\) can be factorised into \(n\) linear factors in \(\mathbb{C}\):
 205
 206    \[\implies P(z)=a_n(z-\alpha_1)(z-\alpha_2)(z-\alpha_3)\dots(z-\alpha_n)\]
 207    \[\text{ where } \alpha_1,\alpha_2,\alpha_3,\dots,\alpha_n \in \mathbb{C}\]
 208
 209    \subsection*{Argand planes}
 210
 211    \begin{center}\begin{tikzpicture}[scale=2]
 212      \draw [->] (-0.2,0) -- (1.5,0) node [right]  {$\operatorname{Re}(z)$};
 213      \draw [->] (0,-0.2) -- (0,1.5) node [above] {$\operatorname{Im}(z)$};
 214      \coordinate (P) at (1,1);
 215      \coordinate (a) at (1,0);
 216      \coordinate (b) at (0,1);
 217      \coordinate (O) at (0,0);
 218      \draw (0,0) -- (P) node[pos=0.5, above left]{\(r\)} node[pos=1, right]{\(\begin{aligned}z&=a+bi\\&=r\operatorname{cis}\theta\end{aligned}\)};
 219        \draw [gray, dashed] (1,1) -- (1,0) node[black, pos=1, below]{\(a\)};
 220        \draw [gray, dashed] (1,1) -- (0,1) node[black, pos=1, left]{\(b\)};
 221        \begin{scope}
 222          \path[clip] (O) -- (P) -- (a);
 223          \fill[red, opacity=0.5, draw=black] (O) circle (2mm);
 224          \node at ($(O)+(20:3mm)$) {$\theta$};
 225        \end{scope}
 226        \filldraw (P) circle (0.5pt);
 227    \end{tikzpicture}\end{center}
 228
 229    \begin{itemize}
 230      \item{Multiplication by \(i \implies\) CCW rotation of \(\frac{\pi}{2}\)}
 231      \item{Addition: \(z_1 + z_2 \equiv\) \overrightharp{\(Oz_1\)} + \overrightharp{\(Oz_2\)}}
 232    \end{itemize}
 233
 234    \subsection*{Sketching complex graphs}
 235
 236    \subsubsection*{Linear}
 237
 238    \begin{itemize}
 239      \item{\(\operatorname{Re}(z)=c\) or \(\operatorname{Im}(z)=c\) (perpendicular bisector)}
 240      \item{\(\operatorname{Im}(z)=m\operatorname{Re}(z)\)}
 241      \item{\(|z+a|=|z+b| \implies 2(a-b)x=b^2-a^2\)\\Geometric: equidistant from \(a,b\)}
 242    \end{itemize}
 243
 244    \subsubsection*{Circles}
 245
 246    \begin{itemize}
 247      \item \(|z-z_1|^2=c^2|z_2+2|^2\)
 248      \item \(|z-(a+bi)|=c \implies (x-a)^2+_(y-b)^2=c^2\)
 249    \end{itemize}
 250
 251    \noindent \textbf{Loci} \qquad \(\operatorname{Arg}(z)<\theta\)
 252
 253    \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
 254      \draw [->] (0,0) -- (1,0) node [right]  {$\operatorname{Re}(z)$};
 255      \draw [->] (0,-0.5) -- (0,1) node [above] {$\operatorname{Im}(z)$};
 256      \draw [<-, dashed, thick, blue] (-1,0) -- (0,0);
 257      \draw [->, thick, blue] (0,0) -- (1,1);
 258      \fill [gray, opacity=0.2, domain=-1:1, variable=\x] (-1,-0.5) -- (-1,0) -- (0, 0) -- (1,1) -- (1,-0.5) -- cycle;
 259      \begin{scope}
 260        \path[clip] (0,0) -- (1,1) -- (1,0);
 261        \fill[red, opacity=0.5, draw=black] (0,0) circle (2mm);
 262        \node at ($(0,0)+(20:3mm)$) {$\frac{\pi}{4}$};
 263      \end{scope}
 264      \node [font=\footnotesize] at (0.5,-0.25) {\(\operatorname{Arg}(z)\le\frac{\pi}{4}\)};
 265      \node [blue, mydot] {};
 266    \end{tikzpicture}\end{center}
 267
 268    \noindent \textbf{Rays} \qquad \(\operatorname{Arg}(z-b)=\theta\)
 269
 270    \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
 271      \draw [->] (-0.75,0) -- (1.5,0) node [right]  {$\operatorname{Re}(z)$};
 272      \draw [->] (0,-1) -- (0,1) node [above] {$\operatorname{Im}(z)$};
 273      \draw [->, thick, brown] (-0.25,0) -- (-0.75,-1);
 274      \node [above, font=\footnotesize] at (-0.25,0) {\(\frac{1}{4}\)};
 275      \begin{scope}
 276        \path[clip] (-0.25,0) -- (-0.75,-1) -- (0,0);
 277        \fill[orange, opacity=0.5, draw=black] (-0.25,0) circle (2mm);
 278      \end{scope}
 279      \node at (-0.08,-0.3) {\(\frac{\pi}{8}\)};
 280      \node [font=\footnotesize, left] at (-0.75,-1) {\(\operatorname{Arg}(z+\frac{1}{4})=\frac{\pi}{8}\)};
 281      \node [brown, mydot] at (-0.25,0) {};
 282      \draw [<->, thick, green] (0,-1) -- (1.5,0.5) node [pos=0.25, black, font=\footnotesize, right] {\(|z-2|=|z-(1+i)|\)};
 283      \node [left, font=\footnotesize] at (0,-1) {\(-1\)};
 284      \node [below, font=\footnotesize] at (1,0) {\(1\)};
 285    \end{tikzpicture}\end{center}
 286
 287    \section{Vectors}
 288    \begin{center}\begin{tikzpicture}
 289      \draw [->] (-0.5,0) -- (3,0) node [right]  {\(x\)};
 290      \draw [->] (0,-0.5) -- (0,3) node [above] {\(y\)};
 291      \draw [orange, ->, thick] (0.5,0.5) -- (2.5,2.5) node [pos=0.5, above] {\(\vec{u}\)};
 292      \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
 293        \draw [gray, dashed, thick] (0.5,0.5) -- (2.5,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, below]{\(x\vec{i}\)};
 294        \draw [gray, dashed, thick] (2.5,0.5) -- (2.5,2.5) node [pos=0.5] {\midarrow};
 295      \end{scope}
 296      \node[black, right] at (2.5,1.5) {\(y\vec{j}\)};
 297    \end{tikzpicture}\end{center}
 298    \subsection*{Column notation}
 299
 300    \[\begin{bmatrix}x\\ y \end{bmatrix} \iff x\boldsymbol{i} + y\boldsymbol{j}\]
 301      \(\begin{bmatrix}x_2-x_1\\ y_2-y_1 \end{bmatrix}\) \quad between \(A(x_1,y_1), \> B(x_2,y_2)\)
 302
 303        \subsection*{Scalar multiplication}
 304
 305        \[k\cdot (x\boldsymbol{i}+y\boldsymbol{j})=kx\boldsymbol{i}+ky\boldsymbol{j}\]
 306
 307        \noindent For \(k \in \mathbb{R}^-\), direction is reversed
 308
 309        \subsection*{Vector addition}
 310        \begin{center}\begin{tikzpicture}[scale=1]
 311          \coordinate (A) at (0,0);
 312          \coordinate (B) at (2,2);
 313          \draw [->, thick, red] (0,0) -- (2,2) node [pos=0.5, below right] {\(\vec{u}=2\vec{i}+2\vec{j}\)};
 314          \draw [->, thick, blue] (2,2) -- (1,4) node [pos=0.5, above right] {\(\vec{v}=-\vec{i}+2\vec{j}\)};
 315          \draw [->, thick, orange] (0,0) -- (1,4) node [pos=0.5, left] {\(\vec{u}+\vec{v}=\vec{i}+4\vec{j}\)};
 316        \end{tikzpicture}\end{center}
 317
 318        \[(x\boldsymbol{i}+y\boldsymbol{j}) \pm (a\boldsymbol{i}+b\boldsymbol{j})=(x \pm a)\boldsymbol{i}+(y \pm b)\boldsymbol{j}\]
 319
 320        \begin{itemize}
 321          \item Draw each vector head to tail then join lines
 322          \item Addition is commutative (parallelogram)
 323          \item \(\boldsymbol{u}-\boldsymbol{v}=\boldsymbol{u}+(-\boldsymbol{v}) \implies \overrightharp{AB}=\boldsymbol{b}-\boldsymbol{a}\)
 324        \end{itemize}
 325
 326        \subsection*{Magnitude}
 327
 328        \[|(x\boldsymbol{i} + y\boldsymbol{j})|=\sqrt{x^2+y^2}\]
 329
 330        \subsection*{Parallel vectors}
 331
 332        \[\boldsymbol{u} || \boldsymbol{v} \iff \boldsymbol{u} = k \boldsymbol{v} \text{ where } k \in \mathbb{R} \setminus \{0\}\]
 333
 334        For parallel vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\):\\
 335        \[\boldsymbol{a \cdot b}=\begin{cases}
 336          |\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction}\\
 337          -|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions}
 338        \end{cases}\]
 339        %\includegraphics[width=0.2,height=\textheight]{graphics/parallelogram-vectors.jpg}
 340        %\includegraphics[width=1]{graphics/vector-subtraction.jpg}
 341
 342        \subsection*{Perpendicular vectors}
 343
 344        \[\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b} = 0\ \quad \text{(since \(\cos 90 = 0\))}\]
 345
 346        \subsection*{Unit vector \(|\hat{\boldsymbol{a}}|=1\)}
 347        \[\begin{split}\hat{\boldsymbol{a}} & = {\frac{1}{|\boldsymbol{a}|}}\boldsymbol{a} \\ & = \boldsymbol{a} \cdot {|\boldsymbol{a}|}\end{split}\]
 348
 349          \subsection*{Scalar product \(\boldsymbol{a} \cdot \boldsymbol{b}\)}
 350
 351
 352          \begin{center}\begin{tikzpicture}[scale=2]
 353            \draw [->] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{b}\)};
 354            \draw [->] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{a}\)};
 355            \begin{scope}
 356              \path[clip] (1,0.5) -- (1,0) -- (0,0);
 357              \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
 358              \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
 359            \end{scope}
 360          \end{tikzpicture}\end{center}
 361          \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*}
 362            \noindent\colorbox{cas}{On CAS: \texttt{dotP({[}a\ b\ c{]},\ {[}d\ e\ f{]})}}
 363
 364            \subsubsection*{Properties}
 365
 366            \begin{enumerate}
 367              \item
 368                \(k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k\boldsymbol{b})\)
 369              \item
 370                \(\boldsymbol{a \cdot 0}=0\)
 371              \item
 372                \(\boldsymbol{a} \cdot (\boldsymbol{b} + \boldsymbol{c})=\boldsymbol{a} \cdot \boldsymbol{b} + \boldsymbol{a} \cdot \boldsymbol{c}\)
 373              \item
 374                \(\boldsymbol{i \cdot i} = \boldsymbol{j \cdot j} = \boldsymbol{k \cdot k}= 1\)
 375              \item
 376                \(\boldsymbol{a} \cdot \boldsymbol{b} = 0 \quad \implies \quad \boldsymbol{a} \perp \boldsymbol{b}\)
 377              \item
 378                \(\boldsymbol{a \cdot a} = |\boldsymbol{a}|^2 = a^2\)
 379            \end{enumerate}
 380
 381            \subsection*{Angle between vectors}
 382
 383            \[\cos \theta = \frac{\boldsymbol{a} \cdot \boldsymbol{b}}{|\boldsymbol{a}| |\boldsymbol{b}|} = \frac{a_1 b_1 + a_2 b_2}{|\boldsymbol{a}| |\boldsymbol{b}|}\]
 384
 385            \noindent \colorbox{cas}{On CAS:} \texttt{angle([a b c], [a b c])}
 386
 387            (Action \(\rightarrow\) Vector \(\rightarrow\)Angle)
 388
 389            \subsection*{Angle between vector and axis}
 390
 391            \noindent For\(\boldsymbol{a} = a_1 \boldsymbol{i} + a_2 \boldsymbol{j} + a_3 \boldsymbol{k}\)
 392            which makes angles \(\alpha, \beta, \gamma\) with positive side of
 393            \(x, y, z\) axes:
 394            \[\cos \alpha = \frac{a_1}{|\boldsymbol{a}|}, \quad \cos \beta = \frac{a_2}{|\boldsymbol{a}|}, \quad \cos \gamma = \frac{a_3}{|\boldsymbol{a}|}\]
 395
 396            \noindent \colorbox{cas}{On CAS:} \texttt{angle({[}a\ b\ c{]},\ {[}1\ 0\ 0{]})}\\for angle
 397            between \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) and
 398            \(x\)-axis
 399
 400            \subsection*{Projections \& resolutes}
 401
 402            \begin{tikzpicture}[scale=3]
 403              \draw [->, purple] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{a}\)};
 404              \draw [->, orange] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{u}\)};
 405              \draw [->, blue] (1,0) -- (2,0) node [pos=0.5, below] {\(\boldsymbol{b}\)};
 406              \begin{scope}
 407                \path[clip] (1,0.5) -- (1,0) -- (0,0);
 408                \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
 409                \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
 410              \end{scope}
 411              \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
 412                \draw [gray, dashed, thick] (1,0) -- (1,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, right, rotate=-90]{\(\boldsymbol{w}\)};
 413              \end{scope}
 414              \draw (0,0) coordinate (O)
 415              (1,0) coordinate (A)
 416              (1,0.5) coordinate (B)
 417              pic [draw,red,angle radius=2mm] {right angle = O--A--B};
 418            \end{tikzpicture}
 419
 420            \subsubsection*{\(\parallel\boldsymbol{b}\) (vector projection/resolute)}
 421
 422            \begin{align*}
 423              \boldsymbol{u} & = \frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|^2}\boldsymbol{b} \\
 424              & = \left(\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\right)\left(\frac{\boldsymbol{b}}{|\boldsymbol{b}|}\right) \\
 425              & = (\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}
 426            \end{align*}
 427
 428            \subsubsection*{\(\perp\boldsymbol{b}\) (perpendicular projection)}
 429            \[\boldsymbol{w} = \boldsymbol{a} - \boldsymbol{u}\]
 430
 431            \subsubsection*{\(|\boldsymbol{u}|\) (scalar projection/resolute)}
 432            \begin{align*}
 433              s &= |\boldsymbol{u}|\\
 434              &= \boldsymbol{a} \cdot \hat{\boldsymbol{b}}\\
 435              &=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\\
 436              &= |\boldsymbol{a}| \cos \theta
 437            \end{align*}
 438
 439            \subsubsection*{Rectangular (\(\parallel,\perp\)) components}
 440
 441            \[\boldsymbol{a}=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}+\left(\boldsymbol{a}-\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}\right)\]
 442
 443
 444            \subsection*{Vector proofs}
 445
 446            \textbf{Concurrent:} intersection of \(\ge\) 3 lines
 447
 448            \begin{tikzpicture}
 449              \draw [blue] (0,0) -- (1,1);
 450              \draw [red] (1,0) -- (0,1);
 451              \draw [brown] (0.4,0) -- (0.6,1);
 452              \filldraw (0.5,0.5) circle (2pt);
 453            \end{tikzpicture}
 454
 455            \subsubsection*{Collinear points}
 456
 457            \(\ge\) 3 points lie on the same line
 458
 459            \begin{tikzpicture}
 460              \draw [purple] (0,0) -- (4,1);
 461              \filldraw (2,0.5) circle (2pt) node [above] {\(C\)};
 462              \filldraw (1,0.25) circle (2pt) node [above] {\(A\)};
 463              \filldraw (3,0.75) circle (2pt) node [above] {\(B\)};
 464              \coordinate (O) at (2.8,-0.2);
 465              \node at (O) [below] {\(O\)}; 
 466              \begin{scope}[->, orange, thick] 
 467                \draw (O) -- (2,0.5) node [pos=0.5, above, font=\footnotesize, black] {\(\boldsymbol{c}\)};
 468                \draw (O) -- (1,0.25) node [pos=0.5, below, font=\footnotesize, black] {\(\boldsymbol{a}\)};
 469                \draw (O) -- (3,0.75) node [pos=0.5, right, font=\footnotesize, black] {\(\boldsymbol{b}\)};
 470              \end{scope}
 471            \end{tikzpicture}
 472
 473            \begin{align*}
 474              \text{e.g. Prove that}\\
 475              \overrightharp{AC}=m\overrightharp{AB} \iff \boldsymbol{c}&=(1-m)\boldsymbol{a}+m\boldsymbol{b}\\
 476              \implies \boldsymbol{c} &= \overrightharp{OA} + \overrightharp{AC}\\
 477              &= \overrightharp{OA} + m\overrightharp{AB}\\
 478              &=\boldsymbol{a}+m(\boldsymbol{b}-\boldsymbol{a})\\
 479              &=\boldsymbol{a}+m\boldsymbol{b}-m\boldsymbol{a}\\
 480              &=(1-m)\boldsymbol{a}+m{b}
 481            \end{align*}
 482            \begin{align*}
 483              \text{Also, } \implies \overrightharp{OC} &= \lambda \vec{OA} + \mu \overrightharp{OB} \\
 484              \text{where } \lambda + \mu &= 1\\
 485              \text{If } C \text{ lies along } \overrightharp{AB}, & \implies 0 < \mu < 1
 486            \end{align*}
 487
 488
 489            \subsubsection*{Parallelograms}
 490
 491            \begin{center}\begin{tikzpicture}
 492              \coordinate (O) at (0,0) node [below left] {\(O\)};
 493              \coordinate (A) at (4,0);
 494              \coordinate (B) at (6,2);
 495              \coordinate (C) at (2,2);
 496              \coordinate (D) at (6,0);
 497
 498              \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (O)--(A) node [below left] {\(A\)};
 499              \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (A)--(B) node [above right] {\(B\)};
 500              \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (B)--(C) node [above left] {\(C\)};
 501              \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (C)--(O);
 502
 503              \draw [gray, dashed] (O) -- (B) node [pos=0.75] {\(\diagdown\diagdown\)} node [pos=0.25] {\(\diagdown\diagdown\)};
 504              \draw [gray, dashed] (A) -- (C) node [pos=0.75] {\(\diagup\)} node [pos=0.25] {\(\diagup\)};
 505              \begin{scope}
 506                \path[clip] (C) -- (A) -- (O);
 507                \fill[orange, opacity=0.5, draw=black] (0,0) circle (4mm);
 508                \node at ($(0,0)+(20:8mm)$) {\(\theta\)};
 509              \end{scope}
 510              \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] {\(|\boldsymbol{c}|\cos\theta\)};
 511              \draw pic [draw,thick,red,angle radius=2mm] {right angle=O--D--B};
 512            \end{tikzpicture}\end{center}
 513
 514            \begin{itemize}
 515              \item
 516                Diagonals \(\overrightharp{OB}, \overrightharp{AC}\) bisect each other
 517              \item
 518                If diagonals are equal length, it is a rectangle
 519              \item
 520                \(|\overrightharp{OB}|^2 + |\overrightharp{CA}|^2 = |\overrightharp{OA}|^2 + |\overrightharp{AB}|^2 + |\overrightharp{CB}|^2 + |\overrightharp{OC}|^2\)
 521              \item
 522                Area \(=\boldsymbol{c} \cdot \boldsymbol{a}\)
 523            \end{itemize}
 524
 525            \subsubsection*{Useful vector properties}
 526
 527            \begin{itemize}
 528              \item
 529                \(\boldsymbol{a} \parallel \boldsymbol{b} \implies \boldsymbol{b}=k\boldsymbol{a}\) for some
 530                \(k \in \mathbb{R} \setminus \{0\}\)
 531              \item
 532                If \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are parallel with at
 533                least one point in common, then they lie on the same straight line
 534              \item
 535                \(\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b}=0\)
 536              \item
 537                \(\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2\)
 538            \end{itemize}
 539
 540            \subsection*{Linear dependence}
 541
 542            \(\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\) are linearly dependent if they are \(\nparallel\) and:
 543            \begin{align*}
 544              0&=k\boldsymbol{a}+l\boldsymbol{b}+m\boldsymbol{c}\\
 545              \therefore \boldsymbol{c} &= m\boldsymbol{a} + n\boldsymbol{b} \quad \text{(simultaneous)}
 546            \end{align*}
 547
 548            \noindent \(\boldsymbol{a}, \boldsymbol{b},\) and \(\boldsymbol{c}\) are linearly
 549            independent if no vector in the set is expressible as a linear
 550            combination of other vectors in set, or if they are parallel.
 551
 552            \subsection*{Three-dimensional vectors}
 553
 554            Right-hand rule for axes: \(z\) is up or out of page.
 555
 556            \tdplotsetmaincoords{60}{120} 
 557            \begin{center}\begin{tikzpicture} [scale=3, tdplot_main_coords, axis/.style={->,thick}, 
 558              vector/.style={-stealth,red,very thick}, 
 559              vector guide/.style={dashed,gray,thick}]
 560
 561              %standard tikz coordinate definition using x, y, z coords
 562              \coordinate (O) at (0,0,0);
 563
 564              %tikz-3dplot coordinate definition using x, y, z coords
 565
 566              \pgfmathsetmacro{\ax}{1}
 567              \pgfmathsetmacro{\ay}{1}
 568              \pgfmathsetmacro{\az}{1}
 569
 570              \coordinate (P) at (\ax,\ay,\az);
 571
 572              %draw axes
 573              \draw[axis] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$};
 574              \draw[axis] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$};
 575              \draw[axis] (0,0,0) -- (0,0,1) node[anchor=south]{$z$};
 576
 577              %draw a vector from O to P
 578              \draw[vector] (O) -- (P);
 579
 580              %draw guide lines to components
 581              \draw[vector guide]         (O) -- (\ax,\ay,0);
 582              \draw[vector guide] (\ax,\ay,0) -- (P);
 583              \draw[vector guide]         (P) -- (0,0,\az);
 584              \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
 585              \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
 586              \draw[vector guide] (\ax,\ay,0) -- (\ax,0,0);
 587              \node[tdplot_main_coords,above right]
 588              at (\ax,\ay,\az){(\ax, \ay, \az)};
 589            \end{tikzpicture}\end{center}
 590
 591            \subsection*{Parametric vectors}
 592
 593            Parametric equation of line through point \((x_0, y_0, z_0)\) and
 594            parallel to \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) is:
 595
 596            \[\begin{cases}x = x_o + a \cdot t \\ y = y_0 + b \cdot t \\ z = z_0 + c \cdot t\end{cases}\]
 597
 598              \section{Circular functions}
 599
 600              \(\sin(bx)\) or \(\cos(bx)\): period \(=\frac{2\pi}{b}\)
 601
 602              \noindent \(\tan(nx)\): period \(=\frac{\pi}{n}\)\\
 603              \indent\indent\indent asymptotes at \(x=\frac{(2k+1)\pi}{2n} \> \vert \> k \in \mathbb{Z}\)
 604
 605              \subsection*{Reciprocal functions}
 606
 607              \subsubsection*{Cosecant}
 608
 609              \[\operatorname{cosec} \theta = \frac{1}{\sin \theta} \> \vert \> \sin \theta \ne 0\]
 610
 611              \begin{itemize}
 612                \item
 613                  \textbf{Domain} \(= \mathbb{R} \setminus {n\pi : n \in \mathbb{Z}}\)
 614                \item
 615                  \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
 616                \item
 617                  \textbf{Turning points} at
 618                  \(\theta = \frac{(2n + 1)\pi}{2} \> \vert \> n \in \mathbb{Z}\)
 619                \item
 620                  \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
 621              \end{itemize}
 622
 623              \subsubsection*{Secant}
 624
 625\begin{tikzpicture}
 626  \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={\(-2\pi\), \(-\pi\), \(\pi\), \(2\pi\)}]
 627%    \addplot[blue, domain=-6.2830:6.2830,unbounded coords=jump,samples=80] {sec(deg(x))};
 628    \addplot[blue, restrict y to domain=-10:10, domain=-7:7,samples=100] {sec(deg(x))} node [pos=0.93, black, right] {\(\operatorname{sec} x\)};
 629    \addplot[red, dashed, domain=-7:7,samples=100] {cos(deg(x))};
 630    \draw [gray, dotted, thick] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{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:-4.71239,0}|-{rel axis cs:0,0}) -- ({axis cs:-4.71239,0}|-{rel axis cs:0,1});
 633    \draw [gray, dotted, thick] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1});
 634\end{axis}
 635    \node [black] at (7,3.5) {\(\cos x\)};
 636\end{tikzpicture}
 637
 638                \[\operatorname{sec} \theta = \frac{1}{\cos \theta} \> \vert \> \cos \theta \ne 0\]
 639
 640                \begin{itemize}
 641
 642                  \item
 643                    \textbf{Domain}
 644                    \(= \mathbb{R} \setminus \frac{(2n + 1) \pi}{2} : n \in \mathbb{Z}\}\)
 645                  \item
 646                    \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
 647                  \item
 648                    \textbf{Turning points} at
 649                    \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
 650                  \item
 651                    \textbf{Asymptotes} at
 652                    \(\theta = \frac{(2n + 1) \pi}{2} \> \vert \> n \in \mathbb{Z}\)
 653                \end{itemize}
 654
 655                \subsubsection*{Cotangent}
 656
 657\begin{tikzpicture}
 658  \begin{axis}[xmin=-3,xmax=3,ymin=-1.5,ymax=1.5,enlargelimits=true, xtick={-3.1415, -1.5708, 1.5708, 3.1415},xticklabels={\(-\pi\), \(-\frac{\pi}{2}\), \(\frac{\pi}{2}\), \(\pi\)}]
 659    \addplot[blue, smooth, domain=-3:-0.1,unbounded coords=jump,samples=105] {cot(deg(x))} node [pos=0.3, left] {\(\operatorname{cot} x\)};
 660\addplot[blue, smooth, domain=0.1:3,unbounded coords=jump,samples=105] {cot(deg(x))};
 661\addplot[red, smooth, dashed] gnuplot [domain=-1.5:1.5,unbounded coords=jump,samples=105] {tan(x)};
 662\addplot[red, smooth, dashed] gnuplot [domain=-3.5:-1.8,unbounded coords=jump,samples=105] {tan(x)} node [pos=0.5, right] {\(\tan x\)};
 663\addplot[red, smooth, dashed] gnuplot [domain=1.8:3.5,unbounded coords=jump,samples=105] {tan(x)};
 664    \draw [thick, red, dotted] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1});
 665    \draw [thick, blue, dotted] ({axis cs:-3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:-3.1415,0}|-{rel axis cs:0,1});
 666    \draw [thick, blue, dotted] ({axis cs:0,0}|-{rel axis cs:0,0}) -- ({axis cs:0,0}|-{rel axis cs:0,1});
 667    \draw [thick, blue, dotted] ({axis cs:3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:3.1415,0}|-{rel axis cs:0,1});
 668    \draw [thick, red, dotted] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{rel axis cs:0,1});
 669\end{axis}
 670\end{tikzpicture}
 671
 672                  \[\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0\]
 673
 674                  \begin{itemize}
 675
 676                    \item
 677                      \textbf{Domain} \(= \mathbb{R} \setminus \{n \pi: n \in \mathbb{Z}\}\)
 678                    \item
 679                      \textbf{Range} \(= \mathbb{R}\)
 680                    \item
 681                      \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
 682                  \end{itemize}
 683
 684                  \subsubsection*{Symmetry properties}
 685
 686                  \[\begin{split}
 687                    \operatorname{sec} (\pi \pm x) & = -\operatorname{sec} x \\
 688                    \operatorname{sec} (-x) & = \operatorname{sec} x \\
 689                    \operatorname{cosec} (\pi \pm x) & = \mp \operatorname{cosec} x \\
 690                    \operatorname{cosec} (-x) & = - \operatorname{cosec} x \\
 691                    \operatorname{cot} (\pi \pm x) & = \pm \operatorname{cot} x \\
 692                    \operatorname{cot} (-x) & = - \operatorname{cot} x
 693                  \end{split}\]
 694
 695                  \subsubsection*{Complementary properties}
 696
 697                  \[\begin{split}
 698                    \operatorname{sec} \left({\pi \over 2} - x\right) & = \operatorname{cosec} x \\
 699                    \operatorname{cosec} \left({\pi \over 2} - x\right) & = \operatorname{sec} x \\
 700                    \operatorname{cot} \left({\pi \over 2} - x\right) & = \tan x \\
 701                    \tan \left({\pi \over 2} - x\right) & = \operatorname{cot} x
 702                  \end{split}\]
 703
 704                  \subsubsection*{Pythagorean identities}
 705
 706                  \[\begin{split}
 707                    1 + \operatorname{cot}^2 x & = \operatorname{cosec}^2 x, \quad \text{where } \sin x \ne 0 \\
 708                    1 + \tan^2 x & = \operatorname{sec}^2 x, \quad \text{where } \cos x \ne 0
 709                  \end{split}\]
 710
 711                  \subsection*{Compound angle formulas}
 712
 713                  \[\cos(x \pm y) = \cos x + \cos y \mp \sin x \sin y\]
 714                  \[\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y\]
 715                  \[\tan(x \pm y) = {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}\]
 716
 717                  \subsection*{Double angle formulas}
 718
 719                  \[\begin{split}
 720                    \cos 2x &= \cos^2 x - \sin^2 x \\
 721                    & = 1 - 2\sin^2 x \\
 722                    & = 2 \cos^2 x -1
 723                  \end{split}\]
 724
 725                  \[\sin 2x = 2 \sin x \cos x\]
 726
 727                  \[\tan 2x = {{2 \tan x} \over {1 - \tan^2 x}}\]
 728
 729                  \subsection*{Inverse circular functions}
 730
 731                  \begin{tikzpicture}
 732                    \begin{axis}[ymin=-2, ymax=4, xmin=-1.1, xmax=1.1, ytick={-1.5708, 1.5708, 3.14159},yticklabels={$-\frac{\pi}{2}$, $\frac{\pi}{2}$, $\pi$}]
 733                      \addplot[color=red, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {asin(x)} node [pos=0.25, below right] {\(\sin^{-1}x\)};
 734                      \addplot[color=blue, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {acos(x)} node [pos=0.25, below left] {\(\cos^{-1}x\)};
 735                      \addplot[mark=*, red] coordinates {(-1,-1.5708)} node[right, font=\footnotesize]{\((-1,-\frac{\pi}{2})\)} ;
 736                      \addplot[mark=*, red] coordinates {(1,1.5708)} node[left, font=\footnotesize]{\((1,\frac{\pi}{2})\)} ;
 737                      \addplot[mark=*, blue] coordinates {(1,0)};
 738                      \addplot[mark=*, blue] coordinates {(-1,3.1415)} node[right, font=\footnotesize]{\((-1,\pi)\)} ;
 739                    \end{axis}
 740                  \end{tikzpicture}\\
 741
 742                  Inverse functions: \(f(f^{-1}(x)) = x\) (restrict domain)
 743
 744                  \[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y\]
 745                  \hfill where \(\sin y = x, \> y \in [{-\pi \over 2}, {\pi \over 2}]\)
 746
 747                  \[\cos^{-1}: [-1,1] \rightarrow \mathbb{R}, \quad \cos^{-1} x = y\]
 748                  \hfill where \(\cos y = x, \> y \in [0, \pi]\)
 749
 750                  \[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\]
 751                  \hfill where \(\tan y = x, \> y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\)
 752
 753                  \begin{tikzpicture}
 754                    \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={\(-\frac{\pi}{2}\),\(\frac{\pi}{2}\)}]
 755                      \addplot[color=orange, smooth] gnuplot [domain=-35:35, unbounded coords=jump,samples=350] {atan(x)} node [pos=0.5, above left] {\(\tan^{-1}x\)};
 756                      \addplot[->, gray, dotted, thick, domain=-35:35] {1.5708};
 757                      \addplot[->, gray, dotted, thick, domain=-35:35] {-1.5708};
 758                    \end{axis}
 759                  \end{tikzpicture}
 760\columnbreak
 761                  \section{Differential calculus}
 762
 763                  \subsection*{Limits}
 764
 765                  \[\lim_{x \rightarrow a}f(x)\]
 766                  \(L^-,\quad L^+\) \qquad limit from below/above\\
 767                  \(\lim_{x \to a} f(x)\) \quad limit of a point\\
 768
 769                  \noindent For solving \(x\rightarrow\infty\), put all \(x\) terms in denominators\\
 770                  e.g. \[\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2\]
 771
 772                  \subsubsection*{Limit theorems}
 773
 774                  \begin{enumerate}
 775                    \item
 776                      For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\)
 777                    \item
 778                      \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\)
 779                    \item
 780                      \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\)
 781                    \item
 782                      \(\therefore \lim_{x \rightarrow a} c \times f(x)=cF\) where \(c=\) constant
 783                    \item
 784                      \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\)
 785                    \item
 786                      \(f(x)\) is continuous \(\iff L^-=L^+=f(x) \> \forall x\)
 787                  \end{enumerate}
 788
 789                  \subsection*{Gradients of secants and tangents}
 790
 791                  \textbf{Secant (chord)} - line joining two points on curve\\
 792                  \textbf{Tangent} - line that intersects curve at one point
 793
 794                  \subsection*{First principles derivative}
 795
 796                  \[f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={\frac{dy}{dx}}\]
 797
 798                  \subsubsection*{Logarithmic identities}
 799
 800                  \(\log_b (xy)=\log_b x + \log_b y\)\\
 801                  \(\log_b x^n = n \log_b x\)\\
 802                  \(\log_b y^{x^n} = x^n \log_b y\)
 803
 804                  \subsubsection*{Index identities}
 805
 806                  \(b^{m+n}=b^m \cdot b^n\)\\
 807                  \((b^m)^n=b^{m \cdot n}\)\\
 808                  \((b \cdot c)^n = b^n \cdot c^n\)\\
 809                  \({a^m \div a^n} = {a^{m-n}}\)
 810
 811                  \subsection*{Derivative rules}
 812
 813                  \renewcommand{\arraystretch}{1.4}
 814                  \begin{tabularx}{\columnwidth}{rX}
 815                    \hline
 816                    \(f(x)\) & \(f^\prime(x)\)\\
 817                    \hline
 818                    \(\sin x\) & \(\cos x\)\\
 819                    \(\sin ax\) & \(a\cos ax\)\\
 820                    \(\cos x\) & \(-\sin x\)\\
 821                    \(\cos ax\) & \(-a \sin ax\)\\
 822                    \(\tan f(x)\) & \(f^2(x) \sec^2f(x)\)\\
 823                    \(e^x\) & \(e^x\)\\
 824                    \(e^{ax}\) & \(ae^{ax}\)\\
 825                    \(ax^{nx}\) & \(an \cdot e^{nx}\)\\
 826                    \(\log_e x\) & \(\dfrac{1}{x}\)\\
 827                    \(\log_e {ax}\) & \(\dfrac{1}{x}\)\\
 828                    \(\log_e f(x)\) & \(\dfrac{f^\prime (x)}{f(x)}\)\\
 829                    \(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\\
 830                    \(\sin^{-1} x\) & \(\dfrac{1}{\sqrt{1-x^2}}\)\\
 831                    \(\cos^{-1} x\) & \(\dfrac{-1}{sqrt{1-x^2}}\)\\
 832                    \(\tan^{-1} x\) & \(\dfrac{1}{1 + x^2}\)\\
 833                    \(\frac{d}{dy}f(y)\) & \(\dfrac{1}{\frac{dx}{dy}}\) (reciprocal)\\
 834                    \(uv\) & \(u \frac{dv}{dx}+v\frac{du}{dx} (product rule)\)\\
 835                    \(\dfrac{u}{v}\) & \(\dfrac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\) (quotient rule)\\
 836                    \(f(g(x))\) & \(f^\prime(g(x))\cdot g^\prime(x)\)\\
 837                    \hline
 838                  \end{tabularx}
 839
 840                  \subsection*{Reciprocal derivatives}
 841
 842                  \[\frac{1}{\frac{dy}{dx}} = \frac{dx}{dy}\]
 843
 844                  \subsection*{Differentiating \(x=f(y)\)}
 845                  \begin{align*}
 846                    \text{Find }& \frac{dx}{dy}\\
 847                    \text{Then, }\frac{dx}{dy} &= \frac{1}{\frac{dy}{dx}} \\
 848                    \implies {\frac{dy}{dx}} &= \frac{1}{\frac{dx}{dy}}\\
 849                    \therefore {\frac{dy}{dx}} &= \frac{1}{\frac{dx}{dy}}
 850                  \end{align*}
 851
 852                  \subsection*{Second derivative}
 853                  \begin{align*}f(x) \longrightarrow &f^\prime (x) \longrightarrow f^{\prime\prime}(x)\\
 854                  \implies y \longrightarrow &\frac{dy}{dx} \longrightarrow \frac{d^2 y}{dx^2}\end{align*}
 855
 856                  \noindent Order of polynomial \(n\)th derivative decrements each time the derivative is taken
 857
 858                  \subsubsection*{Points of Inflection}
 859
 860                  \emph{Stationary point} - i.e.
 861                  \(f^\prime(x)=0\)\\
 862                  \emph{Point of inflection} - max \(|\)gradient\(|\) (i.e.
 863                  \(f^{\prime\prime} = 0\))
 864
 865
 866                  \pgfplotsset{every axis/.append style={
 867                    axis x line=none,    % put the x axis in the middle
 868                    axis y line=none,    % put the y axis in the middle
 869                  }}
 870                  \begin{table*}[ht]
 871                    \centering
 872                    \begin{tabularx}{\textwidth}{rXXX}
 873                      \hline
 874                      \rowcolor{shade2}
 875                      & \centering\(\dfrac{d^2 y}{dx^2} > 0\)  & \centering \(\dfrac{d^2y}{dx^2}<0\) & \(\dfrac{d^2y}{dx^2}=0\) (inflection) \\
 876                      \hline
 877                      \(\dfrac{dy}{dx}>0\) &
 878                      \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)}&
 879                        \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)}&
 880                          \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}\\
 881                            \hline
 882                            \(\dfrac{dy}{dx}<0\) &
 883                            \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)}&
 884                              \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)}&
 885                                \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}\\
 886                                  \hline
 887                                  \(\dfrac{dy}{dx}=0\)&
 888                                  \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}&
 889                                    \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}\\
 890                                      \hline
 891                    \end{tabularx}
 892                  \end{table*}
 893                  \begin{itemize}
 894                    \item
 895                      if \(f^\prime (a) = 0\) and \(f^{\prime\prime}(a) > 0\), then point
 896                      \((a, f(a))\) is a local min (curve is concave up)
 897                    \item
 898                      if \(f^\prime (a) = 0\) and \(f^{\prime\prime} (a) < 0\), then point
 899                      \((a, f(a))\) is local max (curve is concave down)
 900                    \item
 901                      if \(f^{\prime\prime}(a) = 0\), then point \((a, f(a))\) is a point of
 902                      inflection
 903                    \item
 904                      if also \(f^\prime(a)=0\), then it is a stationary point of inflection
 905                  \end{itemize}
 906
 907                  \subsection*{Implicit Differentiation}
 908
 909                  \noindent Used for differentiating circles etc.
 910
 911                  If \(p\) and \(q\) are expressions in \(x\) and \(y\) such that \(p=q\),
 912                  for all \(x\) and \(y\), then:
 913
 914                  \[{\frac{dp}{dx}} = {\frac{dq}{dx}} \quad \text{and} \quad {\frac{dp}{dy}} = {\frac{dq}{dy}}\]
 915
 916                  \noindent \colorbox{cas}{\textbf{On CAS:}}\\
 917                  Action \(\rightarrow\) Calculation \(\rightarrow\) \texttt{impDiff(y\^{}2+ax=5,\ x,\ y)}\\
 918                  Returns \(y^\prime= \dots\).
 919
 920                  \subsection*{Integration}
 921
 922                  \[\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)\]
 923
 924                  \subsection*{Integral laws}
 925
 926                  \renewcommand{\arraystretch}{1.4}
 927                  \begin{tabularx}{\columnwidth}{rX}
 928                    \hline
 929                    \(f(x)\) & \(\int f(x) \cdot dx\) \\
 930                    \hline
 931                    \(k\) (constant) & \(kx + c\)\\
 932                    \(x^n\) & \(\dfrac{1}{n+1} x^{n+1}\) \\
 933                    \(a x^{-n}\) &\(a \cdot \log_e |x| + c\)\\
 934                    \(\dfrac{1}{ax+b}\) &\(\dfrac{1}{a} \log_e (ax+b) + c\)\\
 935                    \((ax+b)^n\) & \(\dfrac{1}{a(n+1)}(ax+b)^{n-1} + c\>|\>n\ne 1\)\\
 936                    \((ax+b)^{-1}\) & \(\dfrac{1}{a}\log_e |ax+b|+c\)\\
 937                    \(e^{kx}\) & \(\dfrac{1}{k} e^{kx} + c\)\\
 938                    \(e^k\) & \(e^kx + c\)\\
 939                    \(\sin kx\) & \(\dfrac{-1}{k} \cos (kx) + c\)\\
 940                    \(\cos kx\) & \(\dfrac{1}{k} \sin (kx) + c\)\\
 941                    \(\sec^2 kx\) & \(\dfrac{1}{k} \tan(kx) + c\)\\
 942                    \(\dfrac{1}{\sqrt{a^2-x^2}}\) & \(\sin^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
 943                    \(\dfrac{-1}{\sqrt{a^2-x^2}}\) & \(\cos^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
 944                    \(\frac{a}{a^2-x^2}\) & \(\tan^{-1} \frac{x}{a} + c\)\\
 945                    \(\frac{f^\prime (x)}{f(x)}\) & \(\log_e f(x) + c\)\\
 946                    \(\int f(u) \cdot \frac{du}{dx} \cdot dx\) & \(\int f(u) \cdot du\) (substitution)\\
 947                    \(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\
 948                    \hline
 949                  \end{tabularx}
 950
 951                  Note \(\sin^{-1} {x \over a} + \cos^{-1} {x \over a}\) is constant \(\forall x \in (-a, a)\)
 952
 953                  \subsection*{Definite integrals}
 954
 955                  \[\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)\]
 956
 957                  \begin{itemize}
 958
 959                    \item
 960                      Signed area enclosed by\\
 961                      \(\> y=f(x), \quad y=0, \quad x=a, \quad x=b\).
 962                    \item
 963                      \emph{Integrand} is \(f\).
 964                  \end{itemize}
 965
 966                  \subsubsection*{Properties}
 967
 968                  \[\int^b_a f(x) \> dx = \int^c_a f(x) \> dx + \int^b_c f(x) \> dx\]
 969
 970                  \[\int^a_a f(x) \> dx = 0\]
 971
 972                  \[\int^b_a k \cdot f(x) \> dx = k \int^b_a f(x) \> dx\]
 973
 974                  \[\int^b_a f(x) \pm g(x) \> dx = \int^b_a f(x) \> dx \pm \int^b_a g(x) \> dx\]
 975
 976                  \[\int^b_a f(x) \> dx = - \int^a_b f(x) \> dx\]
 977
 978                  \subsection*{Integration by substitution}
 979
 980                  \[\int f(u) {\frac{du}{dx}} \cdot dx = \int f(u) \cdot du\]
 981
 982                  \noindent Note \(f(u)\) must be 1:1 \(\implies\) one \(x\) for each \(y\)
 983                  \begin{align*}\text{e.g. for } y&=\int(2x+1)\sqrt{x+4} \cdot dx\\
 984                    \text{let } u&=x+4\\
 985                    \implies& {\frac{du}{dx}} = 1\\
 986                    \implies& x = u - 4\\
 987                    \text{then } &y=\int (2(u-4)+1)u^{\frac{1}{2}} \cdot du\\
 988                    &\text{(solve as  normal integral)}
 989                  \end{align*}
 990
 991                  \subsubsection*{Definite integrals by substitution}
 992
 993                  For \(\int^b_a f(x) {\frac{du}{dx}} \cdot dx\), evaluate new \(a\) and
 994                  \(b\) for \(f(u) \cdot du\).
 995
 996                  \subsubsection*{Trigonometric integration}
 997
 998                  \[\sin^m x \cos^n x \cdot dx\]
 999
1000                  \paragraph{\textbf{\(m\) is odd:}}
1001                  \(m=2k+1\) where \(k \in \mathbb{Z}\)\\
1002                  \(\implies \sin^{2k+1} x = (\sin^2 z)^k \sin x = (1 - \cos^2 x)^k \sin x\)\\
1003                  Substitute \(u=\cos x\)
1004
1005                  \paragraph{\textbf{\(n\) is odd:}}
1006                  \(n=2k+1\) where \(k \in \mathbb{Z}\)\\
1007                  \(\implies \cos^{2k+1} x = (\cos^2 x)^k \cos x = (1-\sin^2 x)^k \cos x\)\\
1008                  Substitute \(u=\sin x\)
1009
1010                  \paragraph{\textbf{\(m\) and \(n\) are even:}}
1011                  use identities...
1012
1013                  \begin{itemize}
1014
1015                    \item
1016                      \(\sin^2x={1 \over 2}(1-\cos 2x)\)
1017                    \item
1018                      \(\cos^2x={1 \over 2}(1+\cos 2x)\)
1019                    \item
1020                      \(\sin 2x = 2 \sin x \cos x\)
1021                  \end{itemize}
1022
1023                  \subsection*{Partial fractions}
1024
1025                  \colorbox{cas}{On CAS:}\\
1026                  \indent Action \(\rightarrow\) Transformation \(\rightarrow\)
1027                  \texttt{expand/combine}\\
1028                  \indent Interactive \(\rightarrow\) Transformation \(\rightarrow\)
1029                  Expand \(\rightarrow\) Partial
1030
1031                  \subsection*{Graphing integrals on CAS}
1032
1033                  \colorbox{cas}{In main:} Interactive \(\rightarrow\) Calculation \(\rightarrow\)
1034                  \(\int\) (\(\rightarrow\) Definite)\\
1035                  Restrictions: \texttt{Define\ f(x)=..} then \texttt{f(x)\textbar{}x\textgreater{}..}
1036
1037                  \subsection*{Applications of antidifferentiation}
1038
1039                  \begin{itemize}
1040
1041                    \item
1042                      \(x\)-intercepts of \(y=f(x)\) identify \(x\)-coordinates of
1043                      stationary points on \(y=F(x)\)
1044                    \item
1045                      nature of stationary points is determined by sign of \(y=f(x)\) on
1046                      either side of its \(x\)-intercepts
1047                    \item
1048                      if \(f(x)\) is a polynomial of degree \(n\), then \(F(x)\) has degree
1049                      \(n+1\)
1050                  \end{itemize}
1051
1052                  To find stationary points of a function, substitute \(x\) value of given
1053                  point into derivative. Solve for \({\frac{dy}{dx}}=0\). Integrate to find
1054                  original function.
1055
1056                  \subsection*{Solids of revolution}
1057
1058                  Approximate as sum of infinitesimally-thick cylinders
1059
1060                  \subsubsection*{Rotation about \(x\)-axis}
1061
1062                  \begin{align*}
1063                    V &= \int^{x=b}_{x-a} \pi y^2 \> dx \\
1064                    &= \pi \int^b_a (f(x))^2 \> dx
1065                  \end{align*}
1066
1067                  \subsubsection*{Rotation about \(y\)-axis}
1068
1069                  \begin{align*}
1070                    V &= \int^{y=b}_{y=a} \pi x^2 \> dy \\
1071                    &= \pi \int^b_a (f(y))^2 \> dy
1072                  \end{align*}
1073
1074                  \subsubsection*{Regions not bound by \(y=0\)}
1075
1076                  \[V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx\]
1077                  \hfill where \(f(x) > g(x)\)
1078
1079                  \subsection*{Length of a curve}
1080
1081                  \[L = \int^b_a \sqrt{1 + ({\frac{dy}{dx}})^2} \> dx \quad \text{(Cartesian)}\]
1082
1083                  \[L = \int^b_a \sqrt{{\frac{dx}{dt}} + ({\frac{dy}{dt}})^2} \> dt \quad \text{(parametric)}\]
1084
1085                  \noindent \colorbox{cas}{On CAS:}\\
1086                  \indent Evaluate formula,\\
1087                  \indent or Interactive \(\rightarrow\) Calculation
1088                  \(\rightarrow\) Line \(\rightarrow\) \texttt{arcLen}
1089
1090                  \subsection*{Rates}
1091
1092                  \subsubsection*{Gradient at a point on parametric curve}
1093
1094                  \[{\frac{dy}{dx}} = {{\frac{dy}{dt}} \div {\frac{dx}{dt}}} \> \vert \> {\frac{dx}{dt}} \ne 0 \text{ (chain rule)}\]
1095
1096                  \[\frac{d^2}{dx^2} = \frac{d(y^\prime)}{dx} = {\frac{dy^\prime}{dt} \div {\frac{dx}{dt}}} \> \vert \> y^\prime = {\frac{dy}{dx}}\]
1097
1098                  \subsection*{Rational functions}
1099
1100                  \[f(x) = \frac{P(x)}{Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}\]
1101
1102                  \subsubsection*{Addition of ordinates}
1103
1104                  \begin{itemize}
1105
1106                    \item
1107                      when two graphs have the same ordinate, \(y\)-coordinate is double the
1108                      ordinate
1109                    \item
1110                      when two graphs have opposite ordinates, \(y\)-coordinate is 0 i.e.
1111                      (\(x\)-intercept)
1112                    \item
1113                      when one of the ordinates is 0, the resulting ordinate is equal to the
1114                      other ordinate
1115                  \end{itemize}
1116
1117                  \subsection*{Fundamental theorem of calculus}
1118
1119                  If \(f\) is continuous on \([a, b]\), then
1120
1121                  \[\int^b_a f(x) \> dx = F(b) - F(a)\]
1122                  \hfill where \(F = \int f \> dx\)
1123                  
1124                  \subsection*{Differential equations}
1125
1126                  \noindent\textbf{Order} - highest power inside derivative\\
1127                  \textbf{Degree} - highest power of highest derivative\\
1128                  e.g. \({\left(\dfrac{dy^2}{d^2} x\right)}^3\) \qquad order 2, degree 3
1129
1130                  \subsubsection*{Verifying solutions}
1131
1132                  Start with \(y=\dots\), and differentiate. Substitute into original
1133                  equation.
1134
1135                  \subsubsection*{Function of the dependent
1136                  variable}
1137
1138                  If \({\frac{dy}{dx}}=g(y)\), then
1139                  \(\frac{dx}{dy} = 1 \div \frac{dy}{dx} = \frac{1}{g(y)}\). Integrate both sides to solve equation. Only add \(c\) on one side. Express
1140                  \(e^c\) as \(A\).
1141
1142
1143
1144                  \subsubsection*{Mixing problems}
1145
1146                  \[\left(\frac{dm}{dt}\right)_\Sigma = \left(\frac{dm}{dt}\right)_{\text{in}} - \left(\frac{dm}{dt}_{\text{out}}\right)\]
1147
1148                  \subsubsection*{Separation of variables}
1149
1150                  If \({\frac{dy}{dx}}=f(x)g(y)\), then:
1151
1152                  \[\int f(x) \> dx = \int \frac{1}{g(y)} \> dy\]
1153
1154                  \subsubsection*{Euler's method for solving DEs}
1155
1156                  \[\frac{f(x+h) - f(x)}{h} \approx f^\prime (x) \quad \text{for small } h\]
1157
1158                  \[\implies f(x+h) \approx f(x) + hf^\prime(x)\]
1159
1160              
1161    \section{Kinematics \& Mechanics}
1162
1163      \subsection*{Constant acceleration}
1164
1165      \begin{itemize}
1166        \item \textbf{Position} - relative to origin
1167        \item \textbf{Displacement} - relative to starting point
1168      \end{itemize}
1169
1170      \subsubsection*{Velocity-time graphs}
1171
1172      \begin{itemize}
1173        \item Displacement: \textit{signed} area between graph and \(t\) axis
1174        \item Distance travelled: \textit{total} area between graph and \(t\) axis
1175      \end{itemize}
1176
1177      \[ \text{acceleration} = \frac{d^2x}{dt^2} = \frac{dv}{dt} = v\frac{dv}{dx} = \frac{d}{dx}\left(\frac{1}{2}v^2\right) \]
1178
1179        \begin{center}
1180          \renewcommand{\arraystretch}{1}
1181          \begin{tabular}{ l r }
1182              \hline & no \\ \hline
1183              \(v=u+at\) & \(x\) \\
1184              \(v^2 = u^2+2as\) & \(t\) \\
1185              \(s = \frac{1}{2} (v+u)t\) & \(a\) \\
1186              \(s = ut + \frac{1}{2} at^2\) & \(v\) \\
1187              \(s = vt- \frac{1}{2} at^2\) & \(u\) \\ \hline
1188            \end{tabular}
1189        \end{center}
1190
1191        \[ v_{\text{avg}} = \frac{\Delta\text{position}}{\Delta t} \]
1192        \begin{align*}
1193          \text{speed} &= |{\text{velocity}}| \\
1194          &= \sqrt{v_x^2 + v_y^2 + v_z^2}
1195        \end{align*}
1196
1197        \noindent \textbf{Distance travelled between \(t=a \rightarrow t=b\):}
1198        \[= \int^b_a \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \cdot dt \]
1199
1200        \noindent \textbf{Shortest distance between \(\boldsymbol{r}(t_0)\) and \(\boldsymbol{r}(t_1)\):}
1201        \[ = |\boldsymbol{r}(t_1) - \boldsymbol{r}(t_2)| \]
1202
1203      \subsection*{Vector functions}
1204
1205        \[ \boldsymbol{r}(t) = x \boldsymbol{i} + y \boldsymbol{j} + z \boldsymbol{k} \]
1206
1207        \begin{itemize}
1208          \item If \(\boldsymbol{r}(t) \equiv\) position with time, then the graph of endpoints of \(\boldsymbol{r}(t) \equiv\) Cartesian path
1209          \item Domain of \(\boldsymbol{r}(t)\) is the range of \(x(t)\)
1210          \item Range of \(\boldsymbol{r}(t)\) is the range of \(y(t)\)
1211        \end{itemize}
1212
1213      \subsection*{Vector calculus}
1214
1215      \subsubsection*{Derivative}
1216
1217        Let \(\boldsymbol{r}(t)=x(t)\boldsymbol{i} + y(t)\boldsymbol(j)\). If both \(x(t)\) and \(y(t)\) are differentiable, then:
1218        \[ \boldsymbol{r}(t)=x(t)\boldsymbol{i}+y(t)\boldsymbol{j} \]
1219
1220  \end{multicols}
1221\end{document}