06cb97454a4eb7d37b40de70704bc1a950c0cbaa
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  74\fancyhead[LO,LE]{Year 12 Specialist}
  75\fancyhead[CO,CE]{Andrew Lorimer}
  76
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 100\newtcolorbox{cas}{colframe=cas!75!black, fonttitle=\sffamily\bfseries, title=On CAS, left*=3mm}
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 103
 104\begin{document}
 105
 106\title{\vspace{-22mm}Year 12 Specialist\vspace{-4mm}}
 107\author{Andrew Lorimer}
 108\date{}
 109\maketitle
 110\vspace{-9mm}
 111\begin{multicols}{2}
 112
 113  \section{Complex numbers}
 114
 115  \[\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}\]
 116  \begin{align*}
 117    \text{Cartesian form: } & a+bi\\
 118    \text{Polar form: } & r\operatorname{cis}\theta
 119  \end{align*}
 120
 121  \subsection*{Operations}
 122
 123  \begin{tabularx}{\columnwidth}{|r|X|X|}
 124    \hline
 125    \rowcolor{cas}
 126    & \textbf{Cartesian} & \textbf{Polar} \\
 127    \hline
 128    \(z_1 \pm z_2\) & \((a \pm c)(b \pm d)i\) & convert to \(a+bi\)\\
 129    \hline
 130    \(+k \times z\) & \multirow{2}{*}{\(ka \pm kbi\)} & \(kr\operatorname{cis} \theta\)\\
 131    \cline{1-1}\cline{3-3}
 132    \(-k \times z\) & & \(kr \operatorname{cis}(\theta\pm \pi)\)\\
 133    \hline
 134    \(z_1 \cdot z_2\) & \(ac-bd+(ad+bc)i\) & \(r_1r_2 \operatorname{cis}(\theta_1 + \theta_2)\)\\
 135    \hline
 136    \(z_1 \div z_2\) & \((z_1 \overline{z_2}) \div |z_2|^2\) & \(\left(\frac{r_1}{r_2}\right) \operatorname{cis}(\theta_1 - \theta_2)\) \\
 137    \hline
 138  \end{tabularx}
 139
 140  \subsubsection*{Scalar multiplication in polar form}
 141
 142  For \(k \in \mathbb{R}^+\):
 143  \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\theta\]
 144
 145  \noindent For \(k \in \mathbb{R}^-\):
 146  \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\left(\begin{cases}\theta - \pi & |0<\operatorname{Arg}(z)\le \pi \\ \theta + \pi & |-\pi<\operatorname{Arg}(z)\le 0\end{cases}\right)\]
 147
 148    \subsection*{Conjugate}
 149    \vspace{-7mm} \hfill  \colorbox{cas}{\texttt{conjg(a+bi)}}
 150    \begin{align*}
 151      \overline{z} &= a \mp bi\\
 152      &= r \operatorname{cis}(-\theta)
 153    \end{align*}
 154
 155    \subsubsection*{Properties}
 156
 157    \begin{align*}
 158      \overline{z_1 \pm z_2} &= \overline{z_1}\pm\overline{z_2}\\
 159      \overline{z_1 \cdot z_2} &= \overline{z_1}\cdot\overline{z_2}\\
 160      \overline{kz} &= k\overline{z} \> \forall \>  k \in \mathbb{R}\\
 161      z\overline{z} &= (a+bi)(a-bi)\\
 162      &= a^2 + b^2\\
 163      &= |z|^2
 164    \end{align*}
 165
 166    \subsection*{Modulus}
 167
 168    \[|z|=|\vec{Oz}|=\sqrt{a^2 + b^2}\]
 169
 170    \subsubsection*{Properties}
 171
 172    \begin{align*}
 173      |z_1z_2|&=|z_1||z_2|\\
 174      \left|\frac{z_1}{z_2}\right|&=\frac{|z_1|}{|z_2|}\\
 175      |z_1+z_2|&\le|z_1|+|z_2|
 176    \end{align*}
 177
 178    \subsection*{Multiplicative inverse}
 179
 180    \begin{align*}
 181      z^{-1}&=\frac{a-bi}{a^2+b^2}\\
 182      &=\frac{\overline{z}}{|z|^2}a\\
 183      &=r \operatorname{cis}(-\theta)
 184    \end{align*}
 185
 186    \subsection*{Dividing over \(\mathbb{C}\)}
 187
 188    \begin{align*}
 189      \frac{z_1}{z_2}&=z_1z_2^{-1}\\
 190      &=\frac{z_1\overline{z_2}}{|z_2|^2}\\
 191      &=\frac{(a+bi)(c-di)}{c^2+d^2}\\
 192      & \text{then rationalise denominator}
 193    \end{align*}
 194
 195    \subsection*{Polar form}
 196
 197    \[ r \operatorname{cis} \theta = r\left( \cos \theta + i \sin \theta \right) \]
 198
 199    \begin{itemize}
 200      \item{\(r=|z|=\sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}\)}
 201      \item{\(\theta = \operatorname{arg}(z)\) \hfill \colorbox{cas}{\texttt{arg(a+bi)}}}
 202      \item{\(\operatorname{Arg}(z) \in (-\pi,\pi)\) \quad \bf{(principal argument)}}
 203      \item{Multiple representations:\\\(r\operatorname{cis}\theta=r\operatorname{cis}(\theta+2n\pi)\) with \(n \in \mathbb{Z}\) revolutions}
 204      \item{\(\operatorname{cis}\pi=-1,\qquad \operatorname{cis}0=1\)}
 205    \end{itemize}
 206
 207    \begin{cas}
 208      \-\hspace{1em}\verb|compToTrig(a+bi)| \(\iff\) \verb|cExpand{r·cisX}|
 209    \end{cas}
 210
 211    \subsection*{de Moivres' theorem}
 212
 213    \begin{theorembox}{}
 214      \[(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}\]
 215    \end{theorembox}
 216
 217    \subsection*{Complex polynomials}
 218
 219    Include \(\pm\) for all solutions, incl. imaginary
 220
 221    \begin{tabularx}{\columnwidth}{ R{0.55} X  }
 222      \hline
 223      Sum of squares & \(\begin{aligned} 
 224        z^2 + a^2 &= z^2-(ai)^2\\
 225      &= (z+ai)(z-ai) \end{aligned}\) \\
 226      \hline
 227      Sum of cubes & \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)\\
 228      \hline
 229      Division & \(P(z)=D(z)Q(z)+R(z)\) \\
 230      \hline
 231      Remainder theorem & Let \(\alpha \in \mathbb{C}\). Remainder of \(P(z) \div (z-\alpha)\) is \(P(\alpha)\)\\
 232      \hline
 233      Factor theorem & \(z-\alpha\) is a factor of \(P(z) \iff P(\alpha)=0\) for \(\alpha \in \mathbb{C}\)\\
 234      \hline
 235      Conjugate root theorem & \(P(z)=0 \text{ at } z=a\pm bi\) (\(\implies\) both \(z_1\) and \(\overline{z_1}\) are solutions)\\
 236      \hline
 237    \end{tabularx}
 238
 239    \begin{theorembox}{title=Factor theorem}
 240      If \(\beta z + \alpha\) is a factor of \(P(z)\), \\
 241      \-\hspace{1em}then \(P(-\dfrac{\alpha}{\beta})=0\).
 242    \end{theorembox}
 243
 244    \subsection*{\(n\)th roots}
 245
 246    \(n\)th roots of \(z=r\operatorname{cis}\theta\) are:
 247
 248    \[z = r^{\frac{1}{n}} \operatorname{cis}\left(\frac{\theta+2k\pi}{n}\right)\]
 249
 250    \begin{itemize}
 251
 252      \item{Same modulus for all solutions}
 253      \item{Arguments separated by \(\frac{2\pi}{n} \therefore\) there are \(n\) roots}
 254      \item{If one square root is \(a+bi\), the other is \(-a-bi\)}
 255      \item{Give one implicit \(n\)th root \(z_1\), function is \(z=z_1^n\)}
 256      \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 \(\frac{2\pi}{n}\))}
 257    \end{itemize}
 258
 259    \noindent For \(0=az^2+bz+c\), use quadratic formula:
 260
 261    \[z=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]
 262
 263    \subsection*{Fundamental theorem of algebra}
 264
 265    A polynomial of degree \(n\) can be factorised into \(n\) linear factors in \(\mathbb{C}\):
 266
 267    \[\implies P(z)=a_n(z-\alpha_1)(z-\alpha_2)(z-\alpha_3)\dots(z-\alpha_n)\]
 268    \[\text{ where } \alpha_1,\alpha_2,\alpha_3,\dots,\alpha_n \in \mathbb{C}\]
 269
 270    \subsection*{Argand planes}
 271
 272    \begin{center}\begin{tikzpicture}[scale=2]
 273      \draw [->] (-0.2,0) -- (1.5,0) node [right]  {$\operatorname{Re}(z)$};
 274      \draw [->] (0,-0.2) -- (0,1.5) node [above] {$\operatorname{Im}(z)$};
 275      \coordinate (P) at (1,1);
 276      \coordinate (a) at (1,0);
 277      \coordinate (b) at (0,1);
 278      \coordinate (O) at (0,0);
 279      \draw (0,0) -- (P) node[pos=0.5, above left]{\(r\)} node[pos=1, right]{\(\begin{aligned}z&=a+bi\\&=r\operatorname{cis}\theta\end{aligned}\)};
 280        \draw [gray, dashed] (1,1) -- (1,0) node[black, pos=1, below]{\(a\)};
 281        \draw [gray, dashed] (1,1) -- (0,1) node[black, pos=1, left]{\(b\)};
 282        \begin{scope}
 283          \path[clip] (O) -- (P) -- (a);
 284          \fill[red, opacity=0.5, draw=black] (O) circle (2mm);
 285          \node at ($(O)+(20:3mm)$) {$\theta$};
 286        \end{scope}
 287        \filldraw (P) circle (0.5pt);
 288    \end{tikzpicture}\end{center}
 289
 290    \begin{itemize}
 291      \item{Multiplication by \(i \implies\) CCW rotation of \(\frac{\pi}{2}\)}
 292      \item{Addition: \(z_1 + z_2 \equiv\) \overrightharp{\(Oz_1\)} + \overrightharp{\(Oz_2\)}}
 293    \end{itemize}
 294
 295    \subsection*{Sketching complex graphs}
 296
 297    \subsubsection*{Linear}
 298
 299    \begin{itemize}
 300      \item{\(\operatorname{Re}(z)=c\) or \(\operatorname{Im}(z)=c\) (perpendicular bisector)}
 301      \item{\(\operatorname{Im}(z)=m\operatorname{Re}(z)\)}
 302      \item{\(|z+a|=|z+b| \implies 2(a-b)x=b^2-a^2\)\\Geometric: equidistant from \(a,b\)}
 303    \end{itemize}
 304
 305    \subsubsection*{Circles}
 306
 307    \begin{itemize}
 308      \item \(|z-z_1|^2=c^2|z_2+2|^2\)
 309      \item \(|z-(a+bi)|=c \implies (x-a)^2+_(y-b)^2=c^2\)
 310    \end{itemize}
 311
 312    \noindent \textbf{Loci} \qquad \(\operatorname{Arg}(z)<\theta\)
 313
 314    \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
 315      \draw [->] (0,0) -- (1,0) node [right]  {$\operatorname{Re}(z)$};
 316      \draw [->] (0,-0.5) -- (0,1) node [above] {$\operatorname{Im}(z)$};
 317      \draw [<-, dashed, thick, blue] (-1,0) -- (0,0);
 318      \draw [->, thick, blue] (0,0) -- (1,1);
 319      \fill [gray, opacity=0.2, domain=-1:1, variable=\x] (-1,-0.5) -- (-1,0) -- (0, 0) -- (1,1) -- (1,-0.5) -- cycle;
 320      \begin{scope}
 321        \path[clip] (0,0) -- (1,1) -- (1,0);
 322        \fill[red, opacity=0.5, draw=black] (0,0) circle (2mm);
 323        \node at ($(0,0)+(20:3mm)$) {$\frac{\pi}{4}$};
 324      \end{scope}
 325      \node [font=\footnotesize] at (0.5,-0.25) {\(\operatorname{Arg}(z)\le\frac{\pi}{4}\)};
 326      \node [blue, mydot] {};
 327    \end{tikzpicture}\end{center}
 328
 329    \noindent \textbf{Rays} \qquad \(\operatorname{Arg}(z-b)=\theta\)
 330
 331    \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
 332      \draw [->] (-0.75,0) -- (1.5,0) node [right]  {$\operatorname{Re}(z)$};
 333      \draw [->] (0,-1) -- (0,1) node [above] {$\operatorname{Im}(z)$};
 334      \draw [->, thick, brown] (-0.25,0) -- (-0.75,-1);
 335      \node [above, font=\footnotesize] at (-0.25,0) {\(\frac{1}{4}\)};
 336      \begin{scope}
 337        \path[clip] (-0.25,0) -- (-0.75,-1) -- (0,0);
 338        \fill[orange, opacity=0.5, draw=black] (-0.25,0) circle (2mm);
 339      \end{scope}
 340      \node at (-0.08,-0.3) {\(\frac{\pi}{8}\)};
 341      \node [font=\footnotesize, left] at (-0.75,-1) {\(\operatorname{Arg}(z+\frac{1}{4})=\frac{\pi}{8}\)};
 342      \node [brown, mydot] at (-0.25,0) {};
 343      \draw [<->, thick, green] (0,-1) -- (1.5,0.5) node [pos=0.25, black, font=\footnotesize, right] {\(|z-2|=|z-(1+i)|\)};
 344      \node [left, font=\footnotesize] at (0,-1) {\(-1\)};
 345      \node [below, font=\footnotesize] at (1,0) {\(1\)};
 346    \end{tikzpicture}\end{center}
 347
 348    \section{Vectors}
 349    \begin{center}\begin{tikzpicture}
 350      \draw [->] (-0.5,0) -- (3,0) node [right]  {\(x\)};
 351      \draw [->] (0,-0.5) -- (0,3) node [above] {\(y\)};
 352      \draw [orange, ->, thick] (0.5,0.5) -- (2.5,2.5) node [pos=0.5, above] {\(\vec{u}\)};
 353      \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
 354        \draw [gray, dashed, thick] (0.5,0.5) -- (2.5,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, below]{\(x\vec{i}\)};
 355        \draw [gray, dashed, thick] (2.5,0.5) -- (2.5,2.5) node [pos=0.5] {\midarrow};
 356      \end{scope}
 357      \node[black, right] at (2.5,1.5) {\(y\vec{j}\)};
 358    \end{tikzpicture}\end{center}
 359    \subsection*{Column notation}
 360
 361    \[\begin{bmatrix}x\\ y \end{bmatrix} \iff x\boldsymbol{i} + y\boldsymbol{j}\]
 362      \(\begin{bmatrix}x_2-x_1\\ y_2-y_1 \end{bmatrix}\) \quad between \(A(x_1,y_1), \> B(x_2,y_2)\)
 363
 364        \subsection*{Scalar multiplication}
 365
 366        \[k\cdot (x\boldsymbol{i}+y\boldsymbol{j})=kx\boldsymbol{i}+ky\boldsymbol{j}\]
 367
 368        \noindent For \(k \in \mathbb{R}^-\), direction is reversed
 369
 370        \subsection*{Vector addition}
 371        \begin{center}\begin{tikzpicture}[scale=1]
 372          \coordinate (A) at (0,0);
 373          \coordinate (B) at (2,2);
 374          \draw [->, thick, red] (0,0) -- (2,2) node [pos=0.5, below right] {\(\vec{u}=2\vec{i}+2\vec{j}\)};
 375          \draw [->, thick, blue] (2,2) -- (1,4) node [pos=0.5, above right] {\(\vec{v}=-\vec{i}+2\vec{j}\)};
 376          \draw [->, thick, orange] (0,0) -- (1,4) node [pos=0.5, left] {\(\vec{u}+\vec{v}=\vec{i}+4\vec{j}\)};
 377        \end{tikzpicture}\end{center}
 378
 379        \[(x\boldsymbol{i}+y\boldsymbol{j}) \pm (a\boldsymbol{i}+b\boldsymbol{j})=(x \pm a)\boldsymbol{i}+(y \pm b)\boldsymbol{j}\]
 380
 381        \begin{itemize}
 382          \item Draw each vector head to tail then join lines
 383          \item Addition is commutative (parallelogram)
 384          \item \(\boldsymbol{u}-\boldsymbol{v}=\boldsymbol{u}+(-\boldsymbol{v}) \implies \overrightharp{AB}=\boldsymbol{b}-\boldsymbol{a}\)
 385        \end{itemize}
 386
 387        \subsection*{Magnitude}
 388
 389        \[|(x\boldsymbol{i} + y\boldsymbol{j})|=\sqrt{x^2+y^2}\]
 390
 391        \subsection*{Parallel vectors}
 392
 393        \[\boldsymbol{u} || \boldsymbol{v} \iff \boldsymbol{u} = k \boldsymbol{v} \text{ where } k \in \mathbb{R} \setminus \{0\}\]
 394
 395        For parallel vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\):\\
 396        \[\boldsymbol{a \cdot b}=\begin{cases}
 397          |\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction}\\
 398          -|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions}
 399        \end{cases}\]
 400        %\includegraphics[width=0.2,height=\textheight]{graphics/parallelogram-vectors.jpg}
 401        %\includegraphics[width=1]{graphics/vector-subtraction.jpg}
 402
 403        \subsection*{Perpendicular vectors}
 404
 405        \[\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b} = 0\ \quad \text{(since \(\cos 90 = 0\))}\]
 406
 407        \subsection*{Unit vector \(|\hat{\boldsymbol{a}}|=1\)}
 408        \[\begin{split}\hat{\boldsymbol{a}} & = {\frac{1}{|\boldsymbol{a}|}}\boldsymbol{a} \\ & = \boldsymbol{a} \cdot {|\boldsymbol{a}|}\end{split}\]
 409
 410          \subsection*{Scalar product \(\boldsymbol{a} \cdot \boldsymbol{b}\)}
 411
 412
 413          \begin{center}\begin{tikzpicture}[scale=2]
 414            \draw [->] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{b}\)};
 415            \draw [->] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{a}\)};
 416            \begin{scope}
 417              \path[clip] (1,0.5) -- (1,0) -- (0,0);
 418              \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
 419              \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
 420            \end{scope}
 421          \end{tikzpicture}\end{center}
 422          \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*}
 423            \noindent\colorbox{cas}{On CAS: \texttt{dotP({[}a\ b\ c{]},\ {[}d\ e\ f{]})}}
 424
 425            \subsubsection*{Properties}
 426
 427            \begin{enumerate}
 428              \item
 429                \(k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k\boldsymbol{b})\)
 430              \item
 431                \(\boldsymbol{a \cdot 0}=0\)
 432              \item
 433                \(\boldsymbol{a} \cdot (\boldsymbol{b} + \boldsymbol{c})=\boldsymbol{a} \cdot \boldsymbol{b} + \boldsymbol{a} \cdot \boldsymbol{c}\)
 434              \item
 435                \(\boldsymbol{i \cdot i} = \boldsymbol{j \cdot j} = \boldsymbol{k \cdot k}= 1\)
 436              \item
 437                \(\boldsymbol{a} \cdot \boldsymbol{b} = 0 \quad \implies \quad \boldsymbol{a} \perp \boldsymbol{b}\)
 438              \item
 439                \(\boldsymbol{a \cdot a} = |\boldsymbol{a}|^2 = a^2\)
 440            \end{enumerate}
 441
 442            \subsection*{Angle between vectors}
 443
 444            \[\cos \theta = \frac{\boldsymbol{a} \cdot \boldsymbol{b}}{|\boldsymbol{a}| |\boldsymbol{b}|} = \frac{a_1 b_1 + a_2 b_2}{|\boldsymbol{a}| |\boldsymbol{b}|}\]
 445
 446            \noindent \colorbox{cas}{On CAS:} \texttt{angle([a b c], [a b c])}
 447
 448            (Action \(\rightarrow\) Vector \(\rightarrow\)Angle)
 449
 450            \subsection*{Angle between vector and axis}
 451
 452            \noindent For\(\boldsymbol{a} = a_1 \boldsymbol{i} + a_2 \boldsymbol{j} + a_3 \boldsymbol{k}\)
 453            which makes angles \(\alpha, \beta, \gamma\) with positive side of
 454            \(x, y, z\) axes:
 455            \[\cos \alpha = \frac{a_1}{|\boldsymbol{a}|}, \quad \cos \beta = \frac{a_2}{|\boldsymbol{a}|}, \quad \cos \gamma = \frac{a_3}{|\boldsymbol{a}|}\]
 456
 457            \noindent \colorbox{cas}{On CAS:} \texttt{angle({[}a\ b\ c{]},\ {[}1\ 0\ 0{]})}\\for angle
 458            between \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) and
 459            \(x\)-axis
 460
 461            \subsection*{Projections \& resolutes}
 462
 463            \begin{tikzpicture}[scale=3]
 464              \draw [->, purple] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{a}\)};
 465              \draw [->, orange] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{u}\)};
 466              \draw [->, blue] (1,0) -- (2,0) node [pos=0.5, below] {\(\boldsymbol{b}\)};
 467              \begin{scope}
 468                \path[clip] (1,0.5) -- (1,0) -- (0,0);
 469                \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
 470                \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
 471              \end{scope}
 472              \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
 473                \draw [gray, dashed, thick] (1,0) -- (1,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, right, rotate=-90]{\(\boldsymbol{w}\)};
 474              \end{scope}
 475              \draw (0,0) coordinate (O)
 476              (1,0) coordinate (A)
 477              (1,0.5) coordinate (B)
 478              pic [draw,red,angle radius=2mm] {right angle = O--A--B};
 479            \end{tikzpicture}
 480
 481            \subsubsection*{\(\parallel\boldsymbol{b}\) (vector projection/resolute)}
 482
 483            \begin{align*}
 484              \boldsymbol{u} & = \frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|^2}\boldsymbol{b} \\
 485              & = \left(\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\right)\left(\frac{\boldsymbol{b}}{|\boldsymbol{b}|}\right) \\
 486              & = (\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}
 487            \end{align*}
 488
 489            \subsubsection*{\(\perp\boldsymbol{b}\) (perpendicular projection)}
 490            \[\boldsymbol{w} = \boldsymbol{a} - \boldsymbol{u}\]
 491
 492            \subsubsection*{\(|\boldsymbol{u}|\) (scalar projection/resolute)}
 493            \begin{align*}
 494              s &= |\boldsymbol{u}|\\
 495              &= \boldsymbol{a} \cdot \hat{\boldsymbol{b}}\\
 496              &=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\\
 497              &= |\boldsymbol{a}| \cos \theta
 498            \end{align*}
 499
 500            \subsubsection*{Rectangular (\(\parallel,\perp\)) components}
 501
 502            \[\boldsymbol{a}=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}+\left(\boldsymbol{a}-\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}\right)\]
 503
 504
 505            \subsection*{Vector proofs}
 506
 507            \textbf{Concurrent:} intersection of \(\ge\) 3 lines
 508
 509            \begin{tikzpicture}
 510              \draw [blue] (0,0) -- (1,1);
 511              \draw [red] (1,0) -- (0,1);
 512              \draw [brown] (0.4,0) -- (0.6,1);
 513              \filldraw (0.5,0.5) circle (2pt);
 514            \end{tikzpicture}
 515
 516            \subsubsection*{Collinear points}
 517
 518            \(\ge\) 3 points lie on the same line
 519
 520            \begin{tikzpicture}
 521              \draw [purple] (0,0) -- (4,1);
 522              \filldraw (2,0.5) circle (2pt) node [above] {\(C\)};
 523              \filldraw (1,0.25) circle (2pt) node [above] {\(A\)};
 524              \filldraw (3,0.75) circle (2pt) node [above] {\(B\)};
 525              \coordinate (O) at (2.8,-0.2);
 526              \node at (O) [below] {\(O\)}; 
 527              \begin{scope}[->, orange, thick] 
 528                \draw (O) -- (2,0.5) node [pos=0.5, above, font=\footnotesize, black] {\(\boldsymbol{c}\)};
 529                \draw (O) -- (1,0.25) node [pos=0.5, below, font=\footnotesize, black] {\(\boldsymbol{a}\)};
 530                \draw (O) -- (3,0.75) node [pos=0.5, right, font=\footnotesize, black] {\(\boldsymbol{b}\)};
 531              \end{scope}
 532            \end{tikzpicture}
 533
 534            \begin{align*}
 535              \text{e.g. Prove that}\\
 536              \overrightharp{AC}=m\overrightharp{AB} \iff \boldsymbol{c}&=(1-m)\boldsymbol{a}+m\boldsymbol{b}\\
 537              \implies \boldsymbol{c} &= \overrightharp{OA} + \overrightharp{AC}\\
 538              &= \overrightharp{OA} + m\overrightharp{AB}\\
 539              &=\boldsymbol{a}+m(\boldsymbol{b}-\boldsymbol{a})\\
 540              &=\boldsymbol{a}+m\boldsymbol{b}-m\boldsymbol{a}\\
 541              &=(1-m)\boldsymbol{a}+m{b}
 542            \end{align*}
 543            \begin{align*}
 544              \text{Also, } \implies \overrightharp{OC} &= \lambda \vec{OA} + \mu \overrightharp{OB} \\
 545              \text{where } \lambda + \mu &= 1\\
 546              \text{If } C \text{ lies along } \overrightharp{AB}, & \implies 0 < \mu < 1
 547            \end{align*}
 548
 549
 550            \subsubsection*{Parallelograms}
 551
 552            \begin{center}\begin{tikzpicture}
 553              \coordinate (O) at (0,0) node [below left] {\(O\)};
 554              \coordinate (A) at (4,0);
 555              \coordinate (B) at (6,2);
 556              \coordinate (C) at (2,2);
 557              \coordinate (D) at (6,0);
 558
 559              \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (O)--(A) node [below left] {\(A\)};
 560              \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (A)--(B) node [above right] {\(B\)};
 561              \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (B)--(C) node [above left] {\(C\)};
 562              \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (C)--(O);
 563
 564              \draw [gray, dashed] (O) -- (B) node [pos=0.75] {\(\diagdown\diagdown\)} node [pos=0.25] {\(\diagdown\diagdown\)};
 565              \draw [gray, dashed] (A) -- (C) node [pos=0.75] {\(\diagup\)} node [pos=0.25] {\(\diagup\)};
 566              \begin{scope}
 567                \path[clip] (C) -- (A) -- (O);
 568                \fill[orange, opacity=0.5, draw=black] (0,0) circle (4mm);
 569                \node at ($(0,0)+(20:8mm)$) {\(\theta\)};
 570              \end{scope}
 571              \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] {\(|\boldsymbol{c}|\cos\theta\)};
 572              \draw pic [draw,thick,red,angle radius=2mm] {right angle=O--D--B};
 573            \end{tikzpicture}\end{center}
 574
 575            \begin{itemize}
 576              \item
 577                Diagonals \(\overrightharp{OB}, \overrightharp{AC}\) bisect each other
 578              \item
 579                If diagonals are equal length, it is a rectangle
 580              \item
 581                \(|\overrightharp{OB}|^2 + |\overrightharp{CA}|^2 = |\overrightharp{OA}|^2 + |\overrightharp{AB}|^2 + |\overrightharp{CB}|^2 + |\overrightharp{OC}|^2\)
 582              \item
 583                Area \(=\boldsymbol{c} \cdot \boldsymbol{a}\)
 584            \end{itemize}
 585
 586            \subsubsection*{Useful vector properties}
 587
 588            \begin{itemize}
 589              \item
 590                \(\boldsymbol{a} \parallel \boldsymbol{b} \implies \boldsymbol{b}=k\boldsymbol{a}\) for some
 591                \(k \in \mathbb{R} \setminus \{0\}\)
 592              \item
 593                If \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are parallel with at
 594                least one point in common, then they lie on the same straight line
 595              \item
 596                \(\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b}=0\)
 597              \item
 598                \(\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2\)
 599            \end{itemize}
 600
 601            \subsection*{Linear dependence}
 602
 603            \(\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\) are linearly dependent if they are \(\nparallel\) and:
 604            \begin{align*}
 605              0&=k\boldsymbol{a}+l\boldsymbol{b}+m\boldsymbol{c}\\
 606              \therefore \boldsymbol{c} &= m\boldsymbol{a} + n\boldsymbol{b} \quad \text{(simultaneous)}
 607            \end{align*}
 608
 609            \noindent \(\boldsymbol{a}, \boldsymbol{b},\) and \(\boldsymbol{c}\) are linearly
 610            independent if no vector in the set is expressible as a linear
 611            combination of other vectors in set, or if they are parallel.
 612
 613            \subsection*{Three-dimensional vectors}
 614
 615            Right-hand rule for axes: \(z\) is up or out of page.
 616
 617            \tdplotsetmaincoords{60}{120} 
 618            \begin{center}\begin{tikzpicture} [scale=3, tdplot_main_coords, axis/.style={->,thick}, 
 619              vector/.style={-stealth,red,very thick}, 
 620              vector guide/.style={dashed,gray,thick}]
 621
 622              %standard tikz coordinate definition using x, y, z coords
 623              \coordinate (O) at (0,0,0);
 624
 625              %tikz-3dplot coordinate definition using x, y, z coords
 626
 627              \pgfmathsetmacro{\ax}{1}
 628              \pgfmathsetmacro{\ay}{1}
 629              \pgfmathsetmacro{\az}{1}
 630
 631              \coordinate (P) at (\ax,\ay,\az);
 632
 633              %draw axes
 634              \draw[axis] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$};
 635              \draw[axis] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$};
 636              \draw[axis] (0,0,0) -- (0,0,1) node[anchor=south]{$z$};
 637
 638              %draw a vector from O to P
 639              \draw[vector] (O) -- (P);
 640
 641              %draw guide lines to components
 642              \draw[vector guide]         (O) -- (\ax,\ay,0);
 643              \draw[vector guide] (\ax,\ay,0) -- (P);
 644              \draw[vector guide]         (P) -- (0,0,\az);
 645              \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
 646              \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
 647              \draw[vector guide] (\ax,\ay,0) -- (\ax,0,0);
 648              \node[tdplot_main_coords,above right]
 649              at (\ax,\ay,\az){(\ax, \ay, \az)};
 650            \end{tikzpicture}\end{center}
 651
 652            \subsection*{Parametric vectors}
 653
 654            Parametric equation of line through point \((x_0, y_0, z_0)\) and
 655            parallel to \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) is:
 656
 657            \[\begin{cases}x = x_o + a \cdot t \\ y = y_0 + b \cdot t \\ z = z_0 + c \cdot t\end{cases}\]
 658
 659              \section{Circular functions}
 660
 661              \(\sin(bx)\) or \(\cos(bx)\): period \(=\frac{2\pi}{b}\)
 662
 663              \noindent \(\tan(nx)\): period \(=\frac{\pi}{n}\)\\
 664              \indent\indent\indent asymptotes at \(x=\frac{(2k+1)\pi}{2n} \> \vert \> k \in \mathbb{Z}\)
 665
 666              \subsection*{Reciprocal functions}
 667
 668              \subsubsection*{Cosecant}
 669
 670              \[\operatorname{cosec} \theta = \frac{1}{\sin \theta} \> \vert \> \sin \theta \ne 0\]
 671
 672              \begin{itemize}
 673                \item
 674                  \textbf{Domain} \(= \mathbb{R} \setminus {n\pi : n \in \mathbb{Z}}\)
 675                \item
 676                  \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
 677                \item
 678                  \textbf{Turning points} at
 679                  \(\theta = \frac{(2n + 1)\pi}{2} \> \vert \> n \in \mathbb{Z}\)
 680                \item
 681                  \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
 682              \end{itemize}
 683
 684              \subsubsection*{Secant}
 685
 686\begin{tikzpicture}
 687  \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={\(-2\pi\), \(-\pi\), \(\pi\), \(2\pi\)}]
 688%    \addplot[blue, domain=-6.2830:6.2830,unbounded coords=jump,samples=80] {sec(deg(x))};
 689    \addplot[blue, restrict y to domain=-10:10, domain=-7:7,samples=100] {sec(deg(x))} node [pos=0.93, black, right] {\(\operatorname{sec} x\)};
 690    \addplot[red, dashed, domain=-7:7,samples=100] {cos(deg(x))};
 691    \draw [gray, dotted, thick] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{rel axis cs:0,1});
 692    \draw [gray, dotted, thick] ({axis cs:4.71239,0}|-{rel axis cs:0,0}) -- ({axis cs:4.71239,0}|-{rel axis cs:0,1});
 693    \draw [gray, dotted, thick] ({axis cs:-4.71239,0}|-{rel axis cs:0,0}) -- ({axis cs:-4.71239,0}|-{rel axis cs:0,1});
 694    \draw [gray, dotted, thick] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1});
 695\end{axis}
 696    \node [black] at (7,3.5) {\(\cos x\)};
 697\end{tikzpicture}
 698
 699                \[\operatorname{sec} \theta = \frac{1}{\cos \theta} \> \vert \> \cos \theta \ne 0\]
 700
 701                \begin{itemize}
 702
 703                  \item
 704                    \textbf{Domain}
 705                    \(= \mathbb{R} \setminus \frac{(2n + 1) \pi}{2} : n \in \mathbb{Z}\}\)
 706                  \item
 707                    \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
 708                  \item
 709                    \textbf{Turning points} at
 710                    \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
 711                  \item
 712                    \textbf{Asymptotes} at
 713                    \(\theta = \frac{(2n + 1) \pi}{2} \> \vert \> n \in \mathbb{Z}\)
 714                \end{itemize}
 715
 716                \subsubsection*{Cotangent}
 717
 718\begin{tikzpicture}
 719  \begin{axis}[xmin=-3,xmax=3,ymin=-1.5,ymax=1.5,enlargelimits=true, xtick={-3.1415, -1.5708, 1.5708, 3.1415},xticklabels={\(-\pi\), \(-\frac{\pi}{2}\), \(\frac{\pi}{2}\), \(\pi\)}]
 720    \addplot[blue, smooth, domain=-3:-0.1,unbounded coords=jump,samples=105] {cot(deg(x))} node [pos=0.3, left] {\(\operatorname{cot} x\)};
 721\addplot[blue, smooth, domain=0.1:3,unbounded coords=jump,samples=105] {cot(deg(x))};
 722\addplot[red, smooth, dashed] gnuplot [domain=-1.5:1.5,unbounded coords=jump,samples=105] {tan(x)};
 723\addplot[red, smooth, dashed] gnuplot [domain=-3.5:-1.8,unbounded coords=jump,samples=105] {tan(x)} node [pos=0.5, right] {\(\tan x\)};
 724\addplot[red, smooth, dashed] gnuplot [domain=1.8:3.5,unbounded coords=jump,samples=105] {tan(x)};
 725    \draw [thick, red, dotted] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1});
 726    \draw [thick, blue, dotted] ({axis cs:-3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:-3.1415,0}|-{rel axis cs:0,1});
 727    \draw [thick, blue, dotted] ({axis cs:0,0}|-{rel axis cs:0,0}) -- ({axis cs:0,0}|-{rel axis cs:0,1});
 728    \draw [thick, blue, dotted] ({axis cs:3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:3.1415,0}|-{rel axis cs:0,1});
 729    \draw [thick, red, dotted] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{rel axis cs:0,1});
 730\end{axis}
 731\end{tikzpicture}
 732
 733                  \[\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0\]
 734
 735                  \begin{itemize}
 736
 737                    \item
 738                      \textbf{Domain} \(= \mathbb{R} \setminus \{n \pi: n \in \mathbb{Z}\}\)
 739                    \item
 740                      \textbf{Range} \(= \mathbb{R}\)
 741                    \item
 742                      \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
 743                  \end{itemize}
 744
 745                  \subsubsection*{Symmetry properties}
 746
 747                  \[\begin{split}
 748                    \operatorname{sec} (\pi \pm x) & = -\operatorname{sec} x \\
 749                    \operatorname{sec} (-x) & = \operatorname{sec} x \\
 750                    \operatorname{cosec} (\pi \pm x) & = \mp \operatorname{cosec} x \\
 751                    \operatorname{cosec} (-x) & = - \operatorname{cosec} x \\
 752                    \operatorname{cot} (\pi \pm x) & = \pm \operatorname{cot} x \\
 753                    \operatorname{cot} (-x) & = - \operatorname{cot} x
 754                  \end{split}\]
 755
 756                  \subsubsection*{Complementary properties}
 757
 758                  \[\begin{split}
 759                    \operatorname{sec} \left({\pi \over 2} - x\right) & = \operatorname{cosec} x \\
 760                    \operatorname{cosec} \left({\pi \over 2} - x\right) & = \operatorname{sec} x \\
 761                    \operatorname{cot} \left({\pi \over 2} - x\right) & = \tan x \\
 762                    \tan \left({\pi \over 2} - x\right) & = \operatorname{cot} x
 763                  \end{split}\]
 764
 765                  \subsubsection*{Pythagorean identities}
 766
 767                  \[\begin{split}
 768                    1 + \operatorname{cot}^2 x & = \operatorname{cosec}^2 x, \quad \text{where } \sin x \ne 0 \\
 769                    1 + \tan^2 x & = \operatorname{sec}^2 x, \quad \text{where } \cos x \ne 0
 770                  \end{split}\]
 771
 772                  \subsection*{Compound angle formulas}
 773
 774                  \[\cos(x \pm y) = \cos x + \cos y \mp \sin x \sin y\]
 775                  \[\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y\]
 776                  \[\tan(x \pm y) = {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}\]
 777
 778                  \subsection*{Double angle formulas}
 779
 780                  \[\begin{split}
 781                    \cos 2x &= \cos^2 x - \sin^2 x \\
 782                    & = 1 - 2\sin^2 x \\
 783                    & = 2 \cos^2 x -1
 784                  \end{split}\]
 785
 786                  \[\sin 2x = 2 \sin x \cos x\]
 787
 788                  \[\tan 2x = {{2 \tan x} \over {1 - \tan^2 x}}\]
 789
 790                  \subsection*{Inverse circular functions}
 791
 792                  \begin{tikzpicture}
 793                    \begin{axis}[ymin=-2, ymax=4, xmin=-1.1, xmax=1.1, ytick={-1.5708, 1.5708, 3.14159},yticklabels={$-\frac{\pi}{2}$, $\frac{\pi}{2}$, $\pi$}]
 794                      \addplot[color=red, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {asin(x)} node [pos=0.25, below right] {\(\sin^{-1}x\)};
 795                      \addplot[color=blue, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {acos(x)} node [pos=0.25, below left] {\(\cos^{-1}x\)};
 796                      \addplot[mark=*, red] coordinates {(-1,-1.5708)} node[right, font=\footnotesize]{\((-1,-\frac{\pi}{2})\)} ;
 797                      \addplot[mark=*, red] coordinates {(1,1.5708)} node[left, font=\footnotesize]{\((1,\frac{\pi}{2})\)} ;
 798                      \addplot[mark=*, blue] coordinates {(1,0)};
 799                      \addplot[mark=*, blue] coordinates {(-1,3.1415)} node[right, font=\footnotesize]{\((-1,\pi)\)} ;
 800                    \end{axis}
 801                  \end{tikzpicture}\\
 802
 803                  Inverse functions: \(f(f^{-1}(x)) = x\) (restrict domain)
 804
 805                  \[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y\]
 806                  \hfill where \(\sin y = x, \> y \in [{-\pi \over 2}, {\pi \over 2}]\)
 807
 808                  \[\cos^{-1}: [-1,1] \rightarrow \mathbb{R}, \quad \cos^{-1} x = y\]
 809                  \hfill where \(\cos y = x, \> y \in [0, \pi]\)
 810
 811                  \[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\]
 812                  \hfill where \(\tan y = x, \> y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\)
 813
 814                  \begin{tikzpicture}
 815                    \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={\(-\frac{\pi}{2}\),\(\frac{\pi}{2}\)}]
 816                      \addplot[color=orange, smooth] gnuplot [domain=-35:35, unbounded coords=jump,samples=350] {atan(x)} node [pos=0.5, above left] {\(\tan^{-1}x\)};
 817                      \addplot[gray, dotted, thick, domain=-35:35] {1.5708} node [black, font=\footnotesize, below right, pos=0] {\(y=\frac{\pi}{2}\)};
 818                      \addplot[gray, dotted, thick, domain=-35:35] {-1.5708} node [black, font=\footnotesize, above left, pos=1] {\(y=-\frac{\pi}{2}\)};
 819                    \end{axis}
 820                  \end{tikzpicture}
 821\columnbreak
 822                  \section{Differential calculus}
 823
 824                  \[f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={\frac{dy}{dx}}\]
 825
 826                  \subsection*{Limits}
 827
 828                  \[\lim_{x \rightarrow a}f(x)\]
 829                  \(L^-,\quad L^+\) \qquad limit from below/above\\
 830                  \(\lim_{x \to a} f(x)\) \quad limit of a point\\
 831
 832                  \noindent For solving \(x\rightarrow\infty\), put all \(x\) terms in denominators\\
 833                  e.g. \[\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2\]
 834
 835                  \subsubsection*{Limit theorems}
 836
 837                  \begin{enumerate}
 838                    \item
 839                      For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\)
 840                    \item
 841                      \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\)
 842                    \item
 843                      \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\)
 844                    \item
 845                      \(\therefore \lim_{x \rightarrow a} c \times f(x)=cF\) where \(c=\) constant
 846                    \item
 847                      \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\)
 848                    \item
 849                      \(f(x)\) is continuous \(\iff L^-=L^+=f(x) \> \forall x\)
 850                  \end{enumerate}
 851
 852                  \subsection*{Gradients}
 853
 854                  \textbf{Secant (chord)} - line joining two points on curve\\
 855                  \textbf{Tangent} - line that intersects curve at one point
 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                  \subsubsection*{Strictly increasing/decreasing}
 865
 866                  For \(x_2\) and \(x_1\) where \(x_2 > x_1\):
 867
 868                  \textbf{strictly increasing}\\
 869                  \-\hspace{1em}where \(f(x_2) > f(x_1)\) or \(f^\prime(x)>0\)
 870
 871                  \textbf{strictly decreasing}\\
 872                  \hspace{1em}where \(f(x_2) < f(x_1)\) or \(f^\prime(x)<0\)
 873                  \begin{warning}
 874                    Endpoints are included, even where \(\boldsymbol{\frac{dy}{dx}=0}\)
 875                  \end{warning}
 876
 877
 878                  \subsection*{Second derivative}
 879                  \begin{align*}f(x) \longrightarrow &f^\prime (x) \longrightarrow f^{\prime\prime}(x)\\
 880                  \implies y \longrightarrow &\frac{dy}{dx} \longrightarrow \frac{d^2 y}{dx^2}\end{align*}
 881
 882                  \noindent Order of polynomial \(n\)th derivative decrements each time the derivative is taken
 883
 884
 885                  \subsection*{Slope fields}
 886
 887                  \begin{tikzpicture}[declare function={diff(\x,\y) = \x+\y;}]
 888                    \begin{axis}[axis equal, ymin=-4, ymax=4, xmin=-4, xmax=4, ticks=none, enlargelimits=true, ]
 889                      \addplot[thick, orange, domain=-4:2] {e^(x)-x-1};
 890                      \pgfplotsinvokeforeach{-4,...,4}{%
 891                        \draw[gray] ( {#1 -0.1}, {4 - diff(#1, 4) *0.1}) --  ( {#1 +0.1}, {4  + diff(#1, 4) *0.1});
 892                        \draw[gray] ( {#1 -0.1}, {3 - diff(#1, 3) *0.1}) --  ( {#1 +0.1}, {3  + diff(#1, 3) *0.1});
 893                        \draw[gray] ( {#1 -0.1}, {2 - diff(#1, 2) *0.1}) --  ( {#1 +0.1}, {2  + diff(#1, 2) *0.1});
 894                        \draw[gray] ( {#1 -0.1}, {1 - diff(#1, 1) *0.1}) --  ( {#1 +0.1}, {1  + diff(#1, 1) *0.1});
 895                        \draw[gray] ( {#1 -0.1}, {0 - diff(#1, 0) *0.1}) --  ( {#1 +0.1}, {0  + diff(#1, 0) *0.1});
 896                        \draw[gray] ( {#1 -0.1}, {-1 - diff(#1, -1) *0.1}) --  ( {#1 +0.1}, {-1  + diff(#1, -1) *0.1});
 897                        \draw[gray] ( {#1 -0.1}, {-2 - diff(#1, -2) *0.1}) --  ( {#1 +0.1}, {-2  + diff(#1, -2) *0.1});
 898                        \draw[gray] ( {#1 -0.1}, {-3 - diff(#1, -3) *0.1}) --  ( {#1 +0.1}, {-3  + diff(#1, -3) *0.1});
 899                        \draw[gray] ( {#1 -0.1}, {-4 - diff(#1, -4) *0.1}) --  ( {#1 +0.1}, {-4  + diff(#1, -4) *0.1});
 900                      }
 901                    \end{axis}
 902                  \end{tikzpicture}
 903
 904                  \begin{table*}[ht]
 905                    \centering
 906                    \begin{tabularx}{\textwidth}{|r|Y|Y|Y|}
 907                      \hline
 908                      \rowcolor{lblue}
 909                      & \adjustbox{margin=0 1ex, valign=m}{\centering\(\dfrac{d^2 y}{dx^2} > 0\)}  & \adjustbox{margin=0 1ex, valign=m}{\centering \(\dfrac{d^2y}{dx^2}<0\)} & \adjustbox{margin=0 1ex, valign=m}{\(\dfrac{d^2y}{dx^2}=0\) (inflection)} \\
 910                      \hline
 911                      \(\dfrac{dy}{dx}>0\) &
 912                      \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, 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)}&
 913                        \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, 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)}&
 914                          \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, 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}\\
 915                            \hline
 916                            \(\dfrac{dy}{dx}<0\) &
 917                            \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, 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)}&
 918                              \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, 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)}&
 919                                \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, 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}\\
 920                                  \hline
 921                                  \(\dfrac{dy}{dx}=0\)&
 922                                  \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, 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}[axis x line=none, axis y line=none, 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}&
 923                                    \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, 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}[axis x line=none, axis y line=none, 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}\\
 924                                      \hline
 925                    \end{tabularx}
 926                  \end{table*}
 927                  \begin{itemize}
 928                    \item
 929                      \(f^\prime (a) = 0, \> f^{\prime\prime}(a) > 0\) \\
 930                      \textbf{local min} at \((a, f(a))\) (concave up)
 931                    \item
 932                      \(f^\prime (a) = 0, \>  f^{\prime\prime} (a) < 0\) \\
 933                      \textbf{local max} at \((a, f(a))\) (concave down)
 934                    \item
 935                      \(f^{\prime\prime}(a) = 0\) \\
 936                      \textbf{point of inflection} at \((a, f(a))\)
 937                    \item
 938                      \(f^{\prime\prime}(a) = 0, \> f^\prime(a)=0\) \\
 939                      stationary point of inflection at \((a, f(a)\)
 940                  \end{itemize}
 941
 942                  \subsection*{Implicit Differentiation}
 943
 944                  \noindent Used for differentiating circles etc.
 945
 946                  If \(p\) and \(q\) are expressions in \(x\) and \(y\) such that \(p=q\),
 947                  for all \(x\) and \(y\), then:
 948
 949                  \[{\frac{dp}{dx}} = {\frac{dq}{dx}} \quad \text{and} \quad {\frac{dp}{dy}} = {\frac{dq}{dy}}\]
 950
 951                  \begin{cas}
 952                    Action \(\rightarrow\) Calculation \\
 953                      \-\hspace{1em}\texttt{impDiff(y\^{}2+ax=5,\ x,\ y)}
 954                  \end{cas}
 955
 956                  \subsection*{Function of the dependent
 957                  variable}
 958
 959                  If \({\frac{dy}{dx}}=g(y)\), then
 960                  \(\frac{dx}{dy} = 1 \div \frac{dy}{dx} = \frac{1}{g(y)}\). Integrate both sides to solve equation. Only add \(c\) on one side. Express
 961                  \(e^c\) as \(A\).
 962
 963                  \subsection*{Reciprocal derivatives}
 964
 965                  \[\frac{1}{\frac{dy}{dx}} = \frac{dx}{dy}\]
 966
 967                  \subsection*{Differentiating \(x=f(y)\)}
 968                  Find \(\dfrac{dx}{dy}\), then \(\dfrac{dy}{dx} = \dfrac{1}{\left(\dfrac{dx}{dy}\right)}\)
 969
 970                  \subsection*{Parametric equations}
 971
 972
 973                  \begin{align*}
 974                    \dfrac{dy}{dt} &= \dfrac{dy}{dx} \cdot \dfrac{dx}{dt} \\
 975                    \therefore \dfrac{dy}{dx} &= \dfrac{\left(\dfrac{dy}{dt}\right)}{\left(\dfrac{dx}{dt}\right)} \text{ provided } \dfrac{dx}{dt} \ne 0 \\
 976                    \dfrac{d^2y}{dx^2} &= \dfrac{\left(\dfrac{dy^\prime}{dt}\right)}{\left(\dfrac{dx}{dt}\right)} \text{ where } y^\prime = \dfrac{dy}{dx}
 977                  \end{align*}
 978
 979                \subsection*{Integration}
 980
 981                \[\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)\]
 982
 983                  \subsubsection*{Properties}
 984
 985                  \begin{align*}
 986                    \int^b_a f(x) \> dx &= \int^c_a f(x) \> dx + \int^b_c f(x) \> dx \\
 987                    \int^a_a f(x) \> dx &= 0 \\
 988                    \int^b_a k \cdot f(x) \> dx &= k \int^b_a f(x) \> dx \\
 989                    \int^b_a f(x) \pm g(x) \> dx &= \int^b_a f(x) \> dx \pm \int^b_a g(x) \> dx \\
 990                    \int^b_a f(x) \> dx &= - \int^a_b f(x) \> dx \\
 991                  \end{align*}
 992
 993                  \subsection*{Integration by substitution}
 994
 995                  \[\int f(u) {\frac{du}{dx}} \cdot dx = \int f(u) \cdot du\]
 996
 997                  \begin{warning}
 998                    \(\boldsymbol{f(u)}\) must be 1:1 \(\boldsymbol{\implies}\) one \(\boldsymbol{x}\) for each \(\boldsymbol{y}\)
 999                  \end{warning}
1000                  \begin{align*}\text{e.g. for } y&=\int(2x+1)\sqrt{x+4} \cdot dx\\
1001                    \text{let } u&=x+4\\
1002                    \implies& {\frac{du}{dx}} = 1\\
1003                    \implies& x = u - 4\\
1004                    \text{then } &y=\int (2(u-4)+1)u^{\frac{1}{2}} \cdot du\\
1005                    &\text{(solve as  normal integral)}
1006                  \end{align*}
1007
1008                  \subsubsection*{Definite integrals by substitution}
1009
1010                  For \(\int^b_a f(x) {\frac{du}{dx}} \cdot dx\), evaluate new \(a\) and
1011                  \(b\) for \(f(u) \cdot du\).
1012
1013                  \subsubsection*{Trigonometric integration}
1014
1015                  \[\sin^m x \cos^n x \cdot dx\]
1016
1017                  \paragraph{\textbf{\(m\) is odd:}}
1018                  \(m=2k+1\) where \(k \in \mathbb{Z}\)\\
1019                  \(\implies \sin^{2k+1} x = (\sin^2 z)^k \sin x = (1 - \cos^2 x)^k \sin x\)\\
1020                  Substitute \(u=\cos x\)
1021
1022                  \paragraph{\textbf{\(n\) is odd:}}
1023                  \(n=2k+1\) where \(k \in \mathbb{Z}\)\\
1024                  \(\implies \cos^{2k+1} x = (\cos^2 x)^k \cos x = (1-\sin^2 x)^k \cos x\)\\
1025                  Substitute \(u=\sin x\)
1026
1027                  \paragraph{\textbf{\(m\) and \(n\) are even:}}
1028                  use identities...
1029
1030                  \begin{itemize}
1031
1032                    \item
1033                      \(\sin^2x={1 \over 2}(1-\cos 2x)\)
1034                    \item
1035                      \(\cos^2x={1 \over 2}(1+\cos 2x)\)
1036                    \item
1037                      \(\sin 2x = 2 \sin x \cos x\)
1038                  \end{itemize}
1039
1040                  \subsection*{Separation of variables}
1041
1042                  If \({\frac{dy}{dx}}=f(x)g(y)\), then:
1043
1044                  \[\int f(x) \> dx = \int \frac{1}{g(y)} \> dy\]
1045
1046                  \subsection*{Partial fractions}
1047
1048                  To factorise \(f(x) = \frac{\delta}{\alpha \cdot \beta}\):
1049                  \begin{align*}
1050                    \dfrac{\delta}{\alpha \cdot \beta \cdot \gamma} &= \dfrac{A}{\alpha} + \dfrac{B}{\beta} + \dfrac{C}{\gamma} \tag{1} \\
1051                    \text{Multiply by } & (\alpha \cdot \beta \cdot \gamma) \text{:} \\
1052                    \delta &= \beta\gamma A + \alpha\gamma B +\alpha\beta C \tag{2} \\
1053                    \text{Substitute } x &= \{\alpha, \beta, \gamma\} \text{ into (2) to find denominators}
1054                  \end{align*}
1055
1056                  \subsubsection*{Repeated linear factors}
1057
1058                  \[ \dfrac{p(x)}{(x-a)^n} = \dfrac{A_1}{(x-a)} + \dfrac{A_2}{(x-a)^2} + \dots + \dfrac{A_n}{(x-a)^n} \]
1059
1060                  \subsubsection*{Irreducible quadratic factors}
1061
1062                  \[ \text{e.g. } \dfrac{3x-4}{(2x-3)(x^2+5)} = \dfrac{A}{2x-3} + \dfrac{Bx+C}{x^2+5} \]
1063
1064                  \begin{cas}
1065                    Action \(\rightarrow\) Transformation:\\
1066                    \-\hspace{1em} \texttt{expand(..., x)}
1067
1068                    To reverse, use \texttt{combine(...)}
1069                  \end{cas}
1070
1071                  \subsection*{Graphing integrals on CAS}
1072
1073                  \begin{cas}
1074                    \textbf{In main:} Interactive \(\rightarrow\) Calculation \(\rightarrow\) \(\int\)\\
1075                    For restrictions, \texttt{Define\ f(x)=...} then \texttt{f(x)\textbar{}x\textgreater{}...}
1076                  \end{cas}
1077
1078                  \subsection*{Solids of revolution}
1079
1080                  Approximate as sum of infinitesimally-thick cylinders
1081
1082                  \subsubsection*{Rotation about \(\boldsymbol{x}\)-axis}
1083
1084                  \[ V = \pi\int^{x=b}_{x=a} f(x)^2 \> dx \]
1085
1086                  \subsubsection*{Rotation about \(\boldsymbol{y}\)-axis}
1087
1088                  \begin{align*}
1089                    V &= \pi \int^{y=b}_{y=a} x^2 \> dy \\
1090                    &= \pi \int^{y=b}_{y=a} (f(y))^2 \> dy
1091                  \end{align*}
1092
1093                  \subsubsection*{Regions not bound by \(\boldsymbol{y=0}\)}
1094
1095                  \[V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx\]
1096                  \hfill where \(f(x) > g(x)\)
1097
1098                  \subsection*{Length of a curve}
1099
1100                  For length of \(f(x)\) from \(x=a \rightarrow x=b\):
1101                  \begin{align*}
1102                    &\text{Cartesian} \> & L &= \int^b_a \sqrt{1 + \left(\dfrac{dy}{dx}\right)^2} \> dx \\
1103                    &\text{Parametric} \> & L & = \int^b_a \sqrt{\left(\dfrac{dx}{dt}\right)^2 + \left(\dfrac{dy}{dt}\right)^2} \> dt
1104                  \end{align*}
1105
1106                  \begin{cas}
1107                    \begin{enumerate}[label=\alph*), leftmargin=5mm]
1108                      \item Evaluate formula
1109                      \item Interactive \(\rightarrow\) Calculation \(\rightarrow\) Line \(\rightarrow\) \texttt{arcLen}
1110                    \end{enumerate}
1111                  \end{cas}
1112
1113                  \subsection*{Applications of antidifferentiation}
1114
1115                  \begin{itemize}
1116
1117                    \item
1118                      \(x\)-intercepts of \(y=f(x)\) identify \(x\)-coordinates of
1119                      stationary points on \(y=F(x)\)
1120                    \item
1121                      nature of stationary points is determined by sign of \(y=f(x)\) on
1122                      either side of its \(x\)-intercepts
1123                    \item
1124                      if \(f(x)\) is a polynomial of degree \(n\), then \(F(x)\) has degree
1125                      \(n+1\)
1126                  \end{itemize}
1127
1128                  To find stationary points of a function, substitute \(x\) value of given
1129                  point into derivative. Solve for \({\frac{dy}{dx}}=0\). Integrate to find
1130                  original function.
1131
1132                  \subsection*{Rates}
1133
1134                  \subsubsection*{Gradient at a point on parametric curve}
1135
1136                  \[{\frac{dy}{dx}} = {{\frac{dy}{dt}} \div {\frac{dx}{dt}}} \> \vert \> {\frac{dx}{dt}} \ne 0 \text{ (chain rule)}\]
1137
1138                  \[\frac{d^2}{dx^2} = \frac{d(y^\prime)}{dx} = {\frac{dy^\prime}{dt} \div {\frac{dx}{dt}}} \> \vert \> y^\prime = {\frac{dy}{dx}}\]
1139
1140                  \subsection*{Rational functions}
1141
1142                  \[f(x) = \frac{P(x)}{Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}\]
1143
1144                  \subsection*{Fundamental theorem of calculus}
1145
1146                  If \(f\) is continuous on \([a, b]\), then
1147
1148                  \[\int^b_a f(x) \> dx = F(b) - F(a)\]
1149                  \hfill where \(F = \int f \> dx\)
1150                  
1151                  \subsection*{Differential equations}
1152
1153                  \noindent\textbf{Order} - highest power inside derivative\\
1154                  \textbf{Degree} - highest power of highest derivative\\
1155                  e.g. \({\left(\dfrac{dy^2}{d^2} x\right)}^3\) \qquad order 2, degree 3
1156
1157                  \begin{warning}
1158                    To verify solutions, find \(\frac{dy}{dx}\) from \(y\) and substitute into original
1159                  \end{warning}
1160
1161
1162
1163                  \subsubsection*{Mixing problems}
1164
1165                  \[\left(\frac{dm}{dt}\right)_\Sigma = \left(\frac{dm}{dt}\right)_{\text{in}} - \left(\frac{dm}{dt}_{\text{out}}\right)\]
1166
1167                  \subsection*{Euler's method}
1168
1169                  \[\frac{f(x+h) - f(x)}{h} \approx f^\prime (x) \quad \text{for small } h\]
1170
1171                  \[\implies f(x+h) \approx f(x) + hf^\prime(x)\]
1172
1173                  \include{calculus-rules}
1174
1175    \section{Kinematics \& Mechanics}
1176
1177      \subsection*{Constant acceleration}
1178
1179      \begin{itemize}
1180        \item \textbf{Position} - relative to origin
1181        \item \textbf{Displacement} - relative to starting point
1182      \end{itemize}
1183
1184      \subsubsection*{Velocity-time graphs}
1185
1186      \begin{description}[nosep, labelindent=0.5cm, leftmargin=0.5\columnwidth]
1187        \item[Displacement:] \textit{signed} area
1188        \item[Distance travelled:] \textit{total} area
1189      \end{description}
1190
1191      \[ \text{acceleration} = \frac{d^2x}{dt^2} = \frac{dv}{dt} = v\frac{dv}{dx} = \frac{d}{dx}\left(\frac{1}{2}v^2\right) \]
1192
1193        \begin{center}
1194          \renewcommand{\arraystretch}{1}
1195          \begin{tabular}{ l r }
1196            \hline & no \\ \hline
1197            \(v=u+at\) & \(x\) \\
1198            \(v^2 = u^2+2as\) & \(t\) \\
1199            \(s = \frac{1}{2} (v+u)t\) & \(a\) \\
1200            \(s = ut + \frac{1}{2} at^2\) & \(v\) \\
1201            \(s = vt- \frac{1}{2} at^2\) & \(u\) \\ \hline
1202          \end{tabular}
1203        \end{center}
1204
1205        \[ v_{\text{avg}} = \frac{\Delta\text{position}}{\Delta t} \]
1206        \begin{align*}
1207          \text{speed} &= |{\text{velocity}}| \\
1208          &= \sqrt{v_x^2 + v_y^2 + v_z^2}
1209        \end{align*}
1210
1211        \noindent \textbf{Distance travelled between \(t=a \rightarrow t=b\):}
1212        \[= \int^b_a \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \cdot dt \]
1213
1214        \noindent \textbf{Shortest distance between \(\boldsymbol{r}(t_0)\) and \(\boldsymbol{r}(t_1)\):}
1215        \[ = |\boldsymbol{r}(t_1) - \boldsymbol{r}(t_2)| \]
1216
1217      \subsection*{Vector functions}
1218
1219        \[ \boldsymbol{r}(t) = x \boldsymbol{i} + y \boldsymbol{j} + z \boldsymbol{k} \]
1220
1221        \begin{itemize}
1222          \item If \(\boldsymbol{r}(t) \equiv\) position with time, then the graph of endpoints of \(\boldsymbol{r}(t) \equiv\) Cartesian path
1223          \item Domain of \(\boldsymbol{r}(t)\) is the range of \(x(t)\)
1224          \item Range of \(\boldsymbol{r}(t)\) is the range of \(y(t)\)
1225        \end{itemize}
1226
1227      \subsection*{Vector calculus}
1228
1229      \subsubsection*{Derivative}
1230
1231        Let \(\boldsymbol{r}(t)=x(t)\boldsymbol{i} + y(t)\boldsymbol(j)\). If both \(x(t)\) and \(y(t)\) are differentiable, then:
1232        \[ \boldsymbol{r}(t)=x(t)\boldsymbol{i}+y(t)\boldsymbol{j} \]
1233
1234      \subfile{dynamics}
1235      \subfile{statistics}
1236  \end{multicols}
1237\end{document}