785a7fdcb6da91ca2a2b82ef125294daf4e1b9a4
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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    \subsection*{\(n\)th roots}
 240
 241    \(n\)th roots of \(z=r\operatorname{cis}\theta\) are:
 242
 243    \[z = r^{\frac{1}{n}} \operatorname{cis}\left(\frac{\theta+2k\pi}{n}\right)\]
 244
 245    \begin{itemize}
 246
 247      \item{Same modulus for all solutions}
 248      \item{Arguments separated by \(\frac{2\pi}{n} \therefore\) there are \(n\) roots}
 249      \item{If one square root is \(a+bi\), the other is \(-a-bi\)}
 250      \item{Give one implicit \(n\)th root \(z_1\), function is \(z=z_1^n\)}
 251      \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 \(\frac{2\pi}{n}\))}
 252    \end{itemize}
 253
 254    \noindent For \(0=az^2+bz+c\), use quadratic formula:
 255
 256    \[z=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]
 257
 258    \subsection*{Fundamental theorem of algebra}
 259
 260    A polynomial of degree \(n\) can be factorised into \(n\) linear factors in \(\mathbb{C}\):
 261
 262    \[\implies P(z)=a_n(z-\alpha_1)(z-\alpha_2)(z-\alpha_3)\dots(z-\alpha_n)\]
 263    \[\text{ where } \alpha_1,\alpha_2,\alpha_3,\dots,\alpha_n \in \mathbb{C}\]
 264
 265    \subsection*{Argand planes}
 266
 267    \begin{center}\begin{tikzpicture}[scale=2]
 268      \draw [->] (-0.2,0) -- (1.5,0) node [right]  {$\operatorname{Re}(z)$};
 269      \draw [->] (0,-0.2) -- (0,1.5) node [above] {$\operatorname{Im}(z)$};
 270      \coordinate (P) at (1,1);
 271      \coordinate (a) at (1,0);
 272      \coordinate (b) at (0,1);
 273      \coordinate (O) at (0,0);
 274      \draw (0,0) -- (P) node[pos=0.5, above left]{\(r\)} node[pos=1, right]{\(\begin{aligned}z&=a+bi\\&=r\operatorname{cis}\theta\end{aligned}\)};
 275        \draw [gray, dashed] (1,1) -- (1,0) node[black, pos=1, below]{\(a\)};
 276        \draw [gray, dashed] (1,1) -- (0,1) node[black, pos=1, left]{\(b\)};
 277        \begin{scope}
 278          \path[clip] (O) -- (P) -- (a);
 279          \fill[red, opacity=0.5, draw=black] (O) circle (2mm);
 280          \node at ($(O)+(20:3mm)$) {$\theta$};
 281        \end{scope}
 282        \filldraw (P) circle (0.5pt);
 283    \end{tikzpicture}\end{center}
 284
 285    \begin{itemize}
 286      \item{Multiplication by \(i \implies\) CCW rotation of \(\frac{\pi}{2}\)}
 287      \item{Addition: \(z_1 + z_2 \equiv\) \overrightharp{\(Oz_1\)} + \overrightharp{\(Oz_2\)}}
 288    \end{itemize}
 289
 290    \subsection*{Sketching complex graphs}
 291
 292    \subsubsection*{Linear}
 293
 294    \begin{itemize}
 295      \item{\(\operatorname{Re}(z)=c\) or \(\operatorname{Im}(z)=c\) (perpendicular bisector)}
 296      \item{\(\operatorname{Im}(z)=m\operatorname{Re}(z)\)}
 297      \item{\(|z+a|=|z+b| \implies 2(a-b)x=b^2-a^2\)\\Geometric: equidistant from \(a,b\)}
 298    \end{itemize}
 299
 300    \subsubsection*{Circles}
 301
 302    \begin{itemize}
 303      \item \(|z-z_1|^2=c^2|z_2+2|^2\)
 304      \item \(|z-(a+bi)|=c \implies (x-a)^2+_(y-b)^2=c^2\)
 305    \end{itemize}
 306
 307    \noindent \textbf{Loci} \qquad \(\operatorname{Arg}(z)<\theta\)
 308
 309    \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
 310      \draw [->] (0,0) -- (1,0) node [right]  {$\operatorname{Re}(z)$};
 311      \draw [->] (0,-0.5) -- (0,1) node [above] {$\operatorname{Im}(z)$};
 312      \draw [<-, dashed, thick, blue] (-1,0) -- (0,0);
 313      \draw [->, thick, blue] (0,0) -- (1,1);
 314      \fill [gray, opacity=0.2, domain=-1:1, variable=\x] (-1,-0.5) -- (-1,0) -- (0, 0) -- (1,1) -- (1,-0.5) -- cycle;
 315      \begin{scope}
 316        \path[clip] (0,0) -- (1,1) -- (1,0);
 317        \fill[red, opacity=0.5, draw=black] (0,0) circle (2mm);
 318        \node at ($(0,0)+(20:3mm)$) {$\frac{\pi}{4}$};
 319      \end{scope}
 320      \node [font=\footnotesize] at (0.5,-0.25) {\(\operatorname{Arg}(z)\le\frac{\pi}{4}\)};
 321      \node [blue, mydot] {};
 322    \end{tikzpicture}\end{center}
 323
 324    \noindent \textbf{Rays} \qquad \(\operatorname{Arg}(z-b)=\theta\)
 325
 326    \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
 327      \draw [->] (-0.75,0) -- (1.5,0) node [right]  {$\operatorname{Re}(z)$};
 328      \draw [->] (0,-1) -- (0,1) node [above] {$\operatorname{Im}(z)$};
 329      \draw [->, thick, brown] (-0.25,0) -- (-0.75,-1);
 330      \node [above, font=\footnotesize] at (-0.25,0) {\(\frac{1}{4}\)};
 331      \begin{scope}
 332        \path[clip] (-0.25,0) -- (-0.75,-1) -- (0,0);
 333        \fill[orange, opacity=0.5, draw=black] (-0.25,0) circle (2mm);
 334      \end{scope}
 335      \node at (-0.08,-0.3) {\(\frac{\pi}{8}\)};
 336      \node [font=\footnotesize, left] at (-0.75,-1) {\(\operatorname{Arg}(z+\frac{1}{4})=\frac{\pi}{8}\)};
 337      \node [brown, mydot] at (-0.25,0) {};
 338      \draw [<->, thick, green] (0,-1) -- (1.5,0.5) node [pos=0.25, black, font=\footnotesize, right] {\(|z-2|=|z-(1+i)|\)};
 339      \node [left, font=\footnotesize] at (0,-1) {\(-1\)};
 340      \node [below, font=\footnotesize] at (1,0) {\(1\)};
 341    \end{tikzpicture}\end{center}
 342
 343    \section{Vectors}
 344    \begin{center}\begin{tikzpicture}
 345      \draw [->] (-0.5,0) -- (3,0) node [right]  {\(x\)};
 346      \draw [->] (0,-0.5) -- (0,3) node [above] {\(y\)};
 347      \draw [orange, ->, thick] (0.5,0.5) -- (2.5,2.5) node [pos=0.5, above] {\(\vec{u}\)};
 348      \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
 349        \draw [gray, dashed, thick] (0.5,0.5) -- (2.5,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, below]{\(x\vec{i}\)};
 350        \draw [gray, dashed, thick] (2.5,0.5) -- (2.5,2.5) node [pos=0.5] {\midarrow};
 351      \end{scope}
 352      \node[black, right] at (2.5,1.5) {\(y\vec{j}\)};
 353    \end{tikzpicture}\end{center}
 354    \subsection*{Column notation}
 355
 356    \[\begin{bmatrix}x\\ y \end{bmatrix} \iff x\boldsymbol{i} + y\boldsymbol{j}\]
 357      \(\begin{bmatrix}x_2-x_1\\ y_2-y_1 \end{bmatrix}\) \quad between \(A(x_1,y_1), \> B(x_2,y_2)\)
 358
 359        \subsection*{Scalar multiplication}
 360
 361        \[k\cdot (x\boldsymbol{i}+y\boldsymbol{j})=kx\boldsymbol{i}+ky\boldsymbol{j}\]
 362
 363        \noindent For \(k \in \mathbb{R}^-\), direction is reversed
 364
 365        \subsection*{Vector addition}
 366        \begin{center}\begin{tikzpicture}[scale=1]
 367          \coordinate (A) at (0,0);
 368          \coordinate (B) at (2,2);
 369          \draw [->, thick, red] (0,0) -- (2,2) node [pos=0.5, below right] {\(\vec{u}=2\vec{i}+2\vec{j}\)};
 370          \draw [->, thick, blue] (2,2) -- (1,4) node [pos=0.5, above right] {\(\vec{v}=-\vec{i}+2\vec{j}\)};
 371          \draw [->, thick, orange] (0,0) -- (1,4) node [pos=0.5, left] {\(\vec{u}+\vec{v}=\vec{i}+4\vec{j}\)};
 372        \end{tikzpicture}\end{center}
 373
 374        \[(x\boldsymbol{i}+y\boldsymbol{j}) \pm (a\boldsymbol{i}+b\boldsymbol{j})=(x \pm a)\boldsymbol{i}+(y \pm b)\boldsymbol{j}\]
 375
 376        \begin{itemize}
 377          \item Draw each vector head to tail then join lines
 378          \item Addition is commutative (parallelogram)
 379          \item \(\boldsymbol{u}-\boldsymbol{v}=\boldsymbol{u}+(-\boldsymbol{v}) \implies \overrightharp{AB}=\boldsymbol{b}-\boldsymbol{a}\)
 380        \end{itemize}
 381
 382        \subsection*{Magnitude}
 383
 384        \[|(x\boldsymbol{i} + y\boldsymbol{j})|=\sqrt{x^2+y^2}\]
 385
 386        \subsection*{Parallel vectors}
 387
 388        \[\boldsymbol{u} || \boldsymbol{v} \iff \boldsymbol{u} = k \boldsymbol{v} \text{ where } k \in \mathbb{R} \setminus \{0\}\]
 389
 390        For parallel vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\):\\
 391        \[\boldsymbol{a \cdot b}=\begin{cases}
 392          |\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction}\\
 393          -|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions}
 394        \end{cases}\]
 395        %\includegraphics[width=0.2,height=\textheight]{graphics/parallelogram-vectors.jpg}
 396        %\includegraphics[width=1]{graphics/vector-subtraction.jpg}
 397
 398        \subsection*{Perpendicular vectors}
 399
 400        \[\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b} = 0\ \quad \text{(since \(\cos 90 = 0\))}\]
 401
 402        \subsection*{Unit vector \(|\hat{\boldsymbol{a}}|=1\)}
 403        \[\begin{split}\hat{\boldsymbol{a}} & = {\frac{1}{|\boldsymbol{a}|}}\boldsymbol{a} \\ & = \boldsymbol{a} \cdot {|\boldsymbol{a}|}\end{split}\]
 404
 405          \subsection*{Scalar product \(\boldsymbol{a} \cdot \boldsymbol{b}\)}
 406
 407
 408          \begin{center}\begin{tikzpicture}[scale=2]
 409            \draw [->] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{b}\)};
 410            \draw [->] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{a}\)};
 411            \begin{scope}
 412              \path[clip] (1,0.5) -- (1,0) -- (0,0);
 413              \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
 414              \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
 415            \end{scope}
 416          \end{tikzpicture}\end{center}
 417          \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*}
 418            \noindent\colorbox{cas}{On CAS: \texttt{dotP({[}a\ b\ c{]},\ {[}d\ e\ f{]})}}
 419
 420            \subsubsection*{Properties}
 421
 422            \begin{enumerate}
 423              \item
 424                \(k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k\boldsymbol{b})\)
 425              \item
 426                \(\boldsymbol{a \cdot 0}=0\)
 427              \item
 428                \(\boldsymbol{a} \cdot (\boldsymbol{b} + \boldsymbol{c})=\boldsymbol{a} \cdot \boldsymbol{b} + \boldsymbol{a} \cdot \boldsymbol{c}\)
 429              \item
 430                \(\boldsymbol{i \cdot i} = \boldsymbol{j \cdot j} = \boldsymbol{k \cdot k}= 1\)
 431              \item
 432                \(\boldsymbol{a} \cdot \boldsymbol{b} = 0 \quad \implies \quad \boldsymbol{a} \perp \boldsymbol{b}\)
 433              \item
 434                \(\boldsymbol{a \cdot a} = |\boldsymbol{a}|^2 = a^2\)
 435            \end{enumerate}
 436
 437            \subsection*{Angle between vectors}
 438
 439            \[\cos \theta = \frac{\boldsymbol{a} \cdot \boldsymbol{b}}{|\boldsymbol{a}| |\boldsymbol{b}|} = \frac{a_1 b_1 + a_2 b_2}{|\boldsymbol{a}| |\boldsymbol{b}|}\]
 440
 441            \noindent \colorbox{cas}{On CAS:} \texttt{angle([a b c], [a b c])}
 442
 443            (Action \(\rightarrow\) Vector \(\rightarrow\)Angle)
 444
 445            \subsection*{Angle between vector and axis}
 446
 447            \noindent For\(\boldsymbol{a} = a_1 \boldsymbol{i} + a_2 \boldsymbol{j} + a_3 \boldsymbol{k}\)
 448            which makes angles \(\alpha, \beta, \gamma\) with positive side of
 449            \(x, y, z\) axes:
 450            \[\cos \alpha = \frac{a_1}{|\boldsymbol{a}|}, \quad \cos \beta = \frac{a_2}{|\boldsymbol{a}|}, \quad \cos \gamma = \frac{a_3}{|\boldsymbol{a}|}\]
 451
 452            \noindent \colorbox{cas}{On CAS:} \texttt{angle({[}a\ b\ c{]},\ {[}1\ 0\ 0{]})}\\for angle
 453            between \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) and
 454            \(x\)-axis
 455
 456            \subsection*{Projections \& resolutes}
 457
 458            \begin{tikzpicture}[scale=3]
 459              \draw [->, purple] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{a}\)};
 460              \draw [->, orange] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{u}\)};
 461              \draw [->, blue] (1,0) -- (2,0) node [pos=0.5, below] {\(\boldsymbol{b}\)};
 462              \begin{scope}
 463                \path[clip] (1,0.5) -- (1,0) -- (0,0);
 464                \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
 465                \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
 466              \end{scope}
 467              \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
 468                \draw [gray, dashed, thick] (1,0) -- (1,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, right, rotate=-90]{\(\boldsymbol{w}\)};
 469              \end{scope}
 470              \draw (0,0) coordinate (O)
 471              (1,0) coordinate (A)
 472              (1,0.5) coordinate (B)
 473              pic [draw,red,angle radius=2mm] {right angle = O--A--B};
 474            \end{tikzpicture}
 475
 476            \subsubsection*{\(\parallel\boldsymbol{b}\) (vector projection/resolute)}
 477
 478            \begin{align*}
 479              \boldsymbol{u} & = \frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|^2}\boldsymbol{b} \\
 480              & = \left(\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\right)\left(\frac{\boldsymbol{b}}{|\boldsymbol{b}|}\right) \\
 481              & = (\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}
 482            \end{align*}
 483
 484            \subsubsection*{\(\perp\boldsymbol{b}\) (perpendicular projection)}
 485            \[\boldsymbol{w} = \boldsymbol{a} - \boldsymbol{u}\]
 486
 487            \subsubsection*{\(|\boldsymbol{u}|\) (scalar projection/resolute)}
 488            \begin{align*}
 489              s &= |\boldsymbol{u}|\\
 490              &= \boldsymbol{a} \cdot \hat{\boldsymbol{b}}\\
 491              &=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\\
 492              &= |\boldsymbol{a}| \cos \theta
 493            \end{align*}
 494
 495            \subsubsection*{Rectangular (\(\parallel,\perp\)) components}
 496
 497            \[\boldsymbol{a}=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}+\left(\boldsymbol{a}-\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}\right)\]
 498
 499
 500            \subsection*{Vector proofs}
 501
 502            \textbf{Concurrent:} intersection of \(\ge\) 3 lines
 503
 504            \begin{tikzpicture}
 505              \draw [blue] (0,0) -- (1,1);
 506              \draw [red] (1,0) -- (0,1);
 507              \draw [brown] (0.4,0) -- (0.6,1);
 508              \filldraw (0.5,0.5) circle (2pt);
 509            \end{tikzpicture}
 510
 511            \subsubsection*{Collinear points}
 512
 513            \(\ge\) 3 points lie on the same line
 514
 515            \begin{tikzpicture}
 516              \draw [purple] (0,0) -- (4,1);
 517              \filldraw (2,0.5) circle (2pt) node [above] {\(C\)};
 518              \filldraw (1,0.25) circle (2pt) node [above] {\(A\)};
 519              \filldraw (3,0.75) circle (2pt) node [above] {\(B\)};
 520              \coordinate (O) at (2.8,-0.2);
 521              \node at (O) [below] {\(O\)}; 
 522              \begin{scope}[->, orange, thick] 
 523                \draw (O) -- (2,0.5) node [pos=0.5, above, font=\footnotesize, black] {\(\boldsymbol{c}\)};
 524                \draw (O) -- (1,0.25) node [pos=0.5, below, font=\footnotesize, black] {\(\boldsymbol{a}\)};
 525                \draw (O) -- (3,0.75) node [pos=0.5, right, font=\footnotesize, black] {\(\boldsymbol{b}\)};
 526              \end{scope}
 527            \end{tikzpicture}
 528
 529            \begin{align*}
 530              \text{e.g. Prove that}\\
 531              \overrightharp{AC}=m\overrightharp{AB} \iff \boldsymbol{c}&=(1-m)\boldsymbol{a}+m\boldsymbol{b}\\
 532              \implies \boldsymbol{c} &= \overrightharp{OA} + \overrightharp{AC}\\
 533              &= \overrightharp{OA} + m\overrightharp{AB}\\
 534              &=\boldsymbol{a}+m(\boldsymbol{b}-\boldsymbol{a})\\
 535              &=\boldsymbol{a}+m\boldsymbol{b}-m\boldsymbol{a}\\
 536              &=(1-m)\boldsymbol{a}+m{b}
 537            \end{align*}
 538            \begin{align*}
 539              \text{Also, } \implies \overrightharp{OC} &= \lambda \vec{OA} + \mu \overrightharp{OB} \\
 540              \text{where } \lambda + \mu &= 1\\
 541              \text{If } C \text{ lies along } \overrightharp{AB}, & \implies 0 < \mu < 1
 542            \end{align*}
 543
 544
 545            \subsubsection*{Parallelograms}
 546
 547            \begin{center}\begin{tikzpicture}
 548              \coordinate (O) at (0,0) node [below left] {\(O\)};
 549              \coordinate (A) at (4,0);
 550              \coordinate (B) at (6,2);
 551              \coordinate (C) at (2,2);
 552              \coordinate (D) at (6,0);
 553
 554              \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (O)--(A) node [below left] {\(A\)};
 555              \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (A)--(B) node [above right] {\(B\)};
 556              \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (B)--(C) node [above left] {\(C\)};
 557              \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (C)--(O);
 558
 559              \draw [gray, dashed] (O) -- (B) node [pos=0.75] {\(\diagdown\diagdown\)} node [pos=0.25] {\(\diagdown\diagdown\)};
 560              \draw [gray, dashed] (A) -- (C) node [pos=0.75] {\(\diagup\)} node [pos=0.25] {\(\diagup\)};
 561              \begin{scope}
 562                \path[clip] (C) -- (A) -- (O);
 563                \fill[orange, opacity=0.5, draw=black] (0,0) circle (4mm);
 564                \node at ($(0,0)+(20:8mm)$) {\(\theta\)};
 565              \end{scope}
 566              \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] {\(|\boldsymbol{c}|\cos\theta\)};
 567              \draw pic [draw,thick,red,angle radius=2mm] {right angle=O--D--B};
 568            \end{tikzpicture}\end{center}
 569
 570            \begin{itemize}
 571              \item
 572                Diagonals \(\overrightharp{OB}, \overrightharp{AC}\) bisect each other
 573              \item
 574                If diagonals are equal length, it is a rectangle
 575              \item
 576                \(|\overrightharp{OB}|^2 + |\overrightharp{CA}|^2 = |\overrightharp{OA}|^2 + |\overrightharp{AB}|^2 + |\overrightharp{CB}|^2 + |\overrightharp{OC}|^2\)
 577              \item
 578                Area \(=\boldsymbol{c} \cdot \boldsymbol{a}\)
 579            \end{itemize}
 580
 581            \subsubsection*{Useful vector properties}
 582
 583            \begin{itemize}
 584              \item
 585                \(\boldsymbol{a} \parallel \boldsymbol{b} \implies \boldsymbol{b}=k\boldsymbol{a}\) for some
 586                \(k \in \mathbb{R} \setminus \{0\}\)
 587              \item
 588                If \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are parallel with at
 589                least one point in common, then they lie on the same straight line
 590              \item
 591                \(\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b}=0\)
 592              \item
 593                \(\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2\)
 594            \end{itemize}
 595
 596            \subsection*{Linear dependence}
 597
 598            \(\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\) are linearly dependent if they are \(\nparallel\) and:
 599            \begin{align*}
 600              0&=k\boldsymbol{a}+l\boldsymbol{b}+m\boldsymbol{c}\\
 601              \therefore \boldsymbol{c} &= m\boldsymbol{a} + n\boldsymbol{b} \quad \text{(simultaneous)}
 602            \end{align*}
 603
 604            \noindent \(\boldsymbol{a}, \boldsymbol{b},\) and \(\boldsymbol{c}\) are linearly
 605            independent if no vector in the set is expressible as a linear
 606            combination of other vectors in set, or if they are parallel.
 607
 608            \subsection*{Three-dimensional vectors}
 609
 610            Right-hand rule for axes: \(z\) is up or out of page.
 611
 612            \tdplotsetmaincoords{60}{120} 
 613            \begin{center}\begin{tikzpicture} [scale=3, tdplot_main_coords, axis/.style={->,thick}, 
 614              vector/.style={-stealth,red,very thick}, 
 615              vector guide/.style={dashed,gray,thick}]
 616
 617              %standard tikz coordinate definition using x, y, z coords
 618              \coordinate (O) at (0,0,0);
 619
 620              %tikz-3dplot coordinate definition using x, y, z coords
 621
 622              \pgfmathsetmacro{\ax}{1}
 623              \pgfmathsetmacro{\ay}{1}
 624              \pgfmathsetmacro{\az}{1}
 625
 626              \coordinate (P) at (\ax,\ay,\az);
 627
 628              %draw axes
 629              \draw[axis] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$};
 630              \draw[axis] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$};
 631              \draw[axis] (0,0,0) -- (0,0,1) node[anchor=south]{$z$};
 632
 633              %draw a vector from O to P
 634              \draw[vector] (O) -- (P);
 635
 636              %draw guide lines to components
 637              \draw[vector guide]         (O) -- (\ax,\ay,0);
 638              \draw[vector guide] (\ax,\ay,0) -- (P);
 639              \draw[vector guide]         (P) -- (0,0,\az);
 640              \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
 641              \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
 642              \draw[vector guide] (\ax,\ay,0) -- (\ax,0,0);
 643              \node[tdplot_main_coords,above right]
 644              at (\ax,\ay,\az){(\ax, \ay, \az)};
 645            \end{tikzpicture}\end{center}
 646
 647            \subsection*{Parametric vectors}
 648
 649            Parametric equation of line through point \((x_0, y_0, z_0)\) and
 650            parallel to \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) is:
 651
 652            \[\begin{cases}x = x_o + a \cdot t \\ y = y_0 + b \cdot t \\ z = z_0 + c \cdot t\end{cases}\]
 653
 654              \section{Circular functions}
 655
 656              \(\sin(bx)\) or \(\cos(bx)\): period \(=\frac{2\pi}{b}\)
 657
 658              \noindent \(\tan(nx)\): period \(=\frac{\pi}{n}\)\\
 659              \indent\indent\indent asymptotes at \(x=\frac{(2k+1)\pi}{2n} \> \vert \> k \in \mathbb{Z}\)
 660
 661              \subsection*{Reciprocal functions}
 662
 663              \subsubsection*{Cosecant}
 664
 665              \[\operatorname{cosec} \theta = \frac{1}{\sin \theta} \> \vert \> \sin \theta \ne 0\]
 666
 667              \begin{itemize}
 668                \item
 669                  \textbf{Domain} \(= \mathbb{R} \setminus {n\pi : n \in \mathbb{Z}}\)
 670                \item
 671                  \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
 672                \item
 673                  \textbf{Turning points} at
 674                  \(\theta = \frac{(2n + 1)\pi}{2} \> \vert \> n \in \mathbb{Z}\)
 675                \item
 676                  \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
 677              \end{itemize}
 678
 679              \subsubsection*{Secant}
 680
 681\begin{tikzpicture}
 682  \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={\(-2\pi\), \(-\pi\), \(\pi\), \(2\pi\)}]
 683%    \addplot[blue, domain=-6.2830:6.2830,unbounded coords=jump,samples=80] {sec(deg(x))};
 684    \addplot[blue, restrict y to domain=-10:10, domain=-7:7,samples=100] {sec(deg(x))} node [pos=0.93, black, right] {\(\operatorname{sec} x\)};
 685    \addplot[red, dashed, domain=-7:7,samples=100] {cos(deg(x))};
 686    \draw [gray, dotted, thick] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{rel axis cs:0,1});
 687    \draw [gray, dotted, thick] ({axis cs:4.71239,0}|-{rel axis cs:0,0}) -- ({axis cs:4.71239,0}|-{rel axis cs:0,1});
 688    \draw [gray, dotted, thick] ({axis cs:-4.71239,0}|-{rel axis cs:0,0}) -- ({axis cs:-4.71239,0}|-{rel axis cs:0,1});
 689    \draw [gray, dotted, thick] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1});
 690\end{axis}
 691    \node [black] at (7,3.5) {\(\cos x\)};
 692\end{tikzpicture}
 693
 694                \[\operatorname{sec} \theta = \frac{1}{\cos \theta} \> \vert \> \cos \theta \ne 0\]
 695
 696                \begin{itemize}
 697
 698                  \item
 699                    \textbf{Domain}
 700                    \(= \mathbb{R} \setminus \frac{(2n + 1) \pi}{2} : n \in \mathbb{Z}\}\)
 701                  \item
 702                    \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
 703                  \item
 704                    \textbf{Turning points} at
 705                    \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
 706                  \item
 707                    \textbf{Asymptotes} at
 708                    \(\theta = \frac{(2n + 1) \pi}{2} \> \vert \> n \in \mathbb{Z}\)
 709                \end{itemize}
 710
 711                \subsubsection*{Cotangent}
 712
 713\begin{tikzpicture}
 714  \begin{axis}[xmin=-3,xmax=3,ymin=-1.5,ymax=1.5,enlargelimits=true, xtick={-3.1415, -1.5708, 1.5708, 3.1415},xticklabels={\(-\pi\), \(-\frac{\pi}{2}\), \(\frac{\pi}{2}\), \(\pi\)}]
 715    \addplot[blue, smooth, domain=-3:-0.1,unbounded coords=jump,samples=105] {cot(deg(x))} node [pos=0.3, left] {\(\operatorname{cot} x\)};
 716\addplot[blue, smooth, domain=0.1:3,unbounded coords=jump,samples=105] {cot(deg(x))};
 717\addplot[red, smooth, dashed] gnuplot [domain=-1.5:1.5,unbounded coords=jump,samples=105] {tan(x)};
 718\addplot[red, smooth, dashed] gnuplot [domain=-3.5:-1.8,unbounded coords=jump,samples=105] {tan(x)} node [pos=0.5, right] {\(\tan x\)};
 719\addplot[red, smooth, dashed] gnuplot [domain=1.8:3.5,unbounded coords=jump,samples=105] {tan(x)};
 720    \draw [thick, red, dotted] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1});
 721    \draw [thick, blue, dotted] ({axis cs:-3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:-3.1415,0}|-{rel axis cs:0,1});
 722    \draw [thick, blue, dotted] ({axis cs:0,0}|-{rel axis cs:0,0}) -- ({axis cs:0,0}|-{rel axis cs:0,1});
 723    \draw [thick, blue, dotted] ({axis cs:3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:3.1415,0}|-{rel axis cs:0,1});
 724    \draw [thick, red, dotted] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{rel axis cs:0,1});
 725\end{axis}
 726\end{tikzpicture}
 727
 728                  \[\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0\]
 729
 730                  \begin{itemize}
 731
 732                    \item
 733                      \textbf{Domain} \(= \mathbb{R} \setminus \{n \pi: n \in \mathbb{Z}\}\)
 734                    \item
 735                      \textbf{Range} \(= \mathbb{R}\)
 736                    \item
 737                      \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
 738                  \end{itemize}
 739
 740                  \subsubsection*{Symmetry properties}
 741
 742                  \[\begin{split}
 743                    \operatorname{sec} (\pi \pm x) & = -\operatorname{sec} x \\
 744                    \operatorname{sec} (-x) & = \operatorname{sec} x \\
 745                    \operatorname{cosec} (\pi \pm x) & = \mp \operatorname{cosec} x \\
 746                    \operatorname{cosec} (-x) & = - \operatorname{cosec} x \\
 747                    \operatorname{cot} (\pi \pm x) & = \pm \operatorname{cot} x \\
 748                    \operatorname{cot} (-x) & = - \operatorname{cot} x
 749                  \end{split}\]
 750
 751                  \subsubsection*{Complementary properties}
 752
 753                  \[\begin{split}
 754                    \operatorname{sec} \left({\pi \over 2} - x\right) & = \operatorname{cosec} x \\
 755                    \operatorname{cosec} \left({\pi \over 2} - x\right) & = \operatorname{sec} x \\
 756                    \operatorname{cot} \left({\pi \over 2} - x\right) & = \tan x \\
 757                    \tan \left({\pi \over 2} - x\right) & = \operatorname{cot} x
 758                  \end{split}\]
 759
 760                  \subsubsection*{Pythagorean identities}
 761
 762                  \[\begin{split}
 763                    1 + \operatorname{cot}^2 x & = \operatorname{cosec}^2 x, \quad \text{where } \sin x \ne 0 \\
 764                    1 + \tan^2 x & = \operatorname{sec}^2 x, \quad \text{where } \cos x \ne 0
 765                  \end{split}\]
 766
 767                  \subsection*{Compound angle formulas}
 768
 769                  \[\cos(x \pm y) = \cos x + \cos y \mp \sin x \sin y\]
 770                  \[\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y\]
 771                  \[\tan(x \pm y) = {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}\]
 772
 773                  \subsection*{Double angle formulas}
 774
 775                  \[\begin{split}
 776                    \cos 2x &= \cos^2 x - \sin^2 x \\
 777                    & = 1 - 2\sin^2 x \\
 778                    & = 2 \cos^2 x -1
 779                  \end{split}\]
 780
 781                  \[\sin 2x = 2 \sin x \cos x\]
 782
 783                  \[\tan 2x = {{2 \tan x} \over {1 - \tan^2 x}}\]
 784
 785                  \subsection*{Inverse circular functions}
 786
 787                  \begin{tikzpicture}
 788                    \begin{axis}[ymin=-2, ymax=4, xmin=-1.1, xmax=1.1, ytick={-1.5708, 1.5708, 3.14159},yticklabels={$-\frac{\pi}{2}$, $\frac{\pi}{2}$, $\pi$}]
 789                      \addplot[color=red, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {asin(x)} node [pos=0.25, below right] {\(\sin^{-1}x\)};
 790                      \addplot[color=blue, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {acos(x)} node [pos=0.25, below left] {\(\cos^{-1}x\)};
 791                      \addplot[mark=*, red] coordinates {(-1,-1.5708)} node[right, font=\footnotesize]{\((-1,-\frac{\pi}{2})\)} ;
 792                      \addplot[mark=*, red] coordinates {(1,1.5708)} node[left, font=\footnotesize]{\((1,\frac{\pi}{2})\)} ;
 793                      \addplot[mark=*, blue] coordinates {(1,0)};
 794                      \addplot[mark=*, blue] coordinates {(-1,3.1415)} node[right, font=\footnotesize]{\((-1,\pi)\)} ;
 795                    \end{axis}
 796                  \end{tikzpicture}\\
 797
 798                  Inverse functions: \(f(f^{-1}(x)) = x\) (restrict domain)
 799
 800                  \[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y\]
 801                  \hfill where \(\sin y = x, \> y \in [{-\pi \over 2}, {\pi \over 2}]\)
 802
 803                  \[\cos^{-1}: [-1,1] \rightarrow \mathbb{R}, \quad \cos^{-1} x = y\]
 804                  \hfill where \(\cos y = x, \> y \in [0, \pi]\)
 805
 806                  \[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\]
 807                  \hfill where \(\tan y = x, \> y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\)
 808
 809                  \begin{tikzpicture}
 810                    \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={\(-\frac{\pi}{2}\),\(\frac{\pi}{2}\)}]
 811                      \addplot[color=orange, smooth] gnuplot [domain=-35:35, unbounded coords=jump,samples=350] {atan(x)} node [pos=0.5, above left] {\(\tan^{-1}x\)};
 812                      \addplot[gray, dotted, thick, domain=-35:35] {1.5708} node [black, font=\footnotesize, below right, pos=0] {\(y=\frac{\pi}{2}\)};
 813                      \addplot[gray, dotted, thick, domain=-35:35] {-1.5708} node [black, font=\footnotesize, above left, pos=1] {\(y=-\frac{\pi}{2}\)};
 814                    \end{axis}
 815                  \end{tikzpicture}
 816\columnbreak
 817                  \section{Differential calculus}
 818
 819                  \subsection*{Limits}
 820
 821                  \[\lim_{x \rightarrow a}f(x)\]
 822                  \(L^-,\quad L^+\) \qquad limit from below/above\\
 823                  \(\lim_{x \to a} f(x)\) \quad limit of a point\\
 824
 825                  \noindent For solving \(x\rightarrow\infty\), put all \(x\) terms in denominators\\
 826                  e.g. \[\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2\]
 827
 828                  \subsubsection*{Limit theorems}
 829
 830                  \begin{enumerate}
 831                    \item
 832                      For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\)
 833                    \item
 834                      \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\)
 835                    \item
 836                      \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\)
 837                    \item
 838                      \(\therefore \lim_{x \rightarrow a} c \times f(x)=cF\) where \(c=\) constant
 839                    \item
 840                      \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\)
 841                    \item
 842                      \(f(x)\) is continuous \(\iff L^-=L^+=f(x) \> \forall x\)
 843                  \end{enumerate}
 844
 845                  \subsection*{Gradients of secants and tangents}
 846
 847                  \textbf{Secant (chord)} - line joining two points on curve\\
 848                  \textbf{Tangent} - line that intersects curve at one point
 849
 850                  \subsection*{First principles derivative}
 851
 852                  \[f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={\frac{dy}{dx}}\]
 853
 854                  \subsubsection*{Logarithmic identities}
 855
 856                  \(\log_b (xy)=\log_b x + \log_b y\)\\
 857                  \(\log_b x^n = n \log_b x\)\\
 858                  \(\log_b y^{x^n} = x^n \log_b y\)
 859
 860                  \subsubsection*{Index identities}
 861
 862                  \(b^{m+n}=b^m \cdot b^n\)\\
 863                  \((b^m)^n=b^{m \cdot n}\)\\
 864                  \((b \cdot c)^n = b^n \cdot c^n\)\\
 865                  \({a^m \div a^n} = {a^{m-n}}\)
 866
 867                  \subsection*{Reciprocal derivatives}
 868
 869                  \[\frac{1}{\frac{dy}{dx}} = \frac{dx}{dy}\]
 870
 871                  \subsection*{Differentiating \(x=f(y)\)}
 872                  Find \(\dfrac{dx}{dy}\), then \(\dfrac{dy}{dx} = \dfrac{1}{\left(\dfrac{dx}{dy}\right)}\)
 873
 874                  \subsection*{Second derivative}
 875                  \begin{align*}f(x) \longrightarrow &f^\prime (x) \longrightarrow f^{\prime\prime}(x)\\
 876                  \implies y \longrightarrow &\frac{dy}{dx} \longrightarrow \frac{d^2 y}{dx^2}\end{align*}
 877
 878                  \noindent Order of polynomial \(n\)th derivative decrements each time the derivative is taken
 879
 880                  \subsubsection*{Points of Inflection}
 881
 882                  \emph{Stationary point} - i.e.
 883                  \(f^\prime(x)=0\)\\
 884                  \emph{Point of inflection} - max \(|\)gradient\(|\) (i.e.
 885                  \(f^{\prime\prime} = 0\))
 886
 887                  \subsubsection*{Strictly increasing/decreasing}
 888
 889                  For \(x_2\) and \(x_1\) where \(x_2 > x_1\):
 890
 891                  \textbf{strictly increasing}\\
 892                  \-\hspace{1em}where \(f(x_2) > f(x_1)\) or \(f^\prime(x)>0\)
 893
 894                  \textbf{strictly decreasing}\\
 895                  \hspace{1em}where \(f(x_2) < f(x_1)\) or \(f^\prime(x)<0\)
 896                  \begin{warning}
 897                    Endpoints are included, even where \(\boldsymbol{\frac{dy}{dx}=0}\)
 898                  \end{warning}
 899
 900
 901                  \begin{table*}[ht]
 902                    \centering
 903                    \begin{tabularx}{\textwidth}{|r|Y|Y|Y|}
 904                      \hline
 905                      \rowcolor{lblue}
 906                      & \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)} \\
 907                      \hline
 908                      \(\dfrac{dy}{dx}>0\) &
 909                      \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)}&
 910                        \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)}&
 911                          \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}\\
 912                            \hline
 913                            \(\dfrac{dy}{dx}<0\) &
 914                            \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)}&
 915                              \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)}&
 916                                \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}\\
 917                                  \hline
 918                                  \(\dfrac{dy}{dx}=0\)&
 919                                  \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}&
 920                                    \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}\\
 921                                      \hline
 922                    \end{tabularx}
 923                  \end{table*}
 924                  \begin{itemize}
 925                    \item
 926                      \(f^\prime (a) = 0, \> f^{\prime\prime}(a) > 0\) \\
 927                      \textbf{local min} at \((a, f(a))\) (concave up)
 928                    \item
 929                      \(f^\prime (a) = 0, \>  f^{\prime\prime} (a) < 0\) \\
 930                      \textbf{local max} at \((a, f(a))\) (concave down)
 931                    \item
 932                      \(f^{\prime\prime}(a) = 0\) \\
 933                      \textbf{point of inflection} at \((a, f(a))\)
 934                    \item
 935                      \(f^{\prime\prime}(a) = 0, \> f^\prime(a)=0\) \\
 936                      stationary point of inflection at \((a, f(a)\)
 937                  \end{itemize}
 938
 939                  \subsection*{Implicit Differentiation}
 940
 941                  \noindent Used for differentiating circles etc.
 942
 943                  If \(p\) and \(q\) are expressions in \(x\) and \(y\) such that \(p=q\),
 944                  for all \(x\) and \(y\), then:
 945
 946                  \[{\frac{dp}{dx}} = {\frac{dq}{dx}} \quad \text{and} \quad {\frac{dp}{dy}} = {\frac{dq}{dy}}\]
 947
 948                  \begin{cas}
 949                    Action \(\rightarrow\) Calculation \\
 950                      \-\hspace{1em}\texttt{impDiff(y\^{}2+ax=5,\ x,\ y)}
 951                  \end{cas}
 952
 953                  \subsection*{Slope fields}
 954
 955                  \begin{tikzpicture}[declare function={diff(\x,\y) = \x+\y;}]
 956                    \begin{axis}[axis equal, ymin=-4, ymax=4, xmin=-4, xmax=4, ticks=none, enlargelimits=true, ]
 957                      \addplot[thick, orange, domain=-4:2] {e^(x)-x-1};
 958                      \pgfplotsinvokeforeach{-4,...,4}{%
 959                        \draw[gray] ( {#1 -0.1}, {4 - diff(#1, 4) *0.1}) --  ( {#1 +0.1}, {4  + diff(#1, 4) *0.1});
 960                        \draw[gray] ( {#1 -0.1}, {3 - diff(#1, 3) *0.1}) --  ( {#1 +0.1}, {3  + diff(#1, 3) *0.1});
 961                        \draw[gray] ( {#1 -0.1}, {2 - diff(#1, 2) *0.1}) --  ( {#1 +0.1}, {2  + diff(#1, 2) *0.1});
 962                        \draw[gray] ( {#1 -0.1}, {1 - diff(#1, 1) *0.1}) --  ( {#1 +0.1}, {1  + diff(#1, 1) *0.1});
 963                        \draw[gray] ( {#1 -0.1}, {0 - diff(#1, 0) *0.1}) --  ( {#1 +0.1}, {0  + diff(#1, 0) *0.1});
 964                        \draw[gray] ( {#1 -0.1}, {-1 - diff(#1, -1) *0.1}) --  ( {#1 +0.1}, {-1  + diff(#1, -1) *0.1});
 965                        \draw[gray] ( {#1 -0.1}, {-2 - diff(#1, -2) *0.1}) --  ( {#1 +0.1}, {-2  + diff(#1, -2) *0.1});
 966                        \draw[gray] ( {#1 -0.1}, {-3 - diff(#1, -3) *0.1}) --  ( {#1 +0.1}, {-3  + diff(#1, -3) *0.1});
 967                        \draw[gray] ( {#1 -0.1}, {-4 - diff(#1, -4) *0.1}) --  ( {#1 +0.1}, {-4  + diff(#1, -4) *0.1});
 968                      }
 969                    \end{axis}
 970                  \end{tikzpicture}
 971
 972                  \subsection*{Parametric equations}
 973
 974
 975                  \begin{align*}
 976                    \dfrac{dy}{dt} &= \dfrac{dy}{dx} \cdot \dfrac{dx}{dt} \\
 977                    \therefore \dfrac{dy}{dx} &= \dfrac{\left(\dfrac{dy}{dt}\right)}{\left(\dfrac{dx}{dt}\right)} \text{ provided } \dfrac{dx}{dt} \ne 0 \\
 978                    \dfrac{d^2y}{dx^2} &= \dfrac{\left(\dfrac{dy^\prime}{dt}\right)}{\left(\dfrac{dx}{dt}\right)} \text{ where } y^\prime = \dfrac{dy}{dx}
 979                  \end{align*}
 980
 981                \subsection*{Integration}
 982
 983                \[\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)\]
 984
 985                  \subsubsection*{Definite integrals}
 986
 987                  \[\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)\]
 988
 989                  \begin{itemize}
 990
 991                    \item
 992                      Signed area enclosed by\\
 993                      \(\> y=f(x), \quad y=0, \quad x=a, \quad x=b\).
 994                    \item
 995                      \emph{Integrand} is \(f\).
 996                  \end{itemize}
 997
 998                  \subsubsection*{Properties}
 999
1000                  \begin{align*}
1001                    \int^b_a f(x) \> dx &= \int^c_a f(x) \> dx + \int^b_c f(x) \> dx \\
1002                    \int^a_a f(x) \> dx &= 0 \\
1003                    \int^b_a k \cdot f(x) \> dx &= k \int^b_a f(x) \> dx \\
1004                    \int^b_a f(x) \pm g(x) \> dx &= \int^b_a f(x) \> dx \pm \int^b_a g(x) \> dx \\
1005                    \int^b_a f(x) \> dx &= - \int^a_b f(x) \> dx \\
1006                  \end{align*}
1007
1008                  \subsection*{Integration by substitution}
1009
1010                  \[\int f(u) {\frac{du}{dx}} \cdot dx = \int f(u) \cdot du\]
1011
1012                  \begin{warning}
1013                    \(\boldsymbol{f(u)}\) must be 1:1 \(\boldsymbol{\implies}\) one \(\boldsymbol{x}\) for each \(\boldsymbol{y}\)
1014                  \end{warning}
1015                  \begin{align*}\text{e.g. for } y&=\int(2x+1)\sqrt{x+4} \cdot dx\\
1016                    \text{let } u&=x+4\\
1017                    \implies& {\frac{du}{dx}} = 1\\
1018                    \implies& x = u - 4\\
1019                    \text{then } &y=\int (2(u-4)+1)u^{\frac{1}{2}} \cdot du\\
1020                    &\text{(solve as  normal integral)}
1021                  \end{align*}
1022
1023                  \subsubsection*{Definite integrals by substitution}
1024
1025                  For \(\int^b_a f(x) {\frac{du}{dx}} \cdot dx\), evaluate new \(a\) and
1026                  \(b\) for \(f(u) \cdot du\).
1027
1028                  \subsubsection*{Trigonometric integration}
1029
1030                  \[\sin^m x \cos^n x \cdot dx\]
1031
1032                  \paragraph{\textbf{\(m\) is odd:}}
1033                  \(m=2k+1\) where \(k \in \mathbb{Z}\)\\
1034                  \(\implies \sin^{2k+1} x = (\sin^2 z)^k \sin x = (1 - \cos^2 x)^k \sin x\)\\
1035                  Substitute \(u=\cos x\)
1036
1037                  \paragraph{\textbf{\(n\) is odd:}}
1038                  \(n=2k+1\) where \(k \in \mathbb{Z}\)\\
1039                  \(\implies \cos^{2k+1} x = (\cos^2 x)^k \cos x = (1-\sin^2 x)^k \cos x\)\\
1040                  Substitute \(u=\sin x\)
1041
1042                  \paragraph{\textbf{\(m\) and \(n\) are even:}}
1043                  use identities...
1044
1045                  \begin{itemize}
1046
1047                    \item
1048                      \(\sin^2x={1 \over 2}(1-\cos 2x)\)
1049                    \item
1050                      \(\cos^2x={1 \over 2}(1+\cos 2x)\)
1051                    \item
1052                      \(\sin 2x = 2 \sin x \cos x\)
1053                  \end{itemize}
1054
1055                  \subsection*{Partial fractions}
1056
1057                  To factorise \(f(x) = \frac{\delta}{\alpha \cdot \beta}\):
1058                  \begin{align*}
1059                    \dfrac{\delta}{\alpha \cdot \beta \cdot \gamma} &= \dfrac{A}{\alpha} + \dfrac{B}{\beta} + \dfrac{C}{\gamma} \tag{1} \\
1060                    \text{Multiply by } & (\alpha \cdot \beta \cdot \gamma) \text{:} \\
1061                    \delta &= \beta\gamma A + \alpha\gamma B +\alpha\beta C \tag{2} \\
1062                    \text{Substitute } x &= \{\alpha, \beta, \gamma\} \text{ into (2) to find denominators}
1063                  \end{align*}
1064
1065                  \subsubsection*{Repeated linear factors}
1066
1067                  \[ \dfrac{p(x)}{(x-a)^n} = \dfrac{A_1}{(x-a)} + \dfrac{A_2}{(x-a)^2} + \dots + \dfrac{A_n}{(x-a)^n} \]
1068
1069                  \subsubsection*{Irreducible quadratic factors}
1070
1071                  \[ \text{e.g. } \dfrac{3x-4}{(2x-3)(x^2+5)} = \dfrac{A}{2x-3} + \dfrac{Bx+C}{x^2+5} \]
1072
1073                  \begin{cas}
1074                    Action \(\rightarrow\) Transformation:\\
1075                    \-\hspace{1em} \texttt{expand(..., x)}
1076
1077                    To reverse, use \texttt{combine(...)}
1078                  \end{cas}
1079
1080                  \subsection*{Graphing integrals on CAS}
1081
1082                  \begin{cas}
1083                    \textbf{In main:} Interactive \(\rightarrow\) Calculation \(\rightarrow\) \(\int\)\\
1084                    For restrictions, \texttt{Define\ f(x)=...} then \texttt{f(x)\textbar{}x\textgreater{}...}
1085                  \end{cas}
1086
1087                  \subsection*{Applications of antidifferentiation}
1088
1089                  \begin{itemize}
1090
1091                    \item
1092                      \(x\)-intercepts of \(y=f(x)\) identify \(x\)-coordinates of
1093                      stationary points on \(y=F(x)\)
1094                    \item
1095                      nature of stationary points is determined by sign of \(y=f(x)\) on
1096                      either side of its \(x\)-intercepts
1097                    \item
1098                      if \(f(x)\) is a polynomial of degree \(n\), then \(F(x)\) has degree
1099                      \(n+1\)
1100                  \end{itemize}
1101
1102                  To find stationary points of a function, substitute \(x\) value of given
1103                  point into derivative. Solve for \({\frac{dy}{dx}}=0\). Integrate to find
1104                  original function.
1105
1106                  \subsection*{Solids of revolution}
1107
1108                  Approximate as sum of infinitesimally-thick cylinders
1109
1110                  \subsubsection*{Rotation about \(\boldsymbol{x}\)-axis}
1111
1112                  \[ V = \pi\int^{x=b}_{x=a} f(x)^2 \> dx \]
1113
1114                  \subsubsection*{Rotation about \(\boldsymbol{y}\)-axis}
1115
1116                  \begin{align*}
1117                    V &= \pi \int^{y=b}_{y=a} x^2 \> dy \\
1118                    &= \pi \int^{y=b}_{y=a} (f(y))^2 \> dy
1119                  \end{align*}
1120
1121                  \subsubsection*{Regions not bound by \(\boldsymbol{y=0}\)}
1122
1123                  \[V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx\]
1124                  \hfill where \(f(x) > g(x)\)
1125
1126                  \subsection*{Length of a curve}
1127
1128                  For length of \(f(x)\) from \(x=a \rightarrow x=b\):
1129                  \begin{align*}
1130                    &\text{Cartesian} \> & L &= \int^b_a \sqrt{1 + \left(\dfrac{dy}{dx}\right)^2} \> dx \\
1131                    &\text{Parametric} \> & L & = \int^b_a \sqrt{\left(\dfrac{dx}{dt}\right)^2 + \left(\dfrac{dy}{dt}\right)^2} \> dt
1132                  \end{align*}
1133
1134                  \begin{cas}
1135                    \begin{enumerate}[label=\alph*), leftmargin=5mm]
1136                      \item Evaluate formula
1137                      \item Interactive \(\rightarrow\) Calculation \(\rightarrow\) Line \(\rightarrow\) \texttt{arcLen}
1138                    \end{enumerate}
1139                  \end{cas}
1140
1141                  \subsection*{Rates}
1142
1143                  \subsubsection*{Gradient at a point on parametric curve}
1144
1145                  \[{\frac{dy}{dx}} = {{\frac{dy}{dt}} \div {\frac{dx}{dt}}} \> \vert \> {\frac{dx}{dt}} \ne 0 \text{ (chain rule)}\]
1146
1147                  \[\frac{d^2}{dx^2} = \frac{d(y^\prime)}{dx} = {\frac{dy^\prime}{dt} \div {\frac{dx}{dt}}} \> \vert \> y^\prime = {\frac{dy}{dx}}\]
1148
1149                  \subsection*{Rational functions}
1150
1151                  \[f(x) = \frac{P(x)}{Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}\]
1152
1153                  \subsection*{Fundamental theorem of calculus}
1154
1155                  If \(f\) is continuous on \([a, b]\), then
1156
1157                  \[\int^b_a f(x) \> dx = F(b) - F(a)\]
1158                  \hfill where \(F = \int f \> dx\)
1159                  
1160                  \subsection*{Differential equations}
1161
1162                  \noindent\textbf{Order} - highest power inside derivative\\
1163                  \textbf{Degree} - highest power of highest derivative\\
1164                  e.g. \({\left(\dfrac{dy^2}{d^2} x\right)}^3\) \qquad order 2, degree 3
1165
1166                  \begin{warning}
1167                    To verify solutions, find \(\frac{dy}{dx}\) from \(y\) and substitute into original
1168                  \end{warning}
1169
1170                  \subsubsection*{Function of the dependent
1171                  variable}
1172
1173                  If \({\frac{dy}{dx}}=g(y)\), then
1174                  \(\frac{dx}{dy} = 1 \div \frac{dy}{dx} = \frac{1}{g(y)}\). Integrate both sides to solve equation. Only add \(c\) on one side. Express
1175                  \(e^c\) as \(A\).
1176
1177
1178
1179                  \subsubsection*{Mixing problems}
1180
1181                  \[\left(\frac{dm}{dt}\right)_\Sigma = \left(\frac{dm}{dt}\right)_{\text{in}} - \left(\frac{dm}{dt}_{\text{out}}\right)\]
1182
1183                  \subsubsection*{Separation of variables}
1184
1185                  If \({\frac{dy}{dx}}=f(x)g(y)\), then:
1186
1187                  \[\int f(x) \> dx = \int \frac{1}{g(y)} \> dy\]
1188
1189                  \subsubsection*{Euler's method for solving DEs}
1190
1191                  \[\frac{f(x+h) - f(x)}{h} \approx f^\prime (x) \quad \text{for small } h\]
1192
1193                  \[\implies f(x+h) \approx f(x) + hf^\prime(x)\]
1194
1195                  \include{calculus-rules}
1196
1197    \section{Kinematics \& Mechanics}
1198
1199      \subsection*{Constant acceleration}
1200
1201      \begin{itemize}
1202        \item \textbf{Position} - relative to origin
1203        \item \textbf{Displacement} - relative to starting point
1204      \end{itemize}
1205
1206      \subsubsection*{Velocity-time graphs}
1207
1208      \begin{description}[nosep, labelindent=0.5cm, leftmargin=0.5\columnwidth]
1209        \item[Displacement:] \textit{signed} area
1210        \item[Distance travelled:] \textit{total} area
1211      \end{description}
1212
1213      \[ \text{acceleration} = \frac{d^2x}{dt^2} = \frac{dv}{dt} = v\frac{dv}{dx} = \frac{d}{dx}\left(\frac{1}{2}v^2\right) \]
1214
1215        \begin{center}
1216          \renewcommand{\arraystretch}{1}
1217          \begin{tabular}{ l r }
1218            \hline & no \\ \hline
1219            \(v=u+at\) & \(x\) \\
1220            \(v^2 = u^2+2as\) & \(t\) \\
1221            \(s = \frac{1}{2} (v+u)t\) & \(a\) \\
1222            \(s = ut + \frac{1}{2} at^2\) & \(v\) \\
1223            \(s = vt- \frac{1}{2} at^2\) & \(u\) \\ \hline
1224          \end{tabular}
1225        \end{center}
1226
1227        \[ v_{\text{avg}} = \frac{\Delta\text{position}}{\Delta t} \]
1228        \begin{align*}
1229          \text{speed} &= |{\text{velocity}}| \\
1230          &= \sqrt{v_x^2 + v_y^2 + v_z^2}
1231        \end{align*}
1232
1233        \noindent \textbf{Distance travelled between \(t=a \rightarrow t=b\):}
1234        \[= \int^b_a \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \cdot dt \]
1235
1236        \noindent \textbf{Shortest distance between \(\boldsymbol{r}(t_0)\) and \(\boldsymbol{r}(t_1)\):}
1237        \[ = |\boldsymbol{r}(t_1) - \boldsymbol{r}(t_2)| \]
1238
1239      \subsection*{Vector functions}
1240
1241        \[ \boldsymbol{r}(t) = x \boldsymbol{i} + y \boldsymbol{j} + z \boldsymbol{k} \]
1242
1243        \begin{itemize}
1244          \item If \(\boldsymbol{r}(t) \equiv\) position with time, then the graph of endpoints of \(\boldsymbol{r}(t) \equiv\) Cartesian path
1245          \item Domain of \(\boldsymbol{r}(t)\) is the range of \(x(t)\)
1246          \item Range of \(\boldsymbol{r}(t)\) is the range of \(y(t)\)
1247        \end{itemize}
1248
1249      \subsection*{Vector calculus}
1250
1251      \subsubsection*{Derivative}
1252
1253        Let \(\boldsymbol{r}(t)=x(t)\boldsymbol{i} + y(t)\boldsymbol(j)\). If both \(x(t)\) and \(y(t)\) are differentiable, then:
1254        \[ \boldsymbol{r}(t)=x(t)\boldsymbol{i}+y(t)\boldsymbol{j} \]
1255
1256      \subfile{dynamics}
1257      \subfile{statistics}
1258  \end{multicols}
1259\end{document}