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74\fancypagestyle{plain}{\fancyhead[LO,LE]{} \fancyhead[CO,CE]{}} % rm title & author for first page
75\fancyhead[LO,LE]{Year 12 Specialist}
76\fancyhead[CO,CE]{Andrew Lorimer}
77
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104
105\begin{document}
106
107\title{\vspace{-22mm}Year 12 Specialist\vspace{-4mm}}
108\author{Andrew Lorimer}
109\date{}
110\maketitle
111\vspace{-9mm}
112\begin{multicols}{2}
113
114 \section{Complex numbers}
115
116 \[\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}\]
117 \begin{align*}
118 \text{Cartesian form: } & a+bi\\
119 \text{Polar form: } & r\operatorname{cis}\theta
120 \end{align*}
121
122 \subsection*{Operations}
123
124 \begin{tabularx}{\columnwidth}{|r|X|X|}
125 \hline
126 \rowcolor{cas}
127 & \textbf{Cartesian} & \textbf{Polar} \\
128 \hline
129 \(z_1 \pm z_2\) & \((a \pm c)(b \pm d)i\) & convert to \(a+bi\)\\
130 \hline
131 \(+k \times z\) & \multirow{2}{*}{\(ka \pm kbi\)} & \(kr\operatorname{cis} \theta\)\\
132 \cline{1-1}\cline{3-3}
133 \(-k \times z\) & & \(kr \operatorname{cis}(\theta\pm \pi)\)\\
134 \hline
135 \(z_1 \cdot z_2\) & \(ac-bd+(ad+bc)i\) & \(r_1r_2 \operatorname{cis}(\theta_1 + \theta_2)\)\\
136 \hline
137 \(z_1 \div z_2\) & \((z_1 \overline{z_2}) \div |z_2|^2\) & \(\left(\frac{r_1}{r_2}\right) \operatorname{cis}(\theta_1 - \theta_2)\) \\
138 \hline
139 \end{tabularx}
140
141 \subsubsection*{Scalar multiplication in polar form}
142
143 For \(k \in \mathbb{R}^+\):
144 \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\theta\]
145
146 \noindent For \(k \in \mathbb{R}^-\):
147 \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\left(\begin{cases}\theta - \pi & |0<\operatorname{Arg}(z)\le \pi \\ \theta + \pi & |-\pi<\operatorname{Arg}(z)\le 0\end{cases}\right)\]
148
149 \subsection*{Conjugate}
150 \vspace{-7mm} \hfill \colorbox{cas}{\texttt{conjg(a+bi)}}
151 \begin{align*}
152 \overline{z} &= a \mp bi\\
153 &= r \operatorname{cis}(-\theta)
154 \end{align*}
155
156 \subsubsection*{Properties}
157
158 \begin{align*}
159 \overline{z_1 \pm z_2} &= \overline{z_1}\pm\overline{z_2}\\
160 \overline{z_1 \cdot z_2} &= \overline{z_1}\cdot\overline{z_2}\\
161 \overline{kz} &= k\overline{z} \> \forall \> k \in \mathbb{R}\\
162 z\overline{z} &= (a+bi)(a-bi)\\
163 &= a^2 + b^2\\
164 &= |z|^2
165 \end{align*}
166
167 \subsection*{Modulus}
168
169 \[|z|=|\vec{Oz}|=\sqrt{a^2 + b^2}\]
170
171 \subsubsection*{Properties}
172
173 \begin{align*}
174 |z_1z_2|&=|z_1||z_2|\\
175 \left|\frac{z_1}{z_2}\right|&=\frac{|z_1|}{|z_2|}\\
176 |z_1+z_2|&\le|z_1|+|z_2|
177 \end{align*}
178
179 \subsection*{Multiplicative inverse}
180
181 \begin{align*}
182 z^{-1}&=\frac{a-bi}{a^2+b^2}\\
183 &=\frac{\overline{z}}{|z|^2}a\\
184 &=r \operatorname{cis}(-\theta)
185 \end{align*}
186
187 \subsection*{Dividing over \(\mathbb{C}\)}
188
189 \begin{align*}
190 \frac{z_1}{z_2}&=z_1z_2^{-1}\\
191 &=\frac{z_1\overline{z_2}}{|z_2|^2}\\
192 &=\frac{(a+bi)(c-di)}{c^2+d^2}\\
193 & \text{then rationalise denominator}
194 \end{align*}
195
196 \subsection*{Polar form}
197
198 \[ r \operatorname{cis} \theta = r\left( \cos \theta + i \sin \theta \right) \]
199
200 \begin{itemize}
201 \item{\(r=|z|=\sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}\)}
202 \item{\(\theta = \operatorname{arg}(z)\) \hfill \colorbox{cas}{\texttt{arg(a+bi)}}}
203 \item{\(\operatorname{Arg}(z) \in (-\pi,\pi)\) \quad \bf{(principal argument)}}
204 \item{Multiple representations:\\\(r\operatorname{cis}\theta=r\operatorname{cis}(\theta+2n\pi)\) with \(n \in \mathbb{Z}\) revolutions}
205 \item{\(\operatorname{cis}\pi=-1,\qquad \operatorname{cis}0=1\)}
206 \end{itemize}
207
208 \begin{cas}
209 \-\hspace{1em}\verb|compToTrig(a+bi)| \(\iff\) \verb|cExpand{r·cisX}|
210 \end{cas}
211
212 \subsection*{de Moivres' theorem}
213
214 \begin{theorembox}{}
215 \[(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}\]
216 \end{theorembox}
217
218 \subsection*{Complex polynomials}
219
220 Include \(\pm\) for all solutions, incl. imaginary
221
222 \begin{tabularx}{\columnwidth}{ R{0.55} X }
223 \hline
224 Sum of squares & \(\begin{aligned}
225 z^2 + a^2 &= z^2-(ai)^2\\
226 &= (z+ai)(z-ai) \end{aligned}\) \\
227 \hline
228 Sum of cubes & \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)\\
229 \hline
230 Division & \(P(z)=D(z)Q(z)+R(z)\) \\
231 \hline
232 Remainder theorem & Let \(\alpha \in \mathbb{C}\). Remainder of \(P(z) \div (z-\alpha)\) is \(P(\alpha)\)\\
233 \hline
234 Factor theorem & \(z-\alpha\) is a factor of \(P(z) \iff P(\alpha)=0\) for \(\alpha \in \mathbb{C}\)\\
235 \hline
236 Conjugate root theorem & \(P(z)=0 \text{ at } z=a\pm bi\) (\(\implies\) both \(z_1\) and \(\overline{z_1}\) are solutions)\\
237 \hline
238 \end{tabularx}
239
240 \begin{theorembox}{title=Factor theorem}
241 If \(\beta z + \alpha\) is a factor of \(P(z)\), \\
242 \-\hspace{1em}then \(P(-\dfrac{\alpha}{\beta})=0\).
243 \end{theorembox}
244
245 \subsection*{\(n\)th roots}
246
247 \(n\)th roots of \(z=r\operatorname{cis}\theta\) are:
248
249 \[z = r^{\frac{1}{n}} \operatorname{cis}\left(\frac{\theta+2k\pi}{n}\right)\]
250
251 \begin{itemize}
252
253 \item{Same modulus for all solutions}
254 \item{Arguments separated by \(\frac{2\pi}{n} \therefore\) there are \(n\) roots}
255 \item{If one square root is \(a+bi\), the other is \(-a-bi\)}
256 \item{Give one implicit \(n\)th root \(z_1\), function is \(z=z_1^n\)}
257 \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 \(\frac{2\pi}{n}\))}
258 \end{itemize}
259
260 \noindent For \(0=az^2+bz+c\), use quadratic formula:
261
262 \[z=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]
263
264 \subsection*{Fundamental theorem of algebra}
265
266 A polynomial of degree \(n\) can be factorised into \(n\) linear factors in \(\mathbb{C}\):
267
268 \[\implies P(z)=a_n(z-\alpha_1)(z-\alpha_2)(z-\alpha_3)\dots(z-\alpha_n)\]
269 \[\text{ where } \alpha_1,\alpha_2,\alpha_3,\dots,\alpha_n \in \mathbb{C}\]
270
271 \subsection*{Argand planes}
272
273 \begin{center}\begin{tikzpicture}[scale=2]
274 \draw [->] (-0.2,0) -- (1.5,0) node [right] {$\operatorname{Re}(z)$};
275 \draw [->] (0,-0.2) -- (0,1.5) node [above] {$\operatorname{Im}(z)$};
276 \coordinate (P) at (1,1);
277 \coordinate (a) at (1,0);
278 \coordinate (b) at (0,1);
279 \coordinate (O) at (0,0);
280 \draw (0,0) -- (P) node[pos=0.5, above left]{\(r\)} node[pos=1, right]{\(\begin{aligned}z&=a+bi\\&=r\operatorname{cis}\theta\end{aligned}\)};
281 \draw [gray, dashed] (1,1) -- (1,0) node[black, pos=1, below]{\(a\)};
282 \draw [gray, dashed] (1,1) -- (0,1) node[black, pos=1, left]{\(b\)};
283 \begin{scope}
284 \path[clip] (O) -- (P) -- (a);
285 \fill[red, opacity=0.5, draw=black] (O) circle (2mm);
286 \node at ($(O)+(20:3mm)$) {$\theta$};
287 \end{scope}
288 \filldraw (P) circle (0.5pt);
289 \end{tikzpicture}\end{center}
290
291 \begin{itemize}
292 \item{Multiplication by \(i \implies\) CCW rotation of \(\frac{\pi}{2}\)}
293 \item{Addition: \(z_1 + z_2 \equiv\) \overrightharp{\(Oz_1\)} + \overrightharp{\(Oz_2\)}}
294 \end{itemize}
295
296 \subsection*{Sketching complex graphs}
297
298 \subsubsection*{Linear}
299
300 \begin{itemize}
301 \item{\(\operatorname{Re}(z)=c\) or \(\operatorname{Im}(z)=c\) (perpendicular bisector)}
302 \item{\(\operatorname{Im}(z)=m\operatorname{Re}(z)\)}
303 \item{\(|z+a|=|z+b| \implies 2(a-b)x=b^2-a^2\)\\Geometric: equidistant from \(a,b\)}
304 \end{itemize}
305
306 \subsubsection*{Circles}
307
308 \begin{itemize}
309 \item \(|z-z_1|^2=c^2|z_2+2|^2\)
310 \item \(|z-(a+bi)|=c \implies (x-a)^2+_(y-b)^2=c^2\)
311 \end{itemize}
312
313 \noindent \textbf{Loci} \qquad \(\operatorname{Arg}(z)<\theta\)
314
315 \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
316 \draw [->] (0,0) -- (1,0) node [right] {$\operatorname{Re}(z)$};
317 \draw [->] (0,-0.5) -- (0,1) node [above] {$\operatorname{Im}(z)$};
318 \draw [<-, dashed, thick, blue] (-1,0) -- (0,0);
319 \draw [->, thick, blue] (0,0) -- (1,1);
320 \fill [gray, opacity=0.2, domain=-1:1, variable=\x] (-1,-0.5) -- (-1,0) -- (0, 0) -- (1,1) -- (1,-0.5) -- cycle;
321 \begin{scope}
322 \path[clip] (0,0) -- (1,1) -- (1,0);
323 \fill[red, opacity=0.5, draw=black] (0,0) circle (2mm);
324 \node at ($(0,0)+(20:3mm)$) {$\frac{\pi}{4}$};
325 \end{scope}
326 \node [font=\footnotesize] at (0.5,-0.25) {\(\operatorname{Arg}(z)\le\frac{\pi}{4}\)};
327 \node [blue, mydot] {};
328 \end{tikzpicture}\end{center}
329
330 \noindent \textbf{Rays} \qquad \(\operatorname{Arg}(z-b)=\theta\)
331
332 \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
333 \draw [->] (-0.75,0) -- (1.5,0) node [right] {$\operatorname{Re}(z)$};
334 \draw [->] (0,-1) -- (0,1) node [above] {$\operatorname{Im}(z)$};
335 \draw [->, thick, brown] (-0.25,0) -- (-0.75,-1);
336 \node [above, font=\footnotesize] at (-0.25,0) {\(\frac{1}{4}\)};
337 \begin{scope}
338 \path[clip] (-0.25,0) -- (-0.75,-1) -- (0,0);
339 \fill[orange, opacity=0.5, draw=black] (-0.25,0) circle (2mm);
340 \end{scope}
341 \node at (-0.08,-0.3) {\(\frac{\pi}{8}\)};
342 \node [font=\footnotesize, left] at (-0.75,-1) {\(\operatorname{Arg}(z+\frac{1}{4})=\frac{\pi}{8}\)};
343 \node [brown, mydot] at (-0.25,0) {};
344 \draw [<->, thick, green] (0,-1) -- (1.5,0.5) node [pos=0.25, black, font=\footnotesize, right] {\(|z-2|=|z-(1+i)|\)};
345 \node [left, font=\footnotesize] at (0,-1) {\(-1\)};
346 \node [below, font=\footnotesize] at (1,0) {\(1\)};
347 \end{tikzpicture}\end{center}
348
349 \section{Vectors}
350 \begin{center}\begin{tikzpicture}
351 \draw [->] (-0.5,0) -- (3,0) node [right] {\(x\)};
352 \draw [->] (0,-0.5) -- (0,3) node [above] {\(y\)};
353 \draw [orange, ->, thick] (0.5,0.5) -- (2.5,2.5) node [pos=0.5, above] {\(\vec{u}\)};
354 \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
355 \draw [gray, dashed, thick] (0.5,0.5) -- (2.5,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, below]{\(x\vec{i}\)};
356 \draw [gray, dashed, thick] (2.5,0.5) -- (2.5,2.5) node [pos=0.5] {\midarrow};
357 \end{scope}
358 \node[black, right] at (2.5,1.5) {\(y\vec{j}\)};
359 \end{tikzpicture}\end{center}
360 \subsection*{Column notation}
361
362 \[\begin{bmatrix}x\\ y \end{bmatrix} \iff x\boldsymbol{i} + y\boldsymbol{j}\]
363 \(\begin{bmatrix}x_2-x_1\\ y_2-y_1 \end{bmatrix}\) \quad between \(A(x_1,y_1), \> B(x_2,y_2)\)
364
365 \subsection*{Scalar multiplication}
366
367 \[k\cdot (x\boldsymbol{i}+y\boldsymbol{j})=kx\boldsymbol{i}+ky\boldsymbol{j}\]
368
369 \noindent For \(k \in \mathbb{R}^-\), direction is reversed
370
371 \subsection*{Vector addition}
372 \begin{center}\begin{tikzpicture}[scale=1]
373 \coordinate (A) at (0,0);
374 \coordinate (B) at (2,2);
375 \draw [->, thick, red] (0,0) -- (2,2) node [pos=0.5, below right] {\(\vec{u}=2\vec{i}+2\vec{j}\)};
376 \draw [->, thick, blue] (2,2) -- (1,4) node [pos=0.5, above right] {\(\vec{v}=-\vec{i}+2\vec{j}\)};
377 \draw [->, thick, orange] (0,0) -- (1,4) node [pos=0.5, left] {\(\vec{u}+\vec{v}=\vec{i}+4\vec{j}\)};
378 \end{tikzpicture}\end{center}
379
380 \[(x\boldsymbol{i}+y\boldsymbol{j}) \pm (a\boldsymbol{i}+b\boldsymbol{j})=(x \pm a)\boldsymbol{i}+(y \pm b)\boldsymbol{j}\]
381
382 \begin{itemize}
383 \item Draw each vector head to tail then join lines
384 \item Addition is commutative (parallelogram)
385 \item \(\boldsymbol{u}-\boldsymbol{v}=\boldsymbol{u}+(-\boldsymbol{v}) \implies \overrightharp{AB}=\boldsymbol{b}-\boldsymbol{a}\)
386 \end{itemize}
387
388 \subsection*{Magnitude}
389
390 \[|(x\boldsymbol{i} + y\boldsymbol{j})|=\sqrt{x^2+y^2}\]
391
392 \subsection*{Parallel vectors}
393
394 \[\boldsymbol{u} || \boldsymbol{v} \iff \boldsymbol{u} = k \boldsymbol{v} \text{ where } k \in \mathbb{R} \setminus \{0\}\]
395
396 For parallel vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\):\\
397 \[\boldsymbol{a \cdot b}=\begin{cases}
398 |\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction}\\
399 -|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions}
400 \end{cases}\]
401 %\includegraphics[width=0.2,height=\textheight]{graphics/parallelogram-vectors.jpg}
402 %\includegraphics[width=1]{graphics/vector-subtraction.jpg}
403
404 \subsection*{Perpendicular vectors}
405
406 \[\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b} = 0\ \quad \text{(since \(\cos 90 = 0\))}\]
407
408 \subsection*{Unit vector \(|\hat{\boldsymbol{a}}|=1\)}
409 \[\begin{split}\hat{\boldsymbol{a}} & = {\frac{1}{|\boldsymbol{a}|}}\boldsymbol{a} \\ & = \boldsymbol{a} \cdot {|\boldsymbol{a}|}\end{split}\]
410
411 \subsection*{Scalar product \(\boldsymbol{a} \cdot \boldsymbol{b}\)}
412
413
414 \begin{center}\begin{tikzpicture}[scale=2]
415 \draw [->] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{b}\)};
416 \draw [->] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{a}\)};
417 \begin{scope}
418 \path[clip] (1,0.5) -- (1,0) -- (0,0);
419 \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
420 \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
421 \end{scope}
422 \end{tikzpicture}\end{center}
423 \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*}
424 \noindent\colorbox{cas}{On CAS: \texttt{dotP({[}a\ b\ c{]},\ {[}d\ e\ f{]})}}
425
426 \subsubsection*{Properties}
427
428 \begin{enumerate}
429 \item
430 \(k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k\boldsymbol{b})\)
431 \item
432 \(\boldsymbol{a \cdot 0}=0\)
433 \item
434 \(\boldsymbol{a} \cdot (\boldsymbol{b} + \boldsymbol{c})=\boldsymbol{a} \cdot \boldsymbol{b} + \boldsymbol{a} \cdot \boldsymbol{c}\)
435 \item
436 \(\boldsymbol{i \cdot i} = \boldsymbol{j \cdot j} = \boldsymbol{k \cdot k}= 1\)
437 \item
438 \(\boldsymbol{a} \cdot \boldsymbol{b} = 0 \quad \implies \quad \boldsymbol{a} \perp \boldsymbol{b}\)
439 \item
440 \(\boldsymbol{a \cdot a} = |\boldsymbol{a}|^2 = a^2\)
441 \end{enumerate}
442
443 \subsection*{Angle between vectors}
444
445 \[\cos \theta = \frac{\boldsymbol{a} \cdot \boldsymbol{b}}{|\boldsymbol{a}| |\boldsymbol{b}|} = \frac{a_1 b_1 + a_2 b_2}{|\boldsymbol{a}| |\boldsymbol{b}|}\]
446
447 \noindent \colorbox{cas}{On CAS:} \texttt{angle([a b c], [a b c])}
448
449 (Action \(\rightarrow\) Vector \(\rightarrow\)Angle)
450
451 \subsection*{Angle between vector and axis}
452
453 \noindent For\(\boldsymbol{a} = a_1 \boldsymbol{i} + a_2 \boldsymbol{j} + a_3 \boldsymbol{k}\)
454 which makes angles \(\alpha, \beta, \gamma\) with positive side of
455 \(x, y, z\) axes:
456 \[\cos \alpha = \frac{a_1}{|\boldsymbol{a}|}, \quad \cos \beta = \frac{a_2}{|\boldsymbol{a}|}, \quad \cos \gamma = \frac{a_3}{|\boldsymbol{a}|}\]
457
458 \noindent \colorbox{cas}{On CAS:} \texttt{angle({[}a\ b\ c{]},\ {[}1\ 0\ 0{]})}\\for angle
459 between \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) and
460 \(x\)-axis
461
462 \subsection*{Projections \& resolutes}
463
464 \begin{tikzpicture}[scale=3]
465 \draw [->, purple] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{a}\)};
466 \draw [->, orange] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{u}\)};
467 \draw [->, blue] (1,0) -- (2,0) node [pos=0.5, below] {\(\boldsymbol{b}\)};
468 \begin{scope}
469 \path[clip] (1,0.5) -- (1,0) -- (0,0);
470 \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
471 \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
472 \end{scope}
473 \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
474 \draw [gray, dashed, thick] (1,0) -- (1,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, right, rotate=-90]{\(\boldsymbol{w}\)};
475 \end{scope}
476 \draw (0,0) coordinate (O)
477 (1,0) coordinate (A)
478 (1,0.5) coordinate (B)
479 pic [draw,red,angle radius=2mm] {right angle = O--A--B};
480 \end{tikzpicture}
481
482 \subsubsection*{\(\parallel\boldsymbol{b}\) (vector projection/resolute)}
483
484 \begin{align*}
485 \boldsymbol{u} & = \frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|^2}\boldsymbol{b} \\
486 & = \left(\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\right)\left(\frac{\boldsymbol{b}}{|\boldsymbol{b}|}\right) \\
487 & = (\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}
488 \end{align*}
489
490 \subsubsection*{\(\perp\boldsymbol{b}\) (perpendicular projection)}
491 \[\boldsymbol{w} = \boldsymbol{a} - \boldsymbol{u}\]
492
493 \subsubsection*{\(|\boldsymbol{u}|\) (scalar projection/resolute)}
494 \begin{align*}
495 s &= |\boldsymbol{u}|\\
496 &= \boldsymbol{a} \cdot \hat{\boldsymbol{b}}\\
497 &=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\\
498 &= |\boldsymbol{a}| \cos \theta
499 \end{align*}
500
501 \subsubsection*{Rectangular (\(\parallel,\perp\)) components}
502
503 \[\boldsymbol{a}=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}+\left(\boldsymbol{a}-\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}\right)\]
504
505
506 \subsection*{Vector proofs}
507
508 \textbf{Concurrent:} intersection of \(\ge\) 3 lines
509
510 \begin{tikzpicture}
511 \draw [blue] (0,0) -- (1,1);
512 \draw [red] (1,0) -- (0,1);
513 \draw [brown] (0.4,0) -- (0.6,1);
514 \filldraw (0.5,0.5) circle (2pt);
515 \end{tikzpicture}
516
517 \subsubsection*{Collinear points}
518
519 \(\ge\) 3 points lie on the same line
520
521 \begin{tikzpicture}
522 \draw [purple] (0,0) -- (4,1);
523 \filldraw (2,0.5) circle (2pt) node [above] {\(C\)};
524 \filldraw (1,0.25) circle (2pt) node [above] {\(A\)};
525 \filldraw (3,0.75) circle (2pt) node [above] {\(B\)};
526 \coordinate (O) at (2.8,-0.2);
527 \node at (O) [below] {\(O\)};
528 \begin{scope}[->, orange, thick]
529 \draw (O) -- (2,0.5) node [pos=0.5, above, font=\footnotesize, black] {\(\boldsymbol{c}\)};
530 \draw (O) -- (1,0.25) node [pos=0.5, below, font=\footnotesize, black] {\(\boldsymbol{a}\)};
531 \draw (O) -- (3,0.75) node [pos=0.5, right, font=\footnotesize, black] {\(\boldsymbol{b}\)};
532 \end{scope}
533 \end{tikzpicture}
534
535 \begin{align*}
536 \text{e.g. Prove that}\\
537 \overrightharp{AC}=m\overrightharp{AB} \iff \boldsymbol{c}&=(1-m)\boldsymbol{a}+m\boldsymbol{b}\\
538 \implies \boldsymbol{c} &= \overrightharp{OA} + \overrightharp{AC}\\
539 &= \overrightharp{OA} + m\overrightharp{AB}\\
540 &=\boldsymbol{a}+m(\boldsymbol{b}-\boldsymbol{a})\\
541 &=\boldsymbol{a}+m\boldsymbol{b}-m\boldsymbol{a}\\
542 &=(1-m)\boldsymbol{a}+m{b}
543 \end{align*}
544 \begin{align*}
545 \text{Also, } \implies \overrightharp{OC} &= \lambda \vec{OA} + \mu \overrightharp{OB} \\
546 \text{where } \lambda + \mu &= 1\\
547 \text{If } C \text{ lies along } \overrightharp{AB}, & \implies 0 < \mu < 1
548 \end{align*}
549
550
551 \subsubsection*{Parallelograms}
552
553 \begin{center}\begin{tikzpicture}
554 \coordinate (O) at (0,0) node [below left] {\(O\)};
555 \coordinate (A) at (4,0);
556 \coordinate (B) at (6,2);
557 \coordinate (C) at (2,2);
558 \coordinate (D) at (6,0);
559
560 \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (O)--(A) node [below left] {\(A\)};
561 \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (A)--(B) node [above right] {\(B\)};
562 \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (B)--(C) node [above left] {\(C\)};
563 \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (C)--(O);
564
565 \draw [gray, dashed] (O) -- (B) node [pos=0.75] {\(\diagdown\diagdown\)} node [pos=0.25] {\(\diagdown\diagdown\)};
566 \draw [gray, dashed] (A) -- (C) node [pos=0.75] {\(\diagup\)} node [pos=0.25] {\(\diagup\)};
567 \begin{scope}
568 \path[clip] (C) -- (A) -- (O);
569 \fill[orange, opacity=0.5, draw=black] (0,0) circle (4mm);
570 \node at ($(0,0)+(20:8mm)$) {\(\theta\)};
571 \end{scope}
572 \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] {\(|\boldsymbol{c}|\cos\theta\)};
573 \draw pic [draw,thick,red,angle radius=2mm] {right angle=O--D--B};
574 \end{tikzpicture}\end{center}
575
576 \begin{itemize}
577 \item
578 Diagonals \(\overrightharp{OB}, \overrightharp{AC}\) bisect each other
579 \item
580 If diagonals are equal length, it is a rectangle
581 \item
582 \(|\overrightharp{OB}|^2 + |\overrightharp{CA}|^2 = |\overrightharp{OA}|^2 + |\overrightharp{AB}|^2 + |\overrightharp{CB}|^2 + |\overrightharp{OC}|^2\)
583 \item
584 Area \(=\boldsymbol{c} \cdot \boldsymbol{a}\)
585 \end{itemize}
586
587 \subsubsection*{Useful vector properties}
588
589 \begin{itemize}
590 \item
591 \(\boldsymbol{a} \parallel \boldsymbol{b} \implies \boldsymbol{b}=k\boldsymbol{a}\) for some
592 \(k \in \mathbb{R} \setminus \{0\}\)
593 \item
594 If \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are parallel with at
595 least one point in common, then they lie on the same straight line
596 \item
597 \(\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b}=0\)
598 \item
599 \(\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2\)
600 \end{itemize}
601
602 \subsection*{Linear dependence}
603
604 \(\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\) are linearly dependent if they are \(\nparallel\) and:
605 \begin{align*}
606 0&=k\boldsymbol{a}+l\boldsymbol{b}+m\boldsymbol{c}\\
607 \therefore \boldsymbol{c} &= m\boldsymbol{a} + n\boldsymbol{b} \quad \text{(simultaneous)}
608 \end{align*}
609
610 \noindent \(\boldsymbol{a}, \boldsymbol{b},\) and \(\boldsymbol{c}\) are linearly
611 independent if no vector in the set is expressible as a linear
612 combination of other vectors in set, or if they are parallel.
613
614 \subsection*{Three-dimensional vectors}
615
616 Right-hand rule for axes: \(z\) is up or out of page.
617
618 \tdplotsetmaincoords{60}{120}
619 \begin{center}\begin{tikzpicture} [scale=3, tdplot_main_coords, axis/.style={->,thick},
620 vector/.style={-stealth,red,very thick},
621 vector guide/.style={dashed,gray,thick}]
622
623 %standard tikz coordinate definition using x, y, z coords
624 \coordinate (O) at (0,0,0);
625
626 %tikz-3dplot coordinate definition using x, y, z coords
627
628 \pgfmathsetmacro{\ax}{1}
629 \pgfmathsetmacro{\ay}{1}
630 \pgfmathsetmacro{\az}{1}
631
632 \coordinate (P) at (\ax,\ay,\az);
633
634 %draw axes
635 \draw[axis] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$};
636 \draw[axis] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$};
637 \draw[axis] (0,0,0) -- (0,0,1) node[anchor=south]{$z$};
638
639 %draw a vector from O to P
640 \draw[vector] (O) -- (P);
641
642 %draw guide lines to components
643 \draw[vector guide] (O) -- (\ax,\ay,0);
644 \draw[vector guide] (\ax,\ay,0) -- (P);
645 \draw[vector guide] (P) -- (0,0,\az);
646 \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
647 \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
648 \draw[vector guide] (\ax,\ay,0) -- (\ax,0,0);
649 \node[tdplot_main_coords,above right]
650 at (\ax,\ay,\az){(\ax, \ay, \az)};
651 \end{tikzpicture}\end{center}
652
653 \subsection*{Parametric vectors}
654
655 Parametric equation of line through point \((x_0, y_0, z_0)\) and
656 parallel to \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) is:
657
658 \[\begin{cases}x = x_o + a \cdot t \\ y = y_0 + b \cdot t \\ z = z_0 + c \cdot t\end{cases}\]
659
660 \section{Circular functions}
661
662 \(\sin(bx)\) or \(\cos(bx)\): period \(=\frac{2\pi}{b}\)
663
664 \noindent \(\tan(nx)\): period \(=\frac{\pi}{n}\)\\
665 \indent\indent\indent asymptotes at \(x=\frac{(2k+1)\pi}{2n} \> \vert \> k \in \mathbb{Z}\)
666
667 \subsection*{Reciprocal functions}
668
669 \subsubsection*{Cosecant}
670
671 \[\operatorname{cosec} \theta = \frac{1}{\sin \theta} \> \vert \> \sin \theta \ne 0\]
672
673 \begin{itemize}
674 \item
675 \textbf{Domain} \(= \mathbb{R} \setminus {n\pi : n \in \mathbb{Z}}\)
676 \item
677 \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
678 \item
679 \textbf{Turning points} at
680 \(\theta = \frac{(2n + 1)\pi}{2} \> \vert \> n \in \mathbb{Z}\)
681 \item
682 \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
683 \end{itemize}
684
685 \subsubsection*{Secant}
686
687\begin{tikzpicture}
688 \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={\(-2\pi\), \(-\pi\), \(\pi\), \(2\pi\)}]
689% \addplot[blue, domain=-6.2830:6.2830,unbounded coords=jump,samples=80] {sec(deg(x))};
690 \addplot[blue, restrict y to domain=-10:10, domain=-7:7,samples=100] {sec(deg(x))} node [pos=0.93, black, right] {\(\operatorname{sec} x\)};
691 \addplot[red, dashed, domain=-7:7,samples=100] {cos(deg(x))};
692 \draw [gray, dotted, thick] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{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:-4.71239,0}|-{rel axis cs:0,0}) -- ({axis cs:-4.71239,0}|-{rel axis cs:0,1});
695 \draw [gray, dotted, thick] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1});
696\end{axis}
697 \node [black] at (7,3.5) {\(\cos x\)};
698\end{tikzpicture}
699
700 \[\operatorname{sec} \theta = \frac{1}{\cos \theta} \> \vert \> \cos \theta \ne 0\]
701
702 \begin{itemize}
703
704 \item
705 \textbf{Domain}
706 \(= \mathbb{R} \setminus \frac{(2n + 1) \pi}{2} : n \in \mathbb{Z}\}\)
707 \item
708 \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
709 \item
710 \textbf{Turning points} at
711 \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
712 \item
713 \textbf{Asymptotes} at
714 \(\theta = \frac{(2n + 1) \pi}{2} \> \vert \> n \in \mathbb{Z}\)
715 \end{itemize}
716
717 \subsubsection*{Cotangent}
718
719\begin{tikzpicture}
720 \begin{axis}[xmin=-3,xmax=3,ymin=-1.5,ymax=1.5,enlargelimits=true, xtick={-3.1415, -1.5708, 1.5708, 3.1415},xticklabels={\(-\pi\), \(-\frac{\pi}{2}\), \(\frac{\pi}{2}\), \(\pi\)}]
721 \addplot[blue, smooth, domain=-3:-0.1,unbounded coords=jump,samples=105] {cot(deg(x))} node [pos=0.3, left] {\(\operatorname{cot} x\)};
722\addplot[blue, smooth, domain=0.1:3,unbounded coords=jump,samples=105] {cot(deg(x))};
723\addplot[red, smooth, dashed] gnuplot [domain=-1.5:1.5,unbounded coords=jump,samples=105] {tan(x)};
724\addplot[red, smooth, dashed] gnuplot [domain=-3.5:-1.8,unbounded coords=jump,samples=105] {tan(x)} node [pos=0.5, right] {\(\tan x\)};
725\addplot[red, smooth, dashed] gnuplot [domain=1.8:3.5,unbounded coords=jump,samples=105] {tan(x)};
726 \draw [thick, red, dotted] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1});
727 \draw [thick, blue, dotted] ({axis cs:-3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:-3.1415,0}|-{rel axis cs:0,1});
728 \draw [thick, blue, dotted] ({axis cs:0,0}|-{rel axis cs:0,0}) -- ({axis cs:0,0}|-{rel axis cs:0,1});
729 \draw [thick, blue, dotted] ({axis cs:3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:3.1415,0}|-{rel axis cs:0,1});
730 \draw [thick, red, dotted] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{rel axis cs:0,1});
731\end{axis}
732\end{tikzpicture}
733
734 \[\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0\]
735
736 \begin{itemize}
737
738 \item
739 \textbf{Domain} \(= \mathbb{R} \setminus \{n \pi: n \in \mathbb{Z}\}\)
740 \item
741 \textbf{Range} \(= \mathbb{R}\)
742 \item
743 \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
744 \end{itemize}
745
746 \subsubsection*{Symmetry properties}
747
748 \[\begin{split}
749 \operatorname{sec} (\pi \pm x) & = -\operatorname{sec} x \\
750 \operatorname{sec} (-x) & = \operatorname{sec} x \\
751 \operatorname{cosec} (\pi \pm x) & = \mp \operatorname{cosec} x \\
752 \operatorname{cosec} (-x) & = - \operatorname{cosec} x \\
753 \operatorname{cot} (\pi \pm x) & = \pm \operatorname{cot} x \\
754 \operatorname{cot} (-x) & = - \operatorname{cot} x
755 \end{split}\]
756
757 \subsubsection*{Complementary properties}
758
759 \[\begin{split}
760 \operatorname{sec} \left({\pi \over 2} - x\right) & = \operatorname{cosec} x \\
761 \operatorname{cosec} \left({\pi \over 2} - x\right) & = \operatorname{sec} x \\
762 \operatorname{cot} \left({\pi \over 2} - x\right) & = \tan x \\
763 \tan \left({\pi \over 2} - x\right) & = \operatorname{cot} x
764 \end{split}\]
765
766 \subsubsection*{Pythagorean identities}
767
768 \[\begin{split}
769 1 + \operatorname{cot}^2 x & = \operatorname{cosec}^2 x, \quad \text{where } \sin x \ne 0 \\
770 1 + \tan^2 x & = \operatorname{sec}^2 x, \quad \text{where } \cos x \ne 0
771 \end{split}\]
772
773 \subsection*{Compound angle formulas}
774
775 \[\cos(x \pm y) = \cos x + \cos y \mp \sin x \sin y\]
776 \[\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y\]
777 \[\tan(x \pm y) = {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}\]
778
779 \subsection*{Double angle formulas}
780
781 \[\begin{split}
782 \cos 2x &= \cos^2 x - \sin^2 x \\
783 & = 1 - 2\sin^2 x \\
784 & = 2 \cos^2 x -1
785 \end{split}\]
786
787 \[\sin 2x = 2 \sin x \cos x\]
788
789 \[\tan 2x = {{2 \tan x} \over {1 - \tan^2 x}}\]
790
791 \subsection*{Inverse circular functions}
792
793 \begin{tikzpicture}
794 \begin{axis}[ymin=-2, ymax=4, xmin=-1.1, xmax=1.1, ytick={-1.5708, 1.5708, 3.14159},yticklabels={$-\frac{\pi}{2}$, $\frac{\pi}{2}$, $\pi$}]
795 \addplot[color=red, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {asin(x)} node [pos=0.25, below right] {\(\sin^{-1}x\)};
796 \addplot[color=blue, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {acos(x)} node [pos=0.25, below left] {\(\cos^{-1}x\)};
797 \addplot[mark=*, red] coordinates {(-1,-1.5708)} node[right, font=\footnotesize]{\((-1,-\frac{\pi}{2})\)} ;
798 \addplot[mark=*, red] coordinates {(1,1.5708)} node[left, font=\footnotesize]{\((1,\frac{\pi}{2})\)} ;
799 \addplot[mark=*, blue] coordinates {(1,0)};
800 \addplot[mark=*, blue] coordinates {(-1,3.1415)} node[right, font=\footnotesize]{\((-1,\pi)\)} ;
801 \end{axis}
802 \end{tikzpicture}\\
803
804 Inverse functions: \(f(f^{-1}(x)) = x\) (restrict domain)
805
806 \[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y\]
807 \hfill where \(\sin y = x, \> y \in [{-\pi \over 2}, {\pi \over 2}]\)
808
809 \[\cos^{-1}: [-1,1] \rightarrow \mathbb{R}, \quad \cos^{-1} x = y\]
810 \hfill where \(\cos y = x, \> y \in [0, \pi]\)
811
812 \[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\]
813 \hfill where \(\tan y = x, \> y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\)
814
815 \begin{tikzpicture}
816 \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={\(-\frac{\pi}{2}\),\(\frac{\pi}{2}\)}]
817 \addplot[color=orange, smooth] gnuplot [domain=-35:35, unbounded coords=jump,samples=350] {atan(x)} node [pos=0.5, above left] {\(\tan^{-1}x\)};
818 \addplot[gray, dotted, thick, domain=-35:35] {1.5708} node [black, font=\footnotesize, below right, pos=0] {\(y=\frac{\pi}{2}\)};
819 \addplot[gray, dotted, thick, domain=-35:35] {-1.5708} node [black, font=\footnotesize, above left, pos=1] {\(y=-\frac{\pi}{2}\)};
820 \end{axis}
821 \end{tikzpicture}
822
823 \subsection*{Mensuration}
824
825 \begin{tikzpicture}[draw=blue!70,thick]
826 \filldraw[fill=lblue] circle (2cm);
827 \filldraw[fill=white]
828 (320:2cm) node[right] {}
829 -- (220:2cm) node[left] {}
830 arc[start angle=220, end angle=320, radius=2cm]
831 -- cycle;
832 \node {Major Segment};
833 \node at (-90:2) {Minor Segment};
834
835 \begin{scope}[xshift=4.5cm]
836 \draw circle (2cm);
837 \filldraw[fill=lblue]
838 (320:2cm) node[right] {}
839 -- (0,0) node[above] {}
840 -- (220:2cm) node[left] {}
841 arc[start angle=220, end angle=320, radius=2cm]
842 -- cycle;
843 \node at (90:1cm) {Major Sector};
844 \node at (-90:1.5) {Minor Sector};
845 \end{scope}
846 \end{tikzpicture}
847
848 \subsubsection*{Sectors}
849
850 \begin{align*}
851 A &= \pi r^2 \dfrac{\theta}{2\pi} \\
852 &= \dfrac{r^2 \theta}{2}
853 \end{align*}
854
855 \subsubsection*{Segments}
856
857 \[ A = \dfrac{r^2}{2} \left( \theta = \sin \theta \right) \]
858
859 \subsubsection*{Chords}
860
861 \begin{align*}
862 \operatorname{crd} \theta &= \sqrt{(1 - \cos\theta)^2 + \sin^2 \theta} \\
863 &= \sqrt{2 - 2\cos\theta} \\
864 &= 2 \sin \left(\dfrac{\theta}{2}\right)
865 \end{align*}
866
867 \section{Differential calculus}
868
869 \[f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={\frac{dy}{dx}}\]
870
871 \subsection*{Limits}
872
873 \[\lim_{x \rightarrow a}f(x)\]
874 \(L^-,\quad L^+\) \qquad limit from below/above\\
875 \(\lim_{x \to a} f(x)\) \quad limit of a point\\
876
877 \noindent For solving \(x\rightarrow\infty\), put all \(x\) terms in denominators\\
878 e.g. \[\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2\]
879
880 \subsubsection*{Limit theorems}
881
882 \begin{enumerate}
883 \item
884 For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\)
885 \item
886 \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\)
887 \item
888 \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\)
889 \item
890 \(\therefore \lim_{x \rightarrow a} c \times f(x)=cF\) where \(c=\) constant
891 \item
892 \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\)
893 \item
894 \(f(x)\) is continuous \(\iff L^-=L^+=f(x) \> \forall x\)
895 \end{enumerate}
896
897 \subsection*{Gradients}
898
899 \textbf{Secant (chord)} - line joining two points on curve\\
900 \textbf{Tangent} - line that intersects curve at one point
901
902 \subsubsection*{Points of Inflection}
903
904 \emph{Stationary point} - i.e.
905 \(f^\prime(x)=0\)\\
906 \emph{Point of inflection} - max \(|\)gradient\(|\) (i.e.
907 \(f^{\prime\prime} = 0\))
908
909 \subsubsection*{Strictly increasing/decreasing}
910
911 For \(x_2\) and \(x_1\) where \(x_2 > x_1\):
912
913 \textbf{strictly increasing}\\
914 \-\hspace{1em}where \(f(x_2) > f(x_1)\) or \(f^\prime(x)>0\)
915
916 \textbf{strictly decreasing}\\
917 \hspace{1em}where \(f(x_2) < f(x_1)\) or \(f^\prime(x)<0\)
918 \begin{warning}
919 Endpoints are included, even where \(\boldsymbol{\frac{dy}{dx}=0}\)
920 \end{warning}
921
922
923 \subsection*{Second derivative}
924 \begin{align*}f(x) \longrightarrow &f^\prime (x) \longrightarrow f^{\prime\prime}(x)\\
925 \implies y \longrightarrow &\frac{dy}{dx} \longrightarrow \frac{d^2 y}{dx^2}\end{align*}
926
927 \noindent Order of polynomial \(n\)th derivative decrements each time the derivative is taken
928
929
930 \subsection*{Slope fields}
931
932 \begin{tikzpicture}[declare function={diff(\x,\y) = \x+\y;}]
933 \begin{axis}[axis equal, ymin=-4, ymax=4, xmin=-4, xmax=4, ticks=none, enlargelimits=true, ]
934 \addplot[thick, orange, domain=-4:2] {e^(x)-x-1};
935 \pgfplotsinvokeforeach{-4,...,4}{%
936 \draw[gray] ( {#1 -0.1}, {4 - diff(#1, 4) *0.1}) -- ( {#1 +0.1}, {4 + diff(#1, 4) *0.1});
937 \draw[gray] ( {#1 -0.1}, {3 - diff(#1, 3) *0.1}) -- ( {#1 +0.1}, {3 + diff(#1, 3) *0.1});
938 \draw[gray] ( {#1 -0.1}, {2 - diff(#1, 2) *0.1}) -- ( {#1 +0.1}, {2 + diff(#1, 2) *0.1});
939 \draw[gray] ( {#1 -0.1}, {1 - diff(#1, 1) *0.1}) -- ( {#1 +0.1}, {1 + diff(#1, 1) *0.1});
940 \draw[gray] ( {#1 -0.1}, {0 - diff(#1, 0) *0.1}) -- ( {#1 +0.1}, {0 + diff(#1, 0) *0.1});
941 \draw[gray] ( {#1 -0.1}, {-1 - diff(#1, -1) *0.1}) -- ( {#1 +0.1}, {-1 + diff(#1, -1) *0.1});
942 \draw[gray] ( {#1 -0.1}, {-2 - diff(#1, -2) *0.1}) -- ( {#1 +0.1}, {-2 + diff(#1, -2) *0.1});
943 \draw[gray] ( {#1 -0.1}, {-3 - diff(#1, -3) *0.1}) -- ( {#1 +0.1}, {-3 + diff(#1, -3) *0.1});
944 \draw[gray] ( {#1 -0.1}, {-4 - diff(#1, -4) *0.1}) -- ( {#1 +0.1}, {-4 + diff(#1, -4) *0.1});
945 }
946 \end{axis}
947 \end{tikzpicture}
948
949 \begin{table*}[ht]
950 \centering
951 \begin{tabularx}{\textwidth}{|r|Y|Y|Y|}
952 \hline
953 \rowcolor{lblue}
954 & \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)} \\
955 \hline
956 \(\dfrac{dy}{dx}>0\) &
957 \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)}&
958 \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)}&
959 \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}\\
960 \hline
961 \(\dfrac{dy}{dx}<0\) &
962 \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)}&
963 \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)}&
964 \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}\\
965 \hline
966 \(\dfrac{dy}{dx}=0\)&
967 \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}&
968 \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}\\
969 \hline
970 \end{tabularx}
971 \end{table*}
972 \begin{itemize}
973 \item
974 \(f^\prime (a) = 0, \> f^{\prime\prime}(a) > 0\) \\
975 \textbf{local min} at \((a, f(a))\) (concave up)
976 \item
977 \(f^\prime (a) = 0, \> f^{\prime\prime} (a) < 0\) \\
978 \textbf{local max} at \((a, f(a))\) (concave down)
979 \item
980 \(f^{\prime\prime}(a) = 0\) \\
981 \textbf{point of inflection} at \((a, f(a))\)
982 \item
983 \(f^{\prime\prime}(a) = 0, \> f^\prime(a)=0\) \\
984 stationary point of inflection at \((a, f(a)\)
985 \end{itemize}
986
987 \subsection*{Implicit Differentiation}
988
989 \noindent Used for differentiating circles etc.
990
991 If \(p\) and \(q\) are expressions in \(x\) and \(y\) such that \(p=q\),
992 for all \(x\) and \(y\), then:
993
994 \[{\frac{dp}{dx}} = {\frac{dq}{dx}} \quad \text{and} \quad {\frac{dp}{dy}} = {\frac{dq}{dy}}\]
995
996 \begin{cas}
997 Action \(\rightarrow\) Calculation \\
998 \-\hspace{1em}\texttt{impDiff(y\^{}2+ax=5,\ x,\ y)}
999 \end{cas}
1000
1001 \subsection*{Function of the dependent
1002 variable}
1003
1004 If \({\frac{dy}{dx}}=g(y)\), then
1005 \(\frac{dx}{dy} = 1 \div \frac{dy}{dx} = \frac{1}{g(y)}\). Integrate both sides to solve equation. Only add \(c\) on one side. Express
1006 \(e^c\) as \(A\).
1007
1008 \subsection*{Reciprocal derivatives}
1009
1010 \[\frac{1}{\frac{dy}{dx}} = \frac{dx}{dy}\]
1011
1012 \subsection*{Differentiating \(x=f(y)\)}
1013 Find \(\dfrac{dx}{dy}\), then \(\dfrac{dy}{dx} = \dfrac{1}{\left(\dfrac{dx}{dy}\right)}\)
1014
1015 \subsection*{Parametric equations}
1016
1017
1018 \begin{align*}
1019 \dfrac{dy}{dt} &= \dfrac{dy}{dx} \cdot \dfrac{dx}{dt} \\
1020 \therefore \dfrac{dy}{dx} &= \dfrac{\left(\dfrac{dy}{dt}\right)}{\left(\dfrac{dx}{dt}\right)} \text{ provided } \dfrac{dx}{dt} \ne 0 \\
1021 \dfrac{d^2y}{dx^2} &= \dfrac{\left(\dfrac{dy^\prime}{dt}\right)}{\left(\dfrac{dx}{dt}\right)} \text{ where } y^\prime = \dfrac{dy}{dx}
1022 \end{align*}
1023
1024 \subsection*{Integration}
1025
1026 \[\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)\]
1027
1028 \subsubsection*{Properties}
1029
1030 \begin{align*}
1031 \int^b_a f(x) \> dx &= \int^c_a f(x) \> dx + \int^b_c f(x) \> dx \\
1032 \int^a_a f(x) \> dx &= 0 \\
1033 \int^b_a k \cdot f(x) \> dx &= k \int^b_a f(x) \> dx \\
1034 \int^b_a f(x) \pm g(x) \> dx &= \int^b_a f(x) \> dx \pm \int^b_a g(x) \> dx \\
1035 \int^b_a f(x) \> dx &= - \int^a_b f(x) \> dx \\
1036 \end{align*}
1037
1038 \subsection*{Integration by substitution}
1039
1040 \[\int f(u) {\frac{du}{dx}} \cdot dx = \int f(u) \cdot du\]
1041
1042 \begin{warning}
1043 \(\boldsymbol{f(u)}\) must be 1:1 \(\boldsymbol{\implies}\) one \(\boldsymbol{x}\) for each \(\boldsymbol{y}\)
1044 \end{warning}
1045 \begin{align*}\text{e.g. for } y&=\int(2x+1)\sqrt{x+4} \cdot dx\\
1046 \text{let } u&=x+4\\
1047 \implies& {\frac{du}{dx}} = 1\\
1048 \implies& x = u - 4\\
1049 \text{then } &y=\int (2(u-4)+1)u^{\frac{1}{2}} \cdot du\\
1050 &\text{(solve as normal integral)}
1051 \end{align*}
1052
1053 \subsubsection*{Definite integrals by substitution}
1054
1055 For \(\int^b_a f(x) {\frac{du}{dx}} \cdot dx\), evaluate new \(a\) and
1056 \(b\) for \(f(u) \cdot du\).
1057
1058 \subsubsection*{Trigonometric integration}
1059
1060 \[\sin^m x \cos^n x \cdot dx\]
1061
1062 \paragraph{\textbf{\(m\) is odd:}}
1063 \(m=2k+1\) where \(k \in \mathbb{Z}\)\\
1064 \(\implies \sin^{2k+1} x = (\sin^2 z)^k \sin x = (1 - \cos^2 x)^k \sin x\)\\
1065 Substitute \(u=\cos x\)
1066
1067 \paragraph{\textbf{\(n\) is odd:}}
1068 \(n=2k+1\) where \(k \in \mathbb{Z}\)\\
1069 \(\implies \cos^{2k+1} x = (\cos^2 x)^k \cos x = (1-\sin^2 x)^k \cos x\)\\
1070 Substitute \(u=\sin x\)
1071
1072 \paragraph{\textbf{\(m\) and \(n\) are even:}}
1073 use identities...
1074
1075 \begin{itemize}
1076
1077 \item
1078 \(\sin^2x={1 \over 2}(1-\cos 2x)\)
1079 \item
1080 \(\cos^2x={1 \over 2}(1+\cos 2x)\)
1081 \item
1082 \(\sin 2x = 2 \sin x \cos x\)
1083 \end{itemize}
1084
1085 \subsection*{Separation of variables}
1086
1087 If \({\frac{dy}{dx}}=f(x)g(y)\), then:
1088
1089 \[\int f(x) \> dx = \int \frac{1}{g(y)} \> dy\]
1090
1091 \subsection*{Partial fractions}
1092
1093 To factorise \(f(x) = \frac{\delta}{\alpha \cdot \beta}\):
1094 \begin{align*}
1095 \dfrac{\delta}{\alpha \cdot \beta \cdot \gamma} &= \dfrac{A}{\alpha} + \dfrac{B}{\beta} + \dfrac{C}{\gamma} \tag{1} \\
1096 \text{Multiply by } & (\alpha \cdot \beta \cdot \gamma) \text{:} \\
1097 \delta &= \beta\gamma A + \alpha\gamma B +\alpha\beta C \tag{2} \\
1098 \text{Substitute } x &= \{\alpha, \beta, \gamma\} \text{ into (2) to find denominators}
1099 \end{align*}
1100
1101 \subsubsection*{Repeated linear factors}
1102
1103 \[ \dfrac{p(x)}{(x-a)^n} = \dfrac{A_1}{(x-a)} + \dfrac{A_2}{(x-a)^2} + \dots + \dfrac{A_n}{(x-a)^n} \]
1104
1105 \subsubsection*{Irreducible quadratic factors}
1106
1107 \[ \text{e.g. } \dfrac{3x-4}{(2x-3)(x^2+5)} = \dfrac{A}{2x-3} + \dfrac{Bx+C}{x^2+5} \]
1108
1109 \begin{cas}
1110 Action \(\rightarrow\) Transformation:\\
1111 \-\hspace{1em} \texttt{expand(..., x)}
1112
1113 To reverse, use \texttt{combine(...)}
1114 \end{cas}
1115
1116 \subsection*{Integrating \(\boldsymbol{\dfrac{dy}{dx} = g(y)}\)}
1117
1118 \[ \text{if } \dfrac{dy}{dx} = g(y), \text{ then } x = \int{\dfrac{1}{g(y)}} \> dy \]
1119
1120 \subsection*{Graphing integrals on CAS}
1121
1122 \begin{cas}
1123 \textbf{In main:} Interactive \(\rightarrow\) Calculation \(\rightarrow\) \(\int\)\\
1124 For restrictions, \texttt{Define\ f(x)=...} then \texttt{f(x)\textbar{}x\textgreater{}...}
1125 \end{cas}
1126
1127 \subsection*{Solids of revolution}
1128
1129 Approximate as sum of infinitesimally-thick cylinders
1130
1131 \subsubsection*{Rotation about \(\boldsymbol{x}\)-axis}
1132
1133 \[ V = \pi\int^{x=b}_{x=a} f(x)^2 \> dx \]
1134
1135 \subsubsection*{Rotation about \(\boldsymbol{y}\)-axis}
1136
1137 \begin{align*}
1138 V &= \pi \int^{y=b}_{y=a} x^2 \> dy \\
1139 &= \pi \int^{y=b}_{y=a} (f(y))^2 \> dy
1140 \end{align*}
1141
1142 \subsubsection*{Regions not bound by \(\boldsymbol{y=0}\)}
1143
1144 \[V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx\]
1145 \hfill where \(f(x) > g(x)\)
1146
1147 \subsection*{Length of a curve}
1148
1149 For length of \(f(x)\) from \(x=a \rightarrow x=b\):
1150 \begin{align*}
1151 &\text{Cartesian} \> & L &= \int^b_a \sqrt{1 + \left(\dfrac{dy}{dx}\right)^2} \> dx \\
1152 &\text{Parametric} \> & L & = \int^b_a \sqrt{\left(\dfrac{dx}{dt}\right)^2 + \left(\dfrac{dy}{dt}\right)^2} \> dt
1153 \end{align*}
1154
1155 \begin{cas}
1156 \begin{enumerate}[label=\alph*), leftmargin=5mm]
1157 \item Evaluate formula
1158 \item Interactive \(\rightarrow\) Calculation \(\rightarrow\) Line \(\rightarrow\) \texttt{arcLen}
1159 \end{enumerate}
1160 \end{cas}
1161
1162 \subsection*{Applications of antidifferentiation}
1163
1164 \begin{itemize}
1165
1166 \item
1167 \(x\)-intercepts of \(y=f(x)\) identify \(x\)-coordinates of
1168 stationary points on \(y=F(x)\)
1169 \item
1170 nature of stationary points is determined by sign of \(y=f(x)\) on
1171 either side of its \(x\)-intercepts
1172 \item
1173 if \(f(x)\) is a polynomial of degree \(n\), then \(F(x)\) has degree
1174 \(n+1\)
1175 \end{itemize}
1176
1177 To find stationary points of a function, substitute \(x\) value of given
1178 point into derivative. Solve for \({\frac{dy}{dx}}=0\). Integrate to find
1179 original function.
1180
1181 \subsection*{Rates}
1182
1183 \subsubsection*{Gradient at a point on parametric curve}
1184
1185 \[{\frac{dy}{dx}} = {{\frac{dy}{dt}} \div {\frac{dx}{dt}}} \> \vert \> {\frac{dx}{dt}} \ne 0 \text{ (chain rule)}\]
1186
1187 \[\frac{d^2}{dx^2} = \frac{d(y^\prime)}{dx} = {\frac{dy^\prime}{dt} \div {\frac{dx}{dt}}} \> \vert \> y^\prime = {\frac{dy}{dx}}\]
1188
1189 \subsection*{Rational functions}
1190
1191 \[f(x) = \frac{P(x)}{Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}\]
1192
1193 \subsection*{Fundamental theorem of calculus}
1194
1195 If \(f\) is continuous on \([a, b]\), then
1196
1197 \[\int^b_a f(x) \> dx = F(b) - F(a)\]
1198 \hfill where \(F = \int f \> dx\)
1199
1200 \subsection*{Differential equations}
1201
1202 \noindent\textbf{Order} - highest power inside derivative\\
1203 \textbf{Degree} - highest power of highest derivative\\
1204 e.g. \({\left(\dfrac{dy^2}{d^2} x\right)}^3\) \qquad order 2, degree 3
1205
1206 \begin{warning}
1207 To verify solutions, find \(\frac{dy}{dx}\) from \(y\) and substitute into original
1208 \end{warning}
1209
1210
1211
1212 \subsubsection*{Mixing problems}
1213
1214 \[\left(\frac{dm}{dt}\right)_\Sigma = \left(\frac{dm}{dt}\right)_{\text{in}} - \left(\frac{dm}{dt}_{\text{out}}\right)\]
1215
1216 \subsection*{Euler's method}
1217
1218 \[\dfrac{f(x+h) - f(x)}{h} \approx f^\prime (x) \quad \text{for small } h\]
1219
1220 \[\implies f(x+h) \approx f(x) + hf^\prime(x)\]
1221
1222 \begin{theorembox}{}
1223 If \(\dfrac{dy}{dx} = g(x)\) with \(x_0 = a\) and \(y_0 = b\), then:
1224 \begin{align*}
1225 x_{n+1} &= x_n + h \\
1226 y_{n+1} &= y_n + hg(x_n)
1227 \end{align*}
1228 \end{theorembox}
1229
1230
1231
1232 \include{calculus-rules}
1233
1234 \section{Kinematics \& Mechanics}
1235
1236 \subsection*{Constant acceleration}
1237
1238 \begin{itemize}
1239 \item \textbf{Position} - relative to origin
1240 \item \textbf{Displacement} - relative to starting point
1241 \end{itemize}
1242
1243 \subsubsection*{Velocity-time graphs}
1244
1245 \begin{description}[nosep, labelindent=0.5cm, leftmargin=0.5\columnwidth]
1246 \item[Displacement:] \textit{signed} area
1247 \item[Distance travelled:] \textit{total} area
1248 \end{description}
1249
1250 \[ \text{acceleration} = \frac{d^2x}{dt^2} = \frac{dv}{dt} = v\frac{dv}{dx} = \frac{d}{dx}\left(\frac{1}{2}v^2\right) \]
1251
1252 \begin{center}
1253 \renewcommand{\arraystretch}{1}
1254 \begin{tabular}{ l r }
1255 \hline & no \\ \hline
1256 \(v=u+at\) & \(x\) \\
1257 \(v^2 = u^2+2as\) & \(t\) \\
1258 \(s = \frac{1}{2} (v+u)t\) & \(a\) \\
1259 \(s = ut + \frac{1}{2} at^2\) & \(v\) \\
1260 \(s = vt- \frac{1}{2} at^2\) & \(u\) \\ \hline
1261 \end{tabular}
1262 \end{center}
1263
1264 \[ v_{\text{avg}} = \frac{\Delta\text{position}}{\Delta t} \]
1265 \begin{align*}
1266 \text{speed} &= |{\text{velocity}}| \\
1267 &= \sqrt{v_x^2 + v_y^2 + v_z^2}
1268 \end{align*}
1269
1270 \noindent \textbf{Distance travelled between \(t=a \rightarrow t=b\):}
1271 \begin{align*}
1272 &= \int^{b}_{a}{\sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}} \> dt \tag{2D} \\
1273 &= \int^{t=b}_{t=a}{\dfrac{dx}{dt}} \> dt \tag{linear}
1274 \end{align*}
1275
1276 \noindent \textbf{Shortest distance between \(\boldsymbol{r}(t_0)\) and \(\boldsymbol{r}(t_1)\):}
1277 \[ = |\boldsymbol{r}(t_1) - \boldsymbol{r}(t_2)| \]
1278
1279 \subsection*{Vector functions}
1280
1281 \[ \boldsymbol{r}(t) = x \boldsymbol{i} + y \boldsymbol{j} + z \boldsymbol{k} \]
1282
1283 \begin{itemize}
1284 \item If \(\boldsymbol{r}(t) \equiv\) position with time, then the graph of endpoints of \(\boldsymbol{r}(t) \equiv\) Cartesian path
1285 \item Domain of \(\boldsymbol{r}(t)\) is the range of \(x(t)\)
1286 \item Range of \(\boldsymbol{r}(t)\) is the range of \(y(t)\)
1287 \end{itemize}
1288
1289 \subsection*{Vector calculus}
1290
1291 \subsubsection*{Derivative}
1292
1293 Let \(\boldsymbol{r}(t)=x(t)\boldsymbol{i} + y(t)\boldsymbol(j)\). If both \(x(t)\) and \(y(t)\) are differentiable, then:
1294 \[ \boldsymbol{r}(t)=x(t)\boldsymbol{i}+y(t)\boldsymbol{j} \]
1295
1296 \subfile{dynamics}
1297 \subfile{statistics}
1298 \end{multicols}
1299\end{document}