0525085edb4a24ea2b82239d61b478374fc9ce3b
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9\usepackage{tabularx}
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11\usepackage{enumitem}
12\usepackage[obeyspaces]{url}
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14\usepackage{blindtext}
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46\usepgflibrary{arrows.meta}
47\usepackage{fancyhdr}
48\pagestyle{fancy}
49\fancyhead[LO,LE]{Year 12 Specialist}
50\fancyhead[CO,CE]{Andrew Lorimer}
51\usepackage{mathtools}
52\usepackage{xcolor} % used only to show the phantomed stuff
53\renewcommand\hphantom[1]{{\color[gray]{.6}#1}} % comment out!
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75\usepackage{tcolorbox}
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77\newtcolorbox{cas}{colframe=cas!75!black, title=On CAS, left*=3mm}
78\usepackage{keystroke}
79\usepackage{listings}
80\usepackage{mathtools}
81\pgfplotsset{compat=1.16}
82\usepackage{subfiles}
83\usepackage{import}
84\setlength{\parindent}{0pt}
85\begin{document}
86
87\begin{multicols}{2}
88
89 \section{Complex numbers}
90
91 \[\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}\]
92
93 \begin{align*}
94 \text{Cartesian form: } & a+bi\\
95 \text{Polar form: } & r\operatorname{cis}\theta
96 \end{align*}
97
98 \subsection*{Operations}
99
100 \definecolor{shade1}{HTML}{ffffff}
101 \definecolor{shade2}{HTML}{e6f2ff}
102 \definecolor{shade3}{HTML}{cce2ff}
103 \begin{tabularx}{\columnwidth}{r|X|X}
104 & \textbf{Cartesian} & \textbf{Polar} \\
105 \hline
106 \(z_1 \pm z_2\) & \((a \pm c)(b \pm d)i\) & convert to \(a+bi\)\\
107 \hline
108 \(+k \times z\) & \multirow{2}{*}{\(ka \pm kbi\)} & \(kr\operatorname{cis} \theta\)\\
109 \cline{1-1}\cline{3-3}
110 \(-k \times z\) & & \(kr \operatorname{cis}(\theta\pm \pi)\)\\
111 \hline
112 \(z_1 \cdot z_2\) & \(ac-bd+(ad+bc)i\) & \(r_1r_2 \operatorname{cis}(\theta_1 + \theta_2)\)\\
113 \hline
114 \(z_1 \div z_2\) & \((z_1 \overline{z_2}) \div |z_2|^2\) & \(\left(\frac{r_1}{r_2}\right) \operatorname{cis}(\theta_1 - \theta_2)\)
115 \end{tabularx}
116
117 \subsubsection*{Scalar multiplication in polar form}
118
119 For \(k \in \mathbb{R}^+\):
120 \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\theta\]
121
122 \noindent For \(k \in \mathbb{R}^-\):
123 \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\left(\begin{cases}\theta - \pi & |0<\operatorname{Arg}(z)\le \pi \\ \theta + \pi & |-\pi<\operatorname{Arg}(z)\le 0\end{cases}\right)\]
124
125 \subsection*{Conjugate}
126
127 \begin{align*}
128 \overline{z} &= a \mp bi\\
129 &= r \operatorname{cis}(-\theta)
130 \end{align*}
131
132 \noindent \colorbox{cas}{On CAS: \texttt{conjg(a+bi)}}
133
134 \subsubsection*{Properties}
135
136 \begin{align*}
137 \overline{z_1 \pm z_2} &= \overline{z_1}\pm\overline{z_2}\\
138 \overline{z_1 \cdot z_2} &= \overline{z_1}\cdot\overline{z_2}\\
139 \overline{kz} &= k\overline{z} \quad | \quad k \in \mathbb{R}\\
140 z\overline{z} &= (a+bi)(a-bi)\\
141 &= a^2 + b^2\\
142 &= |z|^2
143 \end{align*}
144
145 \subsection*{Modulus}
146
147 \[|z|=|\vec{Oz}|=\sqrt{a^2 + b^2}\]
148
149 \subsubsection*{Properties}
150
151 \begin{align*}
152 |z_1z_2|&=|z_1||z_2|\\
153 \left|\frac{z_1}{z_2}\right|&=\frac{|z_1|}{|z_2|}\\
154 |z_1+z_2|&\le|z_1|+|z_2|
155 \end{align*}
156
157 \subsection*{Multiplicative inverse}
158
159 \begin{align*}
160 z^{-1}&=\frac{a-bi}{a^2+b^2}\\
161 &=\frac{\overline{z}}{|z|^2}a\\
162 &=r \operatorname{cis}(-\theta)
163 \end{align*}
164
165 \subsection*{Dividing over \(\mathbb{C}\)}
166
167 \begin{align*}
168 \frac{z_1}{z_2}&=z_1z_2^{-1}\\
169 &=\frac{z_1\overline{z_2}}{|z_2|^2}\\
170 &=\frac{(a+bi)(c-di)}{c^2+d^2}\\
171 & \qquad \text{(rationalise denominator)}
172 \end{align*}
173
174 \subsection*{Polar form}
175
176 \begin{align*}
177 z&=r\operatorname{cis}\theta\\
178 &=r(\cos \theta + i \sin \theta)
179 \end{align*}
180
181 \begin{itemize}
182 \item{\(r=|z|=\sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}\)}
183 \item{\(\theta = \operatorname{arg}(z)\) \quad \colorbox{cas}{On CAS: \texttt{arg(a+bi)}}}
184 \item{\(\operatorname{Arg}(z) \in (-\pi,\pi)\) \quad \bf{(principal argument)}}
185 \item{\colorbox{cas}{Convert on CAS:}\\ \verb|compToTrig(a+bi)| \(\iff\) \verb|cExpand{r·cisX}|}
186 \item{Multiple representations:\\\(r\operatorname{cis}\theta=r\operatorname{cis}(\theta+2n\pi)\) with \(n \in \mathbb{Z}\) revolutions}
187 \item{\(\operatorname{cis}\pi=-1,\qquad \operatorname{cis}0=1\)}
188 \end{itemize}
189
190 \subsection*{de Moivres' theorem}
191
192 \[(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}\]
193
194 \subsection*{Complex polynomials}
195
196 Include \(\pm\) for all solutions, incl. imaginary
197
198 \begin{tabularx}{\columnwidth}{ R{0.55} X }
199 \hline
200 Sum of squares & \(\begin{aligned}
201 z^2 + a^2 &= z^2-(ai)^2\\
202 &= (z+ai)(z-ai) \end{aligned}\) \\
203 \hline
204 Sum of cubes & \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)\\
205 \hline
206 Division & \(P(z)=D(z)Q(z)+R(z)\) \\
207 \hline
208 Remainder theorem & Let \(\alpha \in \mathbb{C}\). Remainder of \(P(z) \div (z-\alpha)\) is \(P(\alpha)\)\\
209 \hline
210 Factor theorem & \(z-\alpha\) is a factor of \(P(z) \iff P(\alpha)=0\) for \(\alpha \in \mathbb{C}\)\\
211 \hline
212 Conjugate root theorem & \(P(z)=0 \text{ at } z=a\pm bi\) (\(\implies\) both \(z_1\) and \(\overline{z_1}\) are solutions)\\
213 \hline
214 \end{tabularx}
215
216 \subsection*{\(n\)th roots}
217
218 \(n\)th roots of \(z=r\operatorname{cis}\theta\) are:
219
220 \[z = r^{\frac{1}{n}} \operatorname{cis}\left(\frac{\theta+2k\pi}{n}\right)\]
221
222 \begin{itemize}
223
224 \item{Same modulus for all solutions}
225 \item{Arguments separated by \(\frac{2\pi}{n} \therefore\) there are \(n\) roots}
226 \item{If one square root is \(a+bi\), the other is \(-a-bi\)}
227 \item{Give one implicit \(n\)th root \(z_1\), function is \(z=z_1^n\)}
228 \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 \(\frac{2\pi}{n}\))}
229 \end{itemize}
230
231 \noindent For \(0=az^2+bz+c\), use quadratic formula:
232
233 \[z=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]
234
235 \subsection*{Fundamental theorem of algebra}
236
237 A polynomial of degree \(n\) can be factorised into \(n\) linear factors in \(\mathbb{C}\):
238
239 \[\implies P(z)=a_n(z-\alpha_1)(z-\alpha_2)(z-\alpha_3)\dots(z-\alpha_n)\]
240 \[\text{ where } \alpha_1,\alpha_2,\alpha_3,\dots,\alpha_n \in \mathbb{C}\]
241
242 \subsection*{Argand planes}
243
244 \begin{center}\begin{tikzpicture}[scale=2]
245 \draw [->] (-0.2,0) -- (1.5,0) node [right] {$\operatorname{Re}(z)$};
246 \draw [->] (0,-0.2) -- (0,1.5) node [above] {$\operatorname{Im}(z)$};
247 \coordinate (P) at (1,1);
248 \coordinate (a) at (1,0);
249 \coordinate (b) at (0,1);
250 \coordinate (O) at (0,0);
251 \draw (0,0) -- (P) node[pos=0.5, above left]{\(r\)} node[pos=1, right]{\(\begin{aligned}z&=a+bi\\&=r\operatorname{cis}\theta\end{aligned}\)};
252 \draw [gray, dashed] (1,1) -- (1,0) node[black, pos=1, below]{\(a\)};
253 \draw [gray, dashed] (1,1) -- (0,1) node[black, pos=1, left]{\(b\)};
254 \begin{scope}
255 \path[clip] (O) -- (P) -- (a);
256 \fill[red, opacity=0.5, draw=black] (O) circle (2mm);
257 \node at ($(O)+(20:3mm)$) {$\theta$};
258 \end{scope}
259 \filldraw (P) circle (0.5pt);
260 \end{tikzpicture}\end{center}
261
262 \begin{itemize}
263 \item{Multiplication by \(i \implies\) CCW rotation of \(\frac{\pi}{2}\)}
264 \item{Addition: \(z_1 + z_2 \equiv\) \overrightharp{\(Oz_1\)} + \overrightharp{\(Oz_2\)}}
265 \end{itemize}
266
267 \subsection*{Sketching complex graphs}
268
269 \subsubsection*{Linear}
270
271 \begin{itemize}
272 \item{\(\operatorname{Re}(z)=c\) or \(\operatorname{Im}(z)=c\) (perpendicular bisector)}
273 \item{\(\operatorname{Im}(z)=m\operatorname{Re}(z)\)}
274 \item{\(|z+a|=|z+b| \implies 2(a-b)x=b^2-a^2\)\\Geometric: equidistant from \(a,b\)}
275 \end{itemize}
276
277 \subsubsection*{Circles}
278
279 \begin{itemize}
280 \item \(|z-z_1|^2=c^2|z_2+2|^2\)
281 \item \(|z-(a+bi)|=c \implies (x-a)^2+_(y-b)^2=c^2\)
282 \end{itemize}
283
284 \noindent \textbf{Loci} \qquad \(\operatorname{Arg}(z)<\theta\)
285
286 \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
287 \draw [->] (0,0) -- (1,0) node [right] {$\operatorname{Re}(z)$};
288 \draw [->] (0,-0.5) -- (0,1) node [above] {$\operatorname{Im}(z)$};
289 \draw [<-, dashed, thick, blue] (-1,0) -- (0,0);
290 \draw [->, thick, blue] (0,0) -- (1,1);
291 \fill [gray, opacity=0.2, domain=-1:1, variable=\x] (-1,-0.5) -- (-1,0) -- (0, 0) -- (1,1) -- (1,-0.5) -- cycle;
292 \begin{scope}
293 \path[clip] (0,0) -- (1,1) -- (1,0);
294 \fill[red, opacity=0.5, draw=black] (0,0) circle (2mm);
295 \node at ($(0,0)+(20:3mm)$) {$\frac{\pi}{4}$};
296 \end{scope}
297 \node [font=\footnotesize] at (0.5,-0.25) {\(\operatorname{Arg}(z)\le\frac{\pi}{4}\)};
298 \node [blue, mydot] {};
299 \end{tikzpicture}\end{center}
300
301 \noindent \textbf{Rays} \qquad \(\operatorname{Arg}(z-b)=\theta\)
302
303 \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
304 \draw [->] (-0.75,0) -- (1.5,0) node [right] {$\operatorname{Re}(z)$};
305 \draw [->] (0,-1) -- (0,1) node [above] {$\operatorname{Im}(z)$};
306 \draw [->, thick, brown] (-0.25,0) -- (-0.75,-1);
307 \node [above, font=\footnotesize] at (-0.25,0) {\(\frac{1}{4}\)};
308 \begin{scope}
309 \path[clip] (-0.25,0) -- (-0.75,-1) -- (0,0);
310 \fill[orange, opacity=0.5, draw=black] (-0.25,0) circle (2mm);
311 \end{scope}
312 \node at (-0.08,-0.3) {\(\frac{\pi}{8}\)};
313 \node [font=\footnotesize, left] at (-0.75,-1) {\(\operatorname{Arg}(z+\frac{1}{4})=\frac{\pi}{8}\)};
314 \node [brown, mydot] at (-0.25,0) {};
315 \draw [<->, thick, green] (0,-1) -- (1.5,0.5) node [pos=0.25, black, font=\footnotesize, right] {\(|z-2|=|z-(1+i)|\)};
316 \node [left, font=\footnotesize] at (0,-1) {\(-1\)};
317 \node [below, font=\footnotesize] at (1,0) {\(1\)};
318 \end{tikzpicture}\end{center}
319
320 \section{Vectors}
321 \begin{center}\begin{tikzpicture}
322 \draw [->] (-0.5,0) -- (3,0) node [right] {\(x\)};
323 \draw [->] (0,-0.5) -- (0,3) node [above] {\(y\)};
324 \draw [orange, ->, thick] (0.5,0.5) -- (2.5,2.5) node [pos=0.5, above] {\(\vec{u}\)};
325 \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
326 \draw [gray, dashed, thick] (0.5,0.5) -- (2.5,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, below]{\(x\vec{i}\)};
327 \draw [gray, dashed, thick] (2.5,0.5) -- (2.5,2.5) node [pos=0.5] {\midarrow};
328 \end{scope}
329 \node[black, right] at (2.5,1.5) {\(y\vec{j}\)};
330 \end{tikzpicture}\end{center}
331 \subsection*{Column notation}
332
333 \[\begin{bmatrix}x\\ y \end{bmatrix} \iff x\boldsymbol{i} + y\boldsymbol{j}\]
334 \(\begin{bmatrix}x_2-x_1\\ y_2-y_1 \end{bmatrix}\) \quad between \(A(x_1,y_1), \> B(x_2,y_2)\)
335
336 \subsection*{Scalar multiplication}
337
338 \[k\cdot (x\boldsymbol{i}+y\boldsymbol{j})=kx\boldsymbol{i}+ky\boldsymbol{j}\]
339
340 \noindent For \(k \in \mathbb{R}^-\), direction is reversed
341
342 \subsection*{Vector addition}
343 \begin{center}\begin{tikzpicture}[scale=1]
344 \coordinate (A) at (0,0);
345 \coordinate (B) at (2,2);
346 \draw [->, thick, red] (0,0) -- (2,2) node [pos=0.5, below right] {\(\vec{u}=2\vec{i}+2\vec{j}\)};
347 \draw [->, thick, blue] (2,2) -- (1,4) node [pos=0.5, above right] {\(\vec{v}=-\vec{i}+2\vec{j}\)};
348 \draw [->, thick, orange] (0,0) -- (1,4) node [pos=0.5, left] {\(\vec{u}+\vec{v}=\vec{i}+4\vec{j}\)};
349 \end{tikzpicture}\end{center}
350
351 \[(x\boldsymbol{i}+y\boldsymbol{j}) \pm (a\boldsymbol{i}+b\boldsymbol{j})=(x \pm a)\boldsymbol{i}+(y \pm b)\boldsymbol{j}\]
352
353 \begin{itemize}
354 \item Draw each vector head to tail then join lines
355 \item Addition is commutative (parallelogram)
356 \item \(\boldsymbol{u}-\boldsymbol{v}=\boldsymbol{u}+(-\boldsymbol{v}) \implies \overrightharp{AB}=\boldsymbol{b}-\boldsymbol{a}\)
357 \end{itemize}
358
359 \subsection*{Magnitude}
360
361 \[|(x\boldsymbol{i} + y\boldsymbol{j})|=\sqrt{x^2+y^2}\]
362
363 \subsection*{Parallel vectors}
364
365 \[\boldsymbol{u} || \boldsymbol{v} \iff \boldsymbol{u} = k \boldsymbol{v} \text{ where } k \in \mathbb{R} \setminus \{0\}\]
366
367 For parallel vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\):\\
368 \[\boldsymbol{a \cdot b}=\begin{cases}
369 |\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction}\\
370 -|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions}
371 \end{cases}\]
372 %\includegraphics[width=0.2,height=\textheight]{graphics/parallelogram-vectors.jpg}
373 %\includegraphics[width=1]{graphics/vector-subtraction.jpg}
374
375 \subsection*{Perpendicular vectors}
376
377 \[\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b} = 0\ \quad \text{(since \(\cos 90 = 0\))}\]
378
379 \subsection*{Unit vector \(|\hat{\boldsymbol{a}}|=1\)}
380 \[\begin{split}\hat{\boldsymbol{a}} & = {\frac{1}{|\boldsymbol{a}|}}\boldsymbol{a} \\ & = \boldsymbol{a} \cdot {|\boldsymbol{a}|}\end{split}\]
381
382 \subsection*{Scalar product \(\boldsymbol{a} \cdot \boldsymbol{b}\)}
383
384
385 \begin{center}\begin{tikzpicture}[scale=2]
386 \draw [->] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{b}\)};
387 \draw [->] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{a}\)};
388 \begin{scope}
389 \path[clip] (1,0.5) -- (1,0) -- (0,0);
390 \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
391 \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
392 \end{scope}
393 \end{tikzpicture}\end{center}
394 \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*}
395 \noindent\colorbox{cas}{On CAS: \texttt{dotP({[}a\ b\ c{]},\ {[}d\ e\ f{]})}}
396
397 \subsubsection*{Properties}
398
399 \begin{enumerate}
400 \item
401 \(k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k\boldsymbol{b})\)
402 \item
403 \(\boldsymbol{a \cdot 0}=0\)
404 \item
405 \(\boldsymbol{a} \cdot (\boldsymbol{b} + \boldsymbol{c})=\boldsymbol{a} \cdot \boldsymbol{b} + \boldsymbol{a} \cdot \boldsymbol{c}\)
406 \item
407 \(\boldsymbol{i \cdot i} = \boldsymbol{j \cdot j} = \boldsymbol{k \cdot k}= 1\)
408 \item
409 \(\boldsymbol{a} \cdot \boldsymbol{b} = 0 \quad \implies \quad \boldsymbol{a} \perp \boldsymbol{b}\)
410 \item
411 \(\boldsymbol{a \cdot a} = |\boldsymbol{a}|^2 = a^2\)
412 \end{enumerate}
413
414 \subsection*{Angle between vectors}
415
416 \[\cos \theta = \frac{\boldsymbol{a} \cdot \boldsymbol{b}}{|\boldsymbol{a}| |\boldsymbol{b}|} = \frac{a_1 b_1 + a_2 b_2}{|\boldsymbol{a}| |\boldsymbol{b}|}\]
417
418 \noindent \colorbox{cas}{On CAS:} \texttt{angle([a b c], [a b c])}
419
420 (Action \(\rightarrow\) Vector \(\rightarrow\)Angle)
421
422 \subsection*{Angle between vector and axis}
423
424 \noindent For\(\boldsymbol{a} = a_1 \boldsymbol{i} + a_2 \boldsymbol{j} + a_3 \boldsymbol{k}\)
425 which makes angles \(\alpha, \beta, \gamma\) with positive side of
426 \(x, y, z\) axes:
427 \[\cos \alpha = \frac{a_1}{|\boldsymbol{a}|}, \quad \cos \beta = \frac{a_2}{|\boldsymbol{a}|}, \quad \cos \gamma = \frac{a_3}{|\boldsymbol{a}|}\]
428
429 \noindent \colorbox{cas}{On CAS:} \texttt{angle({[}a\ b\ c{]},\ {[}1\ 0\ 0{]})}\\for angle
430 between \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) and
431 \(x\)-axis
432
433 \subsection*{Projections \& resolutes}
434
435 \begin{tikzpicture}[scale=3]
436 \draw [->, purple] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{a}\)};
437 \draw [->, orange] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{u}\)};
438 \draw [->, blue] (1,0) -- (2,0) node [pos=0.5, below] {\(\boldsymbol{b}\)};
439 \begin{scope}
440 \path[clip] (1,0.5) -- (1,0) -- (0,0);
441 \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
442 \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
443 \end{scope}
444 \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
445 \draw [gray, dashed, thick] (1,0) -- (1,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, right, rotate=-90]{\(\boldsymbol{w}\)};
446 \end{scope}
447 \draw (0,0) coordinate (O)
448 (1,0) coordinate (A)
449 (1,0.5) coordinate (B)
450 pic [draw,red,angle radius=2mm] {right angle = O--A--B};
451 \end{tikzpicture}
452
453 \subsubsection*{\(\parallel\boldsymbol{b}\) (vector projection/resolute)}
454
455 \begin{align*}
456 \boldsymbol{u} & = \frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|^2}\boldsymbol{b} \\
457 & = \left(\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\right)\left(\frac{\boldsymbol{b}}{|\boldsymbol{b}|}\right) \\
458 & = (\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}
459 \end{align*}
460
461 \subsubsection*{\(\perp\boldsymbol{b}\) (perpendicular projection)}
462 \[\boldsymbol{w} = \boldsymbol{a} - \boldsymbol{u}\]
463
464 \subsubsection*{\(|\boldsymbol{u}|\) (scalar projection/resolute)}
465 \begin{align*}
466 s &= |\boldsymbol{u}|\\
467 &= \boldsymbol{a} \cdot \hat{\boldsymbol{b}}\\
468 &=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\\
469 &= |\boldsymbol{a}| \cos \theta
470 \end{align*}
471
472 \subsubsection*{Rectangular (\(\parallel,\perp\)) components}
473
474 \[\boldsymbol{a}=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}+\left(\boldsymbol{a}-\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}\right)\]
475
476
477 \subsection*{Vector proofs}
478
479 \textbf{Concurrent:} intersection of \(\ge\) 3 lines
480
481 \begin{tikzpicture}
482 \draw [blue] (0,0) -- (1,1);
483 \draw [red] (1,0) -- (0,1);
484 \draw [brown] (0.4,0) -- (0.6,1);
485 \filldraw (0.5,0.5) circle (2pt);
486 \end{tikzpicture}
487
488 \subsubsection*{Collinear points}
489
490 \(\ge\) 3 points lie on the same line
491
492 \begin{tikzpicture}
493 \draw [purple] (0,0) -- (4,1);
494 \filldraw (2,0.5) circle (2pt) node [above] {\(C\)};
495 \filldraw (1,0.25) circle (2pt) node [above] {\(A\)};
496 \filldraw (3,0.75) circle (2pt) node [above] {\(B\)};
497 \coordinate (O) at (2.8,-0.2);
498 \node at (O) [below] {\(O\)};
499 \begin{scope}[->, orange, thick]
500 \draw (O) -- (2,0.5) node [pos=0.5, above, font=\footnotesize, black] {\(\boldsymbol{c}\)};
501 \draw (O) -- (1,0.25) node [pos=0.5, below, font=\footnotesize, black] {\(\boldsymbol{a}\)};
502 \draw (O) -- (3,0.75) node [pos=0.5, right, font=\footnotesize, black] {\(\boldsymbol{b}\)};
503 \end{scope}
504 \end{tikzpicture}
505
506 \begin{align*}
507 \text{e.g. Prove that}\\
508 \overrightharp{AC}=m\overrightharp{AB} \iff \boldsymbol{c}&=(1-m)\boldsymbol{a}+m\boldsymbol{b}\\
509 \implies \boldsymbol{c} &= \overrightharp{OA} + \overrightharp{AC}\\
510 &= \overrightharp{OA} + m\overrightharp{AB}\\
511 &=\boldsymbol{a}+m(\boldsymbol{b}-\boldsymbol{a})\\
512 &=\boldsymbol{a}+m\boldsymbol{b}-m\boldsymbol{a}\\
513 &=(1-m)\boldsymbol{a}+m{b}
514 \end{align*}
515 \begin{align*}
516 \text{Also, } \implies \overrightharp{OC} &= \lambda \vec{OA} + \mu \overrightharp{OB} \\
517 \text{where } \lambda + \mu &= 1\\
518 \text{If } C \text{ lies along } \overrightharp{AB}, & \implies 0 < \mu < 1
519 \end{align*}
520
521
522 \subsubsection*{Parallelograms}
523
524 \begin{center}\begin{tikzpicture}
525 \coordinate (O) at (0,0) node [below left] {\(O\)};
526 \coordinate (A) at (4,0);
527 \coordinate (B) at (6,2);
528 \coordinate (C) at (2,2);
529 \coordinate (D) at (6,0);
530
531 \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (O)--(A) node [below left] {\(A\)};
532 \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (A)--(B) node [above right] {\(B\)};
533 \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (B)--(C) node [above left] {\(C\)};
534 \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (C)--(O);
535
536 \draw [gray, dashed] (O) -- (B) node [pos=0.75] {\(\diagdown\diagdown\)} node [pos=0.25] {\(\diagdown\diagdown\)};
537 \draw [gray, dashed] (A) -- (C) node [pos=0.75] {\(\diagup\)} node [pos=0.25] {\(\diagup\)};
538 \begin{scope}
539 \path[clip] (C) -- (A) -- (O);
540 \fill[orange, opacity=0.5, draw=black] (0,0) circle (4mm);
541 \node at ($(0,0)+(20:8mm)$) {\(\theta\)};
542 \end{scope}
543 \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] {\(|\boldsymbol{c}|\cos\theta\)};
544 \draw pic [draw,thick,red,angle radius=2mm] {right angle=O--D--B};
545 \end{tikzpicture}\end{center}
546
547 \begin{itemize}
548 \item
549 Diagonals \(\overrightharp{OB}, \overrightharp{AC}\) bisect each other
550 \item
551 If diagonals are equal length, it is a rectangle
552 \item
553 \(|\overrightharp{OB}|^2 + |\overrightharp{CA}|^2 = |\overrightharp{OA}|^2 + |\overrightharp{AB}|^2 + |\overrightharp{CB}|^2 + |\overrightharp{OC}|^2\)
554 \item
555 Area \(=\boldsymbol{c} \cdot \boldsymbol{a}\)
556 \end{itemize}
557
558 \subsubsection*{Useful vector properties}
559
560 \begin{itemize}
561 \item
562 \(\boldsymbol{a} \parallel \boldsymbol{b} \implies \boldsymbol{b}=k\boldsymbol{a}\) for some
563 \(k \in \mathbb{R} \setminus \{0\}\)
564 \item
565 If \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are parallel with at
566 least one point in common, then they lie on the same straight line
567 \item
568 \(\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b}=0\)
569 \item
570 \(\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2\)
571 \end{itemize}
572
573 \subsection*{Linear dependence}
574
575 \(\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\) are linearly dependent if they are \(\nparallel\) and:
576 \begin{align*}
577 0&=k\boldsymbol{a}+l\boldsymbol{b}+m\boldsymbol{c}\\
578 \therefore \boldsymbol{c} &= m\boldsymbol{a} + n\boldsymbol{b} \quad \text{(simultaneous)}
579 \end{align*}
580
581 \noindent \(\boldsymbol{a}, \boldsymbol{b},\) and \(\boldsymbol{c}\) are linearly
582 independent if no vector in the set is expressible as a linear
583 combination of other vectors in set, or if they are parallel.
584
585 \subsection*{Three-dimensional vectors}
586
587 Right-hand rule for axes: \(z\) is up or out of page.
588
589 \tdplotsetmaincoords{60}{120}
590 \begin{center}\begin{tikzpicture} [scale=3, tdplot_main_coords, axis/.style={->,thick},
591 vector/.style={-stealth,red,very thick},
592 vector guide/.style={dashed,gray,thick}]
593
594 %standard tikz coordinate definition using x, y, z coords
595 \coordinate (O) at (0,0,0);
596
597 %tikz-3dplot coordinate definition using x, y, z coords
598
599 \pgfmathsetmacro{\ax}{1}
600 \pgfmathsetmacro{\ay}{1}
601 \pgfmathsetmacro{\az}{1}
602
603 \coordinate (P) at (\ax,\ay,\az);
604
605 %draw axes
606 \draw[axis] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$};
607 \draw[axis] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$};
608 \draw[axis] (0,0,0) -- (0,0,1) node[anchor=south]{$z$};
609
610 %draw a vector from O to P
611 \draw[vector] (O) -- (P);
612
613 %draw guide lines to components
614 \draw[vector guide] (O) -- (\ax,\ay,0);
615 \draw[vector guide] (\ax,\ay,0) -- (P);
616 \draw[vector guide] (P) -- (0,0,\az);
617 \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
618 \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
619 \draw[vector guide] (\ax,\ay,0) -- (\ax,0,0);
620 \node[tdplot_main_coords,above right]
621 at (\ax,\ay,\az){(\ax, \ay, \az)};
622 \end{tikzpicture}\end{center}
623
624 \subsection*{Parametric vectors}
625
626 Parametric equation of line through point \((x_0, y_0, z_0)\) and
627 parallel to \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) is:
628
629 \[\begin{cases}x = x_o + a \cdot t \\ y = y_0 + b \cdot t \\ z = z_0 + c \cdot t\end{cases}\]
630
631 \section{Circular functions}
632
633 \(\sin(bx)\) or \(\cos(bx)\): period \(=\frac{2\pi}{b}\)
634
635 \noindent \(\tan(nx)\): period \(=\frac{\pi}{n}\)\\
636 \indent\indent\indent asymptotes at \(x=\frac{(2k+1)\pi}{2n} \> \vert \> k \in \mathbb{Z}\)
637
638 \subsection*{Reciprocal functions}
639
640 \subsubsection*{Cosecant}
641
642 \[\operatorname{cosec} \theta = \frac{1}{\sin \theta} \> \vert \> \sin \theta \ne 0\]
643
644 \begin{itemize}
645 \item
646 \textbf{Domain} \(= \mathbb{R} \setminus {n\pi : n \in \mathbb{Z}}\)
647 \item
648 \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
649 \item
650 \textbf{Turning points} at
651 \(\theta = \frac{(2n + 1)\pi}{2} \> \vert \> n \in \mathbb{Z}\)
652 \item
653 \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
654 \end{itemize}
655
656 \subsubsection*{Secant}
657
658\begin{tikzpicture}
659 \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={\(-2\pi\), \(-\pi\), \(\pi\), \(2\pi\)}]
660% \addplot[blue, domain=-6.2830:6.2830,unbounded coords=jump,samples=80] {sec(deg(x))};
661 \addplot[blue, restrict y to domain=-10:10, domain=-7:7,samples=100] {sec(deg(x))} node [pos=0.93, black, right] {\(\operatorname{sec} x\)};
662 \addplot[red, dashed, domain=-7:7,samples=100] {cos(deg(x))};
663 \draw [gray, dotted, thick] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{rel axis cs:0,1});
664 \draw [gray, dotted, thick] ({axis cs:4.71239,0}|-{rel axis cs:0,0}) -- ({axis cs:4.71239,0}|-{rel axis cs:0,1});
665 \draw [gray, dotted, thick] ({axis cs:-4.71239,0}|-{rel axis cs:0,0}) -- ({axis cs:-4.71239,0}|-{rel axis cs:0,1});
666 \draw [gray, dotted, thick] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1});
667\end{axis}
668 \node [black] at (7,3.5) {\(\cos x\)};
669\end{tikzpicture}
670
671 \[\operatorname{sec} \theta = \frac{1}{\cos \theta} \> \vert \> \cos \theta \ne 0\]
672
673 \begin{itemize}
674
675 \item
676 \textbf{Domain}
677 \(= \mathbb{R} \setminus \frac{(2n + 1) \pi}{2} : n \in \mathbb{Z}\}\)
678 \item
679 \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
680 \item
681 \textbf{Turning points} at
682 \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
683 \item
684 \textbf{Asymptotes} at
685 \(\theta = \frac{(2n + 1) \pi}{2} \> \vert \> n \in \mathbb{Z}\)
686 \end{itemize}
687
688 \subsubsection*{Cotangent}
689
690\begin{tikzpicture}
691 \begin{axis}[xmin=-3,xmax=3,ymin=-1.5,ymax=1.5,enlargelimits=true, xtick={-3.1415, -1.5708, 1.5708, 3.1415},xticklabels={\(-\pi\), \(-\frac{\pi}{2}\), \(\frac{\pi}{2}\), \(\pi\)}]
692 \addplot[blue, smooth, domain=-3:-0.1,unbounded coords=jump,samples=105] {cot(deg(x))} node [pos=0.3, left] {\(\operatorname{cot} x\)};
693\addplot[blue, smooth, domain=0.1:3,unbounded coords=jump,samples=105] {cot(deg(x))};
694\addplot[red, smooth, dashed] gnuplot [domain=-1.5:1.5,unbounded coords=jump,samples=105] {tan(x)};
695\addplot[red, smooth, dashed] gnuplot [domain=-3.5:-1.8,unbounded coords=jump,samples=105] {tan(x)} node [pos=0.5, right] {\(\tan x\)};
696\addplot[red, smooth, dashed] gnuplot [domain=1.8:3.5,unbounded coords=jump,samples=105] {tan(x)};
697 \draw [thick, red, dotted] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1});
698 \draw [thick, blue, dotted] ({axis cs:-3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:-3.1415,0}|-{rel axis cs:0,1});
699 \draw [thick, blue, dotted] ({axis cs:0,0}|-{rel axis cs:0,0}) -- ({axis cs:0,0}|-{rel axis cs:0,1});
700 \draw [thick, blue, dotted] ({axis cs:3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:3.1415,0}|-{rel axis cs:0,1});
701 \draw [thick, red, dotted] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{rel axis cs:0,1});
702\end{axis}
703\end{tikzpicture}
704
705 \[\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0\]
706
707 \begin{itemize}
708
709 \item
710 \textbf{Domain} \(= \mathbb{R} \setminus \{n \pi: n \in \mathbb{Z}\}\)
711 \item
712 \textbf{Range} \(= \mathbb{R}\)
713 \item
714 \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
715 \end{itemize}
716
717 \subsubsection*{Symmetry properties}
718
719 \[\begin{split}
720 \operatorname{sec} (\pi \pm x) & = -\operatorname{sec} x \\
721 \operatorname{sec} (-x) & = \operatorname{sec} x \\
722 \operatorname{cosec} (\pi \pm x) & = \mp \operatorname{cosec} x \\
723 \operatorname{cosec} (-x) & = - \operatorname{cosec} x \\
724 \operatorname{cot} (\pi \pm x) & = \pm \operatorname{cot} x \\
725 \operatorname{cot} (-x) & = - \operatorname{cot} x
726 \end{split}\]
727
728 \subsubsection*{Complementary properties}
729
730 \[\begin{split}
731 \operatorname{sec} \left({\pi \over 2} - x\right) & = \operatorname{cosec} x \\
732 \operatorname{cosec} \left({\pi \over 2} - x\right) & = \operatorname{sec} x \\
733 \operatorname{cot} \left({\pi \over 2} - x\right) & = \tan x \\
734 \tan \left({\pi \over 2} - x\right) & = \operatorname{cot} x
735 \end{split}\]
736
737 \subsubsection*{Pythagorean identities}
738
739 \[\begin{split}
740 1 + \operatorname{cot}^2 x & = \operatorname{cosec}^2 x, \quad \text{where } \sin x \ne 0 \\
741 1 + \tan^2 x & = \operatorname{sec}^2 x, \quad \text{where } \cos x \ne 0
742 \end{split}\]
743
744 \subsection*{Compound angle formulas}
745
746 \[\cos(x \pm y) = \cos x + \cos y \mp \sin x \sin y\]
747 \[\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y\]
748 \[\tan(x \pm y) = {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}\]
749
750 \subsection*{Double angle formulas}
751
752 \[\begin{split}
753 \cos 2x &= \cos^2 x - \sin^2 x \\
754 & = 1 - 2\sin^2 x \\
755 & = 2 \cos^2 x -1
756 \end{split}\]
757
758 \[\sin 2x = 2 \sin x \cos x\]
759
760 \[\tan 2x = {{2 \tan x} \over {1 - \tan^2 x}}\]
761
762 \subsection*{Inverse circular functions}
763
764 \begin{tikzpicture}
765 \begin{axis}[ymin=-2, ymax=4, xmin=-1.1, xmax=1.1, ytick={-1.5708, 1.5708, 3.14159},yticklabels={$-\frac{\pi}{2}$, $\frac{\pi}{2}$, $\pi$}]
766 \addplot[color=red, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {asin(x)} node [pos=0.25, below right] {\(\sin^{-1}x\)};
767 \addplot[color=blue, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {acos(x)} node [pos=0.25, below left] {\(\cos^{-1}x\)};
768 \addplot[mark=*, red] coordinates {(-1,-1.5708)} node[right, font=\footnotesize]{\((-1,-\frac{\pi}{2})\)} ;
769 \addplot[mark=*, red] coordinates {(1,1.5708)} node[left, font=\footnotesize]{\((1,\frac{\pi}{2})\)} ;
770 \addplot[mark=*, blue] coordinates {(1,0)};
771 \addplot[mark=*, blue] coordinates {(-1,3.1415)} node[right, font=\footnotesize]{\((-1,\pi)\)} ;
772 \end{axis}
773 \end{tikzpicture}\\
774
775 Inverse functions: \(f(f^{-1}(x)) = x\) (restrict domain)
776
777 \[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y\]
778 \hfill where \(\sin y = x, \> y \in [{-\pi \over 2}, {\pi \over 2}]\)
779
780 \[\cos^{-1}: [-1,1] \rightarrow \mathbb{R}, \quad \cos^{-1} x = y\]
781 \hfill where \(\cos y = x, \> y \in [0, \pi]\)
782
783 \[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\]
784 \hfill where \(\tan y = x, \> y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\)
785
786 \begin{tikzpicture}
787 \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={\(-\frac{\pi}{2}\),\(\frac{\pi}{2}\)}]
788 \addplot[color=orange, smooth] gnuplot [domain=-35:35, unbounded coords=jump,samples=350] {atan(x)} node [pos=0.5, above left] {\(\tan^{-1}x\)};
789 \addplot[gray, dotted, thick, domain=-35:35] {1.5708} node [black, font=\footnotesize, below right, pos=0] {\(y=\frac{\pi}{2}\)};
790 \addplot[gray, dotted, thick, domain=-35:35] {-1.5708} node [black, font=\footnotesize, above left, pos=1] {\(y=-\frac{\pi}{2}\)};
791 \end{axis}
792 \end{tikzpicture}
793\columnbreak
794 \section{Differential calculus}
795
796 \subsection*{Limits}
797
798 \[\lim_{x \rightarrow a}f(x)\]
799 \(L^-,\quad L^+\) \qquad limit from below/above\\
800 \(\lim_{x \to a} f(x)\) \quad limit of a point\\
801
802 \noindent For solving \(x\rightarrow\infty\), put all \(x\) terms in denominators\\
803 e.g. \[\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2\]
804
805 \subsubsection*{Limit theorems}
806
807 \begin{enumerate}
808 \item
809 For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\)
810 \item
811 \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\)
812 \item
813 \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\)
814 \item
815 \(\therefore \lim_{x \rightarrow a} c \times f(x)=cF\) where \(c=\) constant
816 \item
817 \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\)
818 \item
819 \(f(x)\) is continuous \(\iff L^-=L^+=f(x) \> \forall x\)
820 \end{enumerate}
821
822 \subsection*{Gradients of secants and tangents}
823
824 \textbf{Secant (chord)} - line joining two points on curve\\
825 \textbf{Tangent} - line that intersects curve at one point
826
827 \subsection*{First principles derivative}
828
829 \[f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={\frac{dy}{dx}}\]
830
831 \subsubsection*{Logarithmic identities}
832
833 \(\log_b (xy)=\log_b x + \log_b y\)\\
834 \(\log_b x^n = n \log_b x\)\\
835 \(\log_b y^{x^n} = x^n \log_b y\)
836
837 \subsubsection*{Index identities}
838
839 \(b^{m+n}=b^m \cdot b^n\)\\
840 \((b^m)^n=b^{m \cdot n}\)\\
841 \((b \cdot c)^n = b^n \cdot c^n\)\\
842 \({a^m \div a^n} = {a^{m-n}}\)
843
844 \subsection*{Derivative rules}
845
846 \renewcommand{\arraystretch}{1.4}
847 \begin{tabularx}{\columnwidth}{rX}
848 \hline
849 \(f(x)\) & \(f^\prime(x)\)\\
850 \hline
851 \(\sin x\) & \(\cos x\)\\
852 \(\sin ax\) & \(a\cos ax\)\\
853 \(\cos x\) & \(-\sin x\)\\
854 \(\cos ax\) & \(-a \sin ax\)\\
855 \(\tan f(x)\) & \(f^2(x) \sec^2f(x)\)\\
856 \(e^x\) & \(e^x\)\\
857 \(e^{ax}\) & \(ae^{ax}\)\\
858 \(ax^{nx}\) & \(an \cdot e^{nx}\)\\
859 \(\log_e x\) & \(\dfrac{1}{x}\)\\
860 \(\log_e {ax}\) & \(\dfrac{1}{x}\)\\
861 \(\log_e f(x)\) & \(\dfrac{f^\prime (x)}{f(x)}\)\\
862 \(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\\
863 \(\sin^{-1} x\) & \(\dfrac{1}{\sqrt{1-x^2}}\)\\
864 \(\cos^{-1} x\) & \(\dfrac{-1}{sqrt{1-x^2}}\)\\
865 \(\tan^{-1} x\) & \(\dfrac{1}{1 + x^2}\)\\
866 \(\frac{d}{dy}f(y)\) & \(\dfrac{1}{\frac{dx}{dy}}\) (reciprocal)\\
867 \(uv\) & \(u \frac{dv}{dx}+v\frac{du}{dx} (product rule)\)\\
868 \(\dfrac{u}{v}\) & \(\dfrac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\) (quotient rule)\\
869 \(f(g(x))\) & \(f^\prime(g(x))\cdot g^\prime(x)\)\\
870 \hline
871 \end{tabularx}
872
873 \subsection*{Reciprocal derivatives}
874
875 \[\frac{1}{\frac{dy}{dx}} = \frac{dx}{dy}\]
876
877 \subsection*{Differentiating \(x=f(y)\)}
878 \begin{align*}
879 \text{Find }& \frac{dx}{dy}\\
880 \text{Then, }\frac{dx}{dy} &= \frac{1}{\frac{dy}{dx}} \\
881 \implies {\frac{dy}{dx}} &= \frac{1}{\frac{dx}{dy}}\\
882 \therefore {\frac{dy}{dx}} &= \frac{1}{\frac{dx}{dy}}
883 \end{align*}
884
885 \subsection*{Second derivative}
886 \begin{align*}f(x) \longrightarrow &f^\prime (x) \longrightarrow f^{\prime\prime}(x)\\
887 \implies y \longrightarrow &\frac{dy}{dx} \longrightarrow \frac{d^2 y}{dx^2}\end{align*}
888
889 \noindent Order of polynomial \(n\)th derivative decrements each time the derivative is taken
890
891 \subsubsection*{Points of Inflection}
892
893 \emph{Stationary point} - i.e.
894 \(f^\prime(x)=0\)\\
895 \emph{Point of inflection} - max \(|\)gradient\(|\) (i.e.
896 \(f^{\prime\prime} = 0\))
897
898
899 \begin{table*}[ht]
900 \centering
901 \begin{tabularx}{\textwidth}{rXXX}
902 \hline
903 \rowcolor{shade2}
904 & \centering\(\dfrac{d^2 y}{dx^2} > 0\) & \centering \(\dfrac{d^2y}{dx^2}<0\) & \(\dfrac{d^2y}{dx^2}=0\) (inflection) \\
905 \hline
906 \(\dfrac{dy}{dx}>0\) &
907 \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)}&
908 \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)}&
909 \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}\\
910 \hline
911 \(\dfrac{dy}{dx}<0\) &
912 \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)}&
913 \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)}&
914 \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}\\
915 \hline
916 \(\dfrac{dy}{dx}=0\)&
917 \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}&
918 \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}\\
919 \hline
920 \end{tabularx}
921 \end{table*}
922 \begin{itemize}
923 \item
924 if \(f^\prime (a) = 0\) and \(f^{\prime\prime}(a) > 0\), then point
925 \((a, f(a))\) is a local min (curve is concave up)
926 \item
927 if \(f^\prime (a) = 0\) and \(f^{\prime\prime} (a) < 0\), then point
928 \((a, f(a))\) is local max (curve is concave down)
929 \item
930 if \(f^{\prime\prime}(a) = 0\), then point \((a, f(a))\) is a point of
931 inflection
932 \item
933 if also \(f^\prime(a)=0\), then it is a stationary point of inflection
934 \end{itemize}
935
936 \subsection*{Implicit Differentiation}
937
938 \noindent Used for differentiating circles etc.
939
940 If \(p\) and \(q\) are expressions in \(x\) and \(y\) such that \(p=q\),
941 for all \(x\) and \(y\), then:
942
943 \[{\frac{dp}{dx}} = {\frac{dq}{dx}} \quad \text{and} \quad {\frac{dp}{dy}} = {\frac{dq}{dy}}\]
944
945 \noindent \colorbox{cas}{\textbf{On CAS:}}\\
946 Action \(\rightarrow\) Calculation \(\rightarrow\) \texttt{impDiff(y\^{}2+ax=5,\ x,\ y)}\\
947 Returns \(y^\prime= \dots\).
948
949 \subsection*{Integration}
950
951 \[\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)\]
952
953 \subsection*{Integral laws}
954
955 \renewcommand{\arraystretch}{1.4}
956 \begin{tabularx}{\columnwidth}{rX}
957 \hline
958 \(f(x)\) & \(\int f(x) \cdot dx\) \\
959 \hline
960 \(k\) (constant) & \(kx + c\)\\
961 \(x^n\) & \(\dfrac{1}{n+1} x^{n+1}\) \\
962 \(a x^{-n}\) &\(a \cdot \log_e |x| + c\)\\
963 \(\dfrac{1}{ax+b}\) &\(\dfrac{1}{a} \log_e (ax+b) + c\)\\
964 \((ax+b)^n\) & \(\dfrac{1}{a(n+1)}(ax+b)^{n-1} + c\>|\>n\ne 1\)\\
965 \((ax+b)^{-1}\) & \(\dfrac{1}{a}\log_e |ax+b|+c\)\\
966 \(e^{kx}\) & \(\dfrac{1}{k} e^{kx} + c\)\\
967 \(e^k\) & \(e^kx + c\)\\
968 \(\sin kx\) & \(\dfrac{-1}{k} \cos (kx) + c\)\\
969 \(\cos kx\) & \(\dfrac{1}{k} \sin (kx) + c\)\\
970 \(\sec^2 kx\) & \(\dfrac{1}{k} \tan(kx) + c\)\\
971 \(\dfrac{1}{\sqrt{a^2-x^2}}\) & \(\sin^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
972 \(\dfrac{-1}{\sqrt{a^2-x^2}}\) & \(\cos^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
973 \(\frac{a}{a^2-x^2}\) & \(\tan^{-1} \frac{x}{a} + c\)\\
974 \(\frac{f^\prime (x)}{f(x)}\) & \(\log_e f(x) + c\)\\
975 \(\int f(u) \cdot \frac{du}{dx} \cdot dx\) & \(\int f(u) \cdot du\) (substitution)\\
976 \(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\
977 \hline
978 \end{tabularx}
979
980 Note \(\sin^{-1} {x \over a} + \cos^{-1} {x \over a}\) is constant \(\forall x \in (-a, a)\)
981
982 \subsection*{Definite integrals}
983
984 \[\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)\]
985
986 \begin{itemize}
987
988 \item
989 Signed area enclosed by\\
990 \(\> y=f(x), \quad y=0, \quad x=a, \quad x=b\).
991 \item
992 \emph{Integrand} is \(f\).
993 \end{itemize}
994
995 \subsubsection*{Properties}
996
997 \[\int^b_a f(x) \> dx = \int^c_a f(x) \> dx + \int^b_c f(x) \> dx\]
998
999 \[\int^a_a f(x) \> dx = 0\]
1000
1001 \[\int^b_a k \cdot f(x) \> dx = k \int^b_a f(x) \> dx\]
1002
1003 \[\int^b_a f(x) \pm g(x) \> dx = \int^b_a f(x) \> dx \pm \int^b_a g(x) \> dx\]
1004
1005 \[\int^b_a f(x) \> dx = - \int^a_b f(x) \> dx\]
1006
1007 \subsection*{Integration by substitution}
1008
1009 \[\int f(u) {\frac{du}{dx}} \cdot dx = \int f(u) \cdot du\]
1010
1011 \noindent Note \(f(u)\) must be 1:1 \(\implies\) one \(x\) for each \(y\)
1012 \begin{align*}\text{e.g. for } y&=\int(2x+1)\sqrt{x+4} \cdot dx\\
1013 \text{let } u&=x+4\\
1014 \implies& {\frac{du}{dx}} = 1\\
1015 \implies& x = u - 4\\
1016 \text{then } &y=\int (2(u-4)+1)u^{\frac{1}{2}} \cdot du\\
1017 &\text{(solve as normal integral)}
1018 \end{align*}
1019
1020 \subsubsection*{Definite integrals by substitution}
1021
1022 For \(\int^b_a f(x) {\frac{du}{dx}} \cdot dx\), evaluate new \(a\) and
1023 \(b\) for \(f(u) \cdot du\).
1024
1025 \subsubsection*{Trigonometric integration}
1026
1027 \[\sin^m x \cos^n x \cdot dx\]
1028
1029 \paragraph{\textbf{\(m\) is odd:}}
1030 \(m=2k+1\) where \(k \in \mathbb{Z}\)\\
1031 \(\implies \sin^{2k+1} x = (\sin^2 z)^k \sin x = (1 - \cos^2 x)^k \sin x\)\\
1032 Substitute \(u=\cos x\)
1033
1034 \paragraph{\textbf{\(n\) is odd:}}
1035 \(n=2k+1\) where \(k \in \mathbb{Z}\)\\
1036 \(\implies \cos^{2k+1} x = (\cos^2 x)^k \cos x = (1-\sin^2 x)^k \cos x\)\\
1037 Substitute \(u=\sin x\)
1038
1039 \paragraph{\textbf{\(m\) and \(n\) are even:}}
1040 use identities...
1041
1042 \begin{itemize}
1043
1044 \item
1045 \(\sin^2x={1 \over 2}(1-\cos 2x)\)
1046 \item
1047 \(\cos^2x={1 \over 2}(1+\cos 2x)\)
1048 \item
1049 \(\sin 2x = 2 \sin x \cos x\)
1050 \end{itemize}
1051
1052 \subsection*{Partial fractions}
1053
1054 \colorbox{cas}{On CAS:}\\
1055 \indent Action \(\rightarrow\) Transformation \(\rightarrow\)
1056 \texttt{expand/combine}\\
1057 \indent Interactive \(\rightarrow\) Transformation \(\rightarrow\)
1058 Expand \(\rightarrow\) Partial
1059
1060 \subsection*{Graphing integrals on CAS}
1061
1062 \colorbox{cas}{In main:} Interactive \(\rightarrow\) Calculation \(\rightarrow\)
1063 \(\int\) (\(\rightarrow\) Definite)\\
1064 Restrictions: \texttt{Define\ f(x)=..} then \texttt{f(x)\textbar{}x\textgreater{}..}
1065
1066 \subsection*{Applications of antidifferentiation}
1067
1068 \begin{itemize}
1069
1070 \item
1071 \(x\)-intercepts of \(y=f(x)\) identify \(x\)-coordinates of
1072 stationary points on \(y=F(x)\)
1073 \item
1074 nature of stationary points is determined by sign of \(y=f(x)\) on
1075 either side of its \(x\)-intercepts
1076 \item
1077 if \(f(x)\) is a polynomial of degree \(n\), then \(F(x)\) has degree
1078 \(n+1\)
1079 \end{itemize}
1080
1081 To find stationary points of a function, substitute \(x\) value of given
1082 point into derivative. Solve for \({\frac{dy}{dx}}=0\). Integrate to find
1083 original function.
1084
1085 \subsection*{Solids of revolution}
1086
1087 Approximate as sum of infinitesimally-thick cylinders
1088
1089 \subsubsection*{Rotation about \(x\)-axis}
1090
1091 \begin{align*}
1092 V &= \int^{x=b}_{x-a} \pi y^2 \> dx \\
1093 &= \pi \int^b_a (f(x))^2 \> dx
1094 \end{align*}
1095
1096 \subsubsection*{Rotation about \(y\)-axis}
1097
1098 \begin{align*}
1099 V &= \int^{y=b}_{y=a} \pi x^2 \> dy \\
1100 &= \pi \int^b_a (f(y))^2 \> dy
1101 \end{align*}
1102
1103 \subsubsection*{Regions not bound by \(y=0\)}
1104
1105 \[V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx\]
1106 \hfill where \(f(x) > g(x)\)
1107
1108 \subsection*{Length of a curve}
1109
1110 \[L = \int^b_a \sqrt{1 + ({\frac{dy}{dx}})^2} \> dx \quad \text{(Cartesian)}\]
1111
1112 \[L = \int^b_a \sqrt{{\frac{dx}{dt}} + ({\frac{dy}{dt}})^2} \> dt \quad \text{(parametric)}\]
1113
1114 \noindent \colorbox{cas}{On CAS:}\\
1115 \indent Evaluate formula,\\
1116 \indent or Interactive \(\rightarrow\) Calculation
1117 \(\rightarrow\) Line \(\rightarrow\) \texttt{arcLen}
1118
1119 \subsection*{Rates}
1120
1121 \subsubsection*{Gradient at a point on parametric curve}
1122
1123 \[{\frac{dy}{dx}} = {{\frac{dy}{dt}} \div {\frac{dx}{dt}}} \> \vert \> {\frac{dx}{dt}} \ne 0 \text{ (chain rule)}\]
1124
1125 \[\frac{d^2}{dx^2} = \frac{d(y^\prime)}{dx} = {\frac{dy^\prime}{dt} \div {\frac{dx}{dt}}} \> \vert \> y^\prime = {\frac{dy}{dx}}\]
1126
1127 \subsection*{Rational functions}
1128
1129 \[f(x) = \frac{P(x)}{Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}\]
1130
1131 \subsubsection*{Addition of ordinates}
1132
1133 \begin{itemize}
1134
1135 \item
1136 when two graphs have the same ordinate, \(y\)-coordinate is double the
1137 ordinate
1138 \item
1139 when two graphs have opposite ordinates, \(y\)-coordinate is 0 i.e.
1140 (\(x\)-intercept)
1141 \item
1142 when one of the ordinates is 0, the resulting ordinate is equal to the
1143 other ordinate
1144 \end{itemize}
1145
1146 \subsection*{Fundamental theorem of calculus}
1147
1148 If \(f\) is continuous on \([a, b]\), then
1149
1150 \[\int^b_a f(x) \> dx = F(b) - F(a)\]
1151 \hfill where \(F = \int f \> dx\)
1152
1153 \subsection*{Differential equations}
1154
1155 \noindent\textbf{Order} - highest power inside derivative\\
1156 \textbf{Degree} - highest power of highest derivative\\
1157 e.g. \({\left(\dfrac{dy^2}{d^2} x\right)}^3\) \qquad order 2, degree 3
1158
1159 \subsubsection*{Verifying solutions}
1160
1161 Start with \(y=\dots\), and differentiate. Substitute into original
1162 equation.
1163
1164 \subsubsection*{Function of the dependent
1165 variable}
1166
1167 If \({\frac{dy}{dx}}=g(y)\), then
1168 \(\frac{dx}{dy} = 1 \div \frac{dy}{dx} = \frac{1}{g(y)}\). Integrate both sides to solve equation. Only add \(c\) on one side. Express
1169 \(e^c\) as \(A\).
1170
1171
1172
1173 \subsubsection*{Mixing problems}
1174
1175 \[\left(\frac{dm}{dt}\right)_\Sigma = \left(\frac{dm}{dt}\right)_{\text{in}} - \left(\frac{dm}{dt}_{\text{out}}\right)\]
1176
1177 \subsubsection*{Separation of variables}
1178
1179 If \({\frac{dy}{dx}}=f(x)g(y)\), then:
1180
1181 \[\int f(x) \> dx = \int \frac{1}{g(y)} \> dy\]
1182
1183 \subsubsection*{Euler's method for solving DEs}
1184
1185 \[\frac{f(x+h) - f(x)}{h} \approx f^\prime (x) \quad \text{for small } h\]
1186
1187 \[\implies f(x+h) \approx f(x) + hf^\prime(x)\]
1188
1189
1190 \section{Kinematics \& Mechanics}
1191
1192 \subsection*{Constant acceleration}
1193
1194 \begin{itemize}
1195 \item \textbf{Position} - relative to origin
1196 \item \textbf{Displacement} - relative to starting point
1197 \end{itemize}
1198
1199 \subsubsection*{Velocity-time graphs}
1200
1201 \begin{itemize}
1202 \item Displacement: \textit{signed} area between graph and \(t\) axis
1203 \item Distance travelled: \textit{total} area between graph and \(t\) axis
1204 \end{itemize}
1205
1206 \[ \text{acceleration} = \frac{d^2x}{dt^2} = \frac{dv}{dt} = v\frac{dv}{dx} = \frac{d}{dx}\left(\frac{1}{2}v^2\right) \]
1207
1208 \begin{center}
1209 \renewcommand{\arraystretch}{1}
1210 \begin{tabular}{ l r }
1211 \hline & no \\ \hline
1212 \(v=u+at\) & \(x\) \\
1213 \(v^2 = u^2+2as\) & \(t\) \\
1214 \(s = \frac{1}{2} (v+u)t\) & \(a\) \\
1215 \(s = ut + \frac{1}{2} at^2\) & \(v\) \\
1216 \(s = vt- \frac{1}{2} at^2\) & \(u\) \\ \hline
1217 \end{tabular}
1218 \end{center}
1219
1220 \[ v_{\text{avg}} = \frac{\Delta\text{position}}{\Delta t} \]
1221 \begin{align*}
1222 \text{speed} &= |{\text{velocity}}| \\
1223 &= \sqrt{v_x^2 + v_y^2 + v_z^2}
1224 \end{align*}
1225
1226 \noindent \textbf{Distance travelled between \(t=a \rightarrow t=b\):}
1227 \[= \int^b_a \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \cdot dt \]
1228
1229 \noindent \textbf{Shortest distance between \(\boldsymbol{r}(t_0)\) and \(\boldsymbol{r}(t_1)\):}
1230 \[ = |\boldsymbol{r}(t_1) - \boldsymbol{r}(t_2)| \]
1231
1232 \subsection*{Vector functions}
1233
1234 \[ \boldsymbol{r}(t) = x \boldsymbol{i} + y \boldsymbol{j} + z \boldsymbol{k} \]
1235
1236 \begin{itemize}
1237 \item If \(\boldsymbol{r}(t) \equiv\) position with time, then the graph of endpoints of \(\boldsymbol{r}(t) \equiv\) Cartesian path
1238 \item Domain of \(\boldsymbol{r}(t)\) is the range of \(x(t)\)
1239 \item Range of \(\boldsymbol{r}(t)\) is the range of \(y(t)\)
1240 \end{itemize}
1241
1242 \subsection*{Vector calculus}
1243
1244 \subsubsection*{Derivative}
1245
1246 Let \(\boldsymbol{r}(t)=x(t)\boldsymbol{i} + y(t)\boldsymbol(j)\). If both \(x(t)\) and \(y(t)\) are differentiable, then:
1247 \[ \boldsymbol{r}(t)=x(t)\boldsymbol{i}+y(t)\boldsymbol{j} \]
1248
1249 \subfile{dynamics}
1250 \subfile{statistics}
1251 \end{multicols}
1252\end{document}