4adcd53d464c205ead2d43104ef40974dc4c6f51
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67\pagestyle{fancy}
68\fancyhead[LO,LE]{Year 12 Specialist}
69\fancyhead[CO,CE]{Andrew Lorimer}
70
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92
93\begin{document}
94
95\begin{multicols}{2}
96
97 \section{Complex numbers}
98
99 \[\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}\]
100
101 \begin{align*}
102 \text{Cartesian form: } & a+bi\\
103 \text{Polar form: } & r\operatorname{cis}\theta
104 \end{align*}
105
106 \subsection*{Operations}
107
108 \definecolor{shade1}{HTML}{ffffff}
109 \definecolor{shade2}{HTML}{e6f2ff}
110 \definecolor{shade3}{HTML}{cce2ff}
111 \begin{tabularx}{\columnwidth}{r|X|X}
112 & \textbf{Cartesian} & \textbf{Polar} \\
113 \hline
114 \(z_1 \pm z_2\) & \((a \pm c)(b \pm d)i\) & convert to \(a+bi\)\\
115 \hline
116 \(+k \times z\) & \multirow{2}{*}{\(ka \pm kbi\)} & \(kr\operatorname{cis} \theta\)\\
117 \cline{1-1}\cline{3-3}
118 \(-k \times z\) & & \(kr \operatorname{cis}(\theta\pm \pi)\)\\
119 \hline
120 \(z_1 \cdot z_2\) & \(ac-bd+(ad+bc)i\) & \(r_1r_2 \operatorname{cis}(\theta_1 + \theta_2)\)\\
121 \hline
122 \(z_1 \div z_2\) & \((z_1 \overline{z_2}) \div |z_2|^2\) & \(\left(\frac{r_1}{r_2}\right) \operatorname{cis}(\theta_1 - \theta_2)\)
123 \end{tabularx}
124
125 \subsubsection*{Scalar multiplication in polar form}
126
127 For \(k \in \mathbb{R}^+\):
128 \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\theta\]
129
130 \noindent For \(k \in \mathbb{R}^-\):
131 \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\left(\begin{cases}\theta - \pi & |0<\operatorname{Arg}(z)\le \pi \\ \theta + \pi & |-\pi<\operatorname{Arg}(z)\le 0\end{cases}\right)\]
132
133 \subsection*{Conjugate}
134
135 \begin{align*}
136 \overline{z} &= a \mp bi\\
137 &= r \operatorname{cis}(-\theta)
138 \end{align*}
139
140 \noindent \colorbox{cas}{On CAS: \texttt{conjg(a+bi)}}
141
142 \subsubsection*{Properties}
143
144 \begin{align*}
145 \overline{z_1 \pm z_2} &= \overline{z_1}\pm\overline{z_2}\\
146 \overline{z_1 \cdot z_2} &= \overline{z_1}\cdot\overline{z_2}\\
147 \overline{kz} &= k\overline{z} \quad | \quad k \in \mathbb{R}\\
148 z\overline{z} &= (a+bi)(a-bi)\\
149 &= a^2 + b^2\\
150 &= |z|^2
151 \end{align*}
152
153 \subsection*{Modulus}
154
155 \[|z|=|\vec{Oz}|=\sqrt{a^2 + b^2}\]
156
157 \subsubsection*{Properties}
158
159 \begin{align*}
160 |z_1z_2|&=|z_1||z_2|\\
161 \left|\frac{z_1}{z_2}\right|&=\frac{|z_1|}{|z_2|}\\
162 |z_1+z_2|&\le|z_1|+|z_2|
163 \end{align*}
164
165 \subsection*{Multiplicative inverse}
166
167 \begin{align*}
168 z^{-1}&=\frac{a-bi}{a^2+b^2}\\
169 &=\frac{\overline{z}}{|z|^2}a\\
170 &=r \operatorname{cis}(-\theta)
171 \end{align*}
172
173 \subsection*{Dividing over \(\mathbb{C}\)}
174
175 \begin{align*}
176 \frac{z_1}{z_2}&=z_1z_2^{-1}\\
177 &=\frac{z_1\overline{z_2}}{|z_2|^2}\\
178 &=\frac{(a+bi)(c-di)}{c^2+d^2}\\
179 & \qquad \text{(rationalise denominator)}
180 \end{align*}
181
182 \subsection*{Polar form}
183
184 \begin{align*}
185 z&=r\operatorname{cis}\theta\\
186 &=r(\cos \theta + i \sin \theta)
187 \end{align*}
188
189 \begin{itemize}
190 \item{\(r=|z|=\sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}\)}
191 \item{\(\theta = \operatorname{arg}(z)\) \quad \colorbox{cas}{On CAS: \texttt{arg(a+bi)}}}
192 \item{\(\operatorname{Arg}(z) \in (-\pi,\pi)\) \quad \bf{(principal argument)}}
193 \item{\colorbox{cas}{Convert on CAS:}\\ \verb|compToTrig(a+bi)| \(\iff\) \verb|cExpand{r·cisX}|}
194 \item{Multiple representations:\\\(r\operatorname{cis}\theta=r\operatorname{cis}(\theta+2n\pi)\) with \(n \in \mathbb{Z}\) revolutions}
195 \item{\(\operatorname{cis}\pi=-1,\qquad \operatorname{cis}0=1\)}
196 \end{itemize}
197
198 \subsection*{de Moivres' theorem}
199
200 \[(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}\]
201
202 \subsection*{Complex polynomials}
203
204 Include \(\pm\) for all solutions, incl. imaginary
205
206 \begin{tabularx}{\columnwidth}{ R{0.55} X }
207 \hline
208 Sum of squares & \(\begin{aligned}
209 z^2 + a^2 &= z^2-(ai)^2\\
210 &= (z+ai)(z-ai) \end{aligned}\) \\
211 \hline
212 Sum of cubes & \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)\\
213 \hline
214 Division & \(P(z)=D(z)Q(z)+R(z)\) \\
215 \hline
216 Remainder theorem & Let \(\alpha \in \mathbb{C}\). Remainder of \(P(z) \div (z-\alpha)\) is \(P(\alpha)\)\\
217 \hline
218 Factor theorem & \(z-\alpha\) is a factor of \(P(z) \iff P(\alpha)=0\) for \(\alpha \in \mathbb{C}\)\\
219 \hline
220 Conjugate root theorem & \(P(z)=0 \text{ at } z=a\pm bi\) (\(\implies\) both \(z_1\) and \(\overline{z_1}\) are solutions)\\
221 \hline
222 \end{tabularx}
223
224 \subsection*{\(n\)th roots}
225
226 \(n\)th roots of \(z=r\operatorname{cis}\theta\) are:
227
228 \[z = r^{\frac{1}{n}} \operatorname{cis}\left(\frac{\theta+2k\pi}{n}\right)\]
229
230 \begin{itemize}
231
232 \item{Same modulus for all solutions}
233 \item{Arguments separated by \(\frac{2\pi}{n} \therefore\) there are \(n\) roots}
234 \item{If one square root is \(a+bi\), the other is \(-a-bi\)}
235 \item{Give one implicit \(n\)th root \(z_1\), function is \(z=z_1^n\)}
236 \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 \(\frac{2\pi}{n}\))}
237 \end{itemize}
238
239 \noindent For \(0=az^2+bz+c\), use quadratic formula:
240
241 \[z=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]
242
243 \subsection*{Fundamental theorem of algebra}
244
245 A polynomial of degree \(n\) can be factorised into \(n\) linear factors in \(\mathbb{C}\):
246
247 \[\implies P(z)=a_n(z-\alpha_1)(z-\alpha_2)(z-\alpha_3)\dots(z-\alpha_n)\]
248 \[\text{ where } \alpha_1,\alpha_2,\alpha_3,\dots,\alpha_n \in \mathbb{C}\]
249
250 \subsection*{Argand planes}
251
252 \begin{center}\begin{tikzpicture}[scale=2]
253 \draw [->] (-0.2,0) -- (1.5,0) node [right] {$\operatorname{Re}(z)$};
254 \draw [->] (0,-0.2) -- (0,1.5) node [above] {$\operatorname{Im}(z)$};
255 \coordinate (P) at (1,1);
256 \coordinate (a) at (1,0);
257 \coordinate (b) at (0,1);
258 \coordinate (O) at (0,0);
259 \draw (0,0) -- (P) node[pos=0.5, above left]{\(r\)} node[pos=1, right]{\(\begin{aligned}z&=a+bi\\&=r\operatorname{cis}\theta\end{aligned}\)};
260 \draw [gray, dashed] (1,1) -- (1,0) node[black, pos=1, below]{\(a\)};
261 \draw [gray, dashed] (1,1) -- (0,1) node[black, pos=1, left]{\(b\)};
262 \begin{scope}
263 \path[clip] (O) -- (P) -- (a);
264 \fill[red, opacity=0.5, draw=black] (O) circle (2mm);
265 \node at ($(O)+(20:3mm)$) {$\theta$};
266 \end{scope}
267 \filldraw (P) circle (0.5pt);
268 \end{tikzpicture}\end{center}
269
270 \begin{itemize}
271 \item{Multiplication by \(i \implies\) CCW rotation of \(\frac{\pi}{2}\)}
272 \item{Addition: \(z_1 + z_2 \equiv\) \overrightharp{\(Oz_1\)} + \overrightharp{\(Oz_2\)}}
273 \end{itemize}
274
275 \subsection*{Sketching complex graphs}
276
277 \subsubsection*{Linear}
278
279 \begin{itemize}
280 \item{\(\operatorname{Re}(z)=c\) or \(\operatorname{Im}(z)=c\) (perpendicular bisector)}
281 \item{\(\operatorname{Im}(z)=m\operatorname{Re}(z)\)}
282 \item{\(|z+a|=|z+b| \implies 2(a-b)x=b^2-a^2\)\\Geometric: equidistant from \(a,b\)}
283 \end{itemize}
284
285 \subsubsection*{Circles}
286
287 \begin{itemize}
288 \item \(|z-z_1|^2=c^2|z_2+2|^2\)
289 \item \(|z-(a+bi)|=c \implies (x-a)^2+_(y-b)^2=c^2\)
290 \end{itemize}
291
292 \noindent \textbf{Loci} \qquad \(\operatorname{Arg}(z)<\theta\)
293
294 \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
295 \draw [->] (0,0) -- (1,0) node [right] {$\operatorname{Re}(z)$};
296 \draw [->] (0,-0.5) -- (0,1) node [above] {$\operatorname{Im}(z)$};
297 \draw [<-, dashed, thick, blue] (-1,0) -- (0,0);
298 \draw [->, thick, blue] (0,0) -- (1,1);
299 \fill [gray, opacity=0.2, domain=-1:1, variable=\x] (-1,-0.5) -- (-1,0) -- (0, 0) -- (1,1) -- (1,-0.5) -- cycle;
300 \begin{scope}
301 \path[clip] (0,0) -- (1,1) -- (1,0);
302 \fill[red, opacity=0.5, draw=black] (0,0) circle (2mm);
303 \node at ($(0,0)+(20:3mm)$) {$\frac{\pi}{4}$};
304 \end{scope}
305 \node [font=\footnotesize] at (0.5,-0.25) {\(\operatorname{Arg}(z)\le\frac{\pi}{4}\)};
306 \node [blue, mydot] {};
307 \end{tikzpicture}\end{center}
308
309 \noindent \textbf{Rays} \qquad \(\operatorname{Arg}(z-b)=\theta\)
310
311 \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
312 \draw [->] (-0.75,0) -- (1.5,0) node [right] {$\operatorname{Re}(z)$};
313 \draw [->] (0,-1) -- (0,1) node [above] {$\operatorname{Im}(z)$};
314 \draw [->, thick, brown] (-0.25,0) -- (-0.75,-1);
315 \node [above, font=\footnotesize] at (-0.25,0) {\(\frac{1}{4}\)};
316 \begin{scope}
317 \path[clip] (-0.25,0) -- (-0.75,-1) -- (0,0);
318 \fill[orange, opacity=0.5, draw=black] (-0.25,0) circle (2mm);
319 \end{scope}
320 \node at (-0.08,-0.3) {\(\frac{\pi}{8}\)};
321 \node [font=\footnotesize, left] at (-0.75,-1) {\(\operatorname{Arg}(z+\frac{1}{4})=\frac{\pi}{8}\)};
322 \node [brown, mydot] at (-0.25,0) {};
323 \draw [<->, thick, green] (0,-1) -- (1.5,0.5) node [pos=0.25, black, font=\footnotesize, right] {\(|z-2|=|z-(1+i)|\)};
324 \node [left, font=\footnotesize] at (0,-1) {\(-1\)};
325 \node [below, font=\footnotesize] at (1,0) {\(1\)};
326 \end{tikzpicture}\end{center}
327
328 \section{Vectors}
329 \begin{center}\begin{tikzpicture}
330 \draw [->] (-0.5,0) -- (3,0) node [right] {\(x\)};
331 \draw [->] (0,-0.5) -- (0,3) node [above] {\(y\)};
332 \draw [orange, ->, thick] (0.5,0.5) -- (2.5,2.5) node [pos=0.5, above] {\(\vec{u}\)};
333 \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
334 \draw [gray, dashed, thick] (0.5,0.5) -- (2.5,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, below]{\(x\vec{i}\)};
335 \draw [gray, dashed, thick] (2.5,0.5) -- (2.5,2.5) node [pos=0.5] {\midarrow};
336 \end{scope}
337 \node[black, right] at (2.5,1.5) {\(y\vec{j}\)};
338 \end{tikzpicture}\end{center}
339 \subsection*{Column notation}
340
341 \[\begin{bmatrix}x\\ y \end{bmatrix} \iff x\boldsymbol{i} + y\boldsymbol{j}\]
342 \(\begin{bmatrix}x_2-x_1\\ y_2-y_1 \end{bmatrix}\) \quad between \(A(x_1,y_1), \> B(x_2,y_2)\)
343
344 \subsection*{Scalar multiplication}
345
346 \[k\cdot (x\boldsymbol{i}+y\boldsymbol{j})=kx\boldsymbol{i}+ky\boldsymbol{j}\]
347
348 \noindent For \(k \in \mathbb{R}^-\), direction is reversed
349
350 \subsection*{Vector addition}
351 \begin{center}\begin{tikzpicture}[scale=1]
352 \coordinate (A) at (0,0);
353 \coordinate (B) at (2,2);
354 \draw [->, thick, red] (0,0) -- (2,2) node [pos=0.5, below right] {\(\vec{u}=2\vec{i}+2\vec{j}\)};
355 \draw [->, thick, blue] (2,2) -- (1,4) node [pos=0.5, above right] {\(\vec{v}=-\vec{i}+2\vec{j}\)};
356 \draw [->, thick, orange] (0,0) -- (1,4) node [pos=0.5, left] {\(\vec{u}+\vec{v}=\vec{i}+4\vec{j}\)};
357 \end{tikzpicture}\end{center}
358
359 \[(x\boldsymbol{i}+y\boldsymbol{j}) \pm (a\boldsymbol{i}+b\boldsymbol{j})=(x \pm a)\boldsymbol{i}+(y \pm b)\boldsymbol{j}\]
360
361 \begin{itemize}
362 \item Draw each vector head to tail then join lines
363 \item Addition is commutative (parallelogram)
364 \item \(\boldsymbol{u}-\boldsymbol{v}=\boldsymbol{u}+(-\boldsymbol{v}) \implies \overrightharp{AB}=\boldsymbol{b}-\boldsymbol{a}\)
365 \end{itemize}
366
367 \subsection*{Magnitude}
368
369 \[|(x\boldsymbol{i} + y\boldsymbol{j})|=\sqrt{x^2+y^2}\]
370
371 \subsection*{Parallel vectors}
372
373 \[\boldsymbol{u} || \boldsymbol{v} \iff \boldsymbol{u} = k \boldsymbol{v} \text{ where } k \in \mathbb{R} \setminus \{0\}\]
374
375 For parallel vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\):\\
376 \[\boldsymbol{a \cdot b}=\begin{cases}
377 |\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction}\\
378 -|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions}
379 \end{cases}\]
380 %\includegraphics[width=0.2,height=\textheight]{graphics/parallelogram-vectors.jpg}
381 %\includegraphics[width=1]{graphics/vector-subtraction.jpg}
382
383 \subsection*{Perpendicular vectors}
384
385 \[\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b} = 0\ \quad \text{(since \(\cos 90 = 0\))}\]
386
387 \subsection*{Unit vector \(|\hat{\boldsymbol{a}}|=1\)}
388 \[\begin{split}\hat{\boldsymbol{a}} & = {\frac{1}{|\boldsymbol{a}|}}\boldsymbol{a} \\ & = \boldsymbol{a} \cdot {|\boldsymbol{a}|}\end{split}\]
389
390 \subsection*{Scalar product \(\boldsymbol{a} \cdot \boldsymbol{b}\)}
391
392
393 \begin{center}\begin{tikzpicture}[scale=2]
394 \draw [->] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{b}\)};
395 \draw [->] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{a}\)};
396 \begin{scope}
397 \path[clip] (1,0.5) -- (1,0) -- (0,0);
398 \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
399 \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
400 \end{scope}
401 \end{tikzpicture}\end{center}
402 \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*}
403 \noindent\colorbox{cas}{On CAS: \texttt{dotP({[}a\ b\ c{]},\ {[}d\ e\ f{]})}}
404
405 \subsubsection*{Properties}
406
407 \begin{enumerate}
408 \item
409 \(k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k\boldsymbol{b})\)
410 \item
411 \(\boldsymbol{a \cdot 0}=0\)
412 \item
413 \(\boldsymbol{a} \cdot (\boldsymbol{b} + \boldsymbol{c})=\boldsymbol{a} \cdot \boldsymbol{b} + \boldsymbol{a} \cdot \boldsymbol{c}\)
414 \item
415 \(\boldsymbol{i \cdot i} = \boldsymbol{j \cdot j} = \boldsymbol{k \cdot k}= 1\)
416 \item
417 \(\boldsymbol{a} \cdot \boldsymbol{b} = 0 \quad \implies \quad \boldsymbol{a} \perp \boldsymbol{b}\)
418 \item
419 \(\boldsymbol{a \cdot a} = |\boldsymbol{a}|^2 = a^2\)
420 \end{enumerate}
421
422 \subsection*{Angle between vectors}
423
424 \[\cos \theta = \frac{\boldsymbol{a} \cdot \boldsymbol{b}}{|\boldsymbol{a}| |\boldsymbol{b}|} = \frac{a_1 b_1 + a_2 b_2}{|\boldsymbol{a}| |\boldsymbol{b}|}\]
425
426 \noindent \colorbox{cas}{On CAS:} \texttt{angle([a b c], [a b c])}
427
428 (Action \(\rightarrow\) Vector \(\rightarrow\)Angle)
429
430 \subsection*{Angle between vector and axis}
431
432 \noindent For\(\boldsymbol{a} = a_1 \boldsymbol{i} + a_2 \boldsymbol{j} + a_3 \boldsymbol{k}\)
433 which makes angles \(\alpha, \beta, \gamma\) with positive side of
434 \(x, y, z\) axes:
435 \[\cos \alpha = \frac{a_1}{|\boldsymbol{a}|}, \quad \cos \beta = \frac{a_2}{|\boldsymbol{a}|}, \quad \cos \gamma = \frac{a_3}{|\boldsymbol{a}|}\]
436
437 \noindent \colorbox{cas}{On CAS:} \texttt{angle({[}a\ b\ c{]},\ {[}1\ 0\ 0{]})}\\for angle
438 between \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) and
439 \(x\)-axis
440
441 \subsection*{Projections \& resolutes}
442
443 \begin{tikzpicture}[scale=3]
444 \draw [->, purple] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{a}\)};
445 \draw [->, orange] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{u}\)};
446 \draw [->, blue] (1,0) -- (2,0) node [pos=0.5, below] {\(\boldsymbol{b}\)};
447 \begin{scope}
448 \path[clip] (1,0.5) -- (1,0) -- (0,0);
449 \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
450 \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
451 \end{scope}
452 \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
453 \draw [gray, dashed, thick] (1,0) -- (1,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, right, rotate=-90]{\(\boldsymbol{w}\)};
454 \end{scope}
455 \draw (0,0) coordinate (O)
456 (1,0) coordinate (A)
457 (1,0.5) coordinate (B)
458 pic [draw,red,angle radius=2mm] {right angle = O--A--B};
459 \end{tikzpicture}
460
461 \subsubsection*{\(\parallel\boldsymbol{b}\) (vector projection/resolute)}
462
463 \begin{align*}
464 \boldsymbol{u} & = \frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|^2}\boldsymbol{b} \\
465 & = \left(\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\right)\left(\frac{\boldsymbol{b}}{|\boldsymbol{b}|}\right) \\
466 & = (\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}
467 \end{align*}
468
469 \subsubsection*{\(\perp\boldsymbol{b}\) (perpendicular projection)}
470 \[\boldsymbol{w} = \boldsymbol{a} - \boldsymbol{u}\]
471
472 \subsubsection*{\(|\boldsymbol{u}|\) (scalar projection/resolute)}
473 \begin{align*}
474 s &= |\boldsymbol{u}|\\
475 &= \boldsymbol{a} \cdot \hat{\boldsymbol{b}}\\
476 &=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\\
477 &= |\boldsymbol{a}| \cos \theta
478 \end{align*}
479
480 \subsubsection*{Rectangular (\(\parallel,\perp\)) components}
481
482 \[\boldsymbol{a}=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}+\left(\boldsymbol{a}-\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}\right)\]
483
484
485 \subsection*{Vector proofs}
486
487 \textbf{Concurrent:} intersection of \(\ge\) 3 lines
488
489 \begin{tikzpicture}
490 \draw [blue] (0,0) -- (1,1);
491 \draw [red] (1,0) -- (0,1);
492 \draw [brown] (0.4,0) -- (0.6,1);
493 \filldraw (0.5,0.5) circle (2pt);
494 \end{tikzpicture}
495
496 \subsubsection*{Collinear points}
497
498 \(\ge\) 3 points lie on the same line
499
500 \begin{tikzpicture}
501 \draw [purple] (0,0) -- (4,1);
502 \filldraw (2,0.5) circle (2pt) node [above] {\(C\)};
503 \filldraw (1,0.25) circle (2pt) node [above] {\(A\)};
504 \filldraw (3,0.75) circle (2pt) node [above] {\(B\)};
505 \coordinate (O) at (2.8,-0.2);
506 \node at (O) [below] {\(O\)};
507 \begin{scope}[->, orange, thick]
508 \draw (O) -- (2,0.5) node [pos=0.5, above, font=\footnotesize, black] {\(\boldsymbol{c}\)};
509 \draw (O) -- (1,0.25) node [pos=0.5, below, font=\footnotesize, black] {\(\boldsymbol{a}\)};
510 \draw (O) -- (3,0.75) node [pos=0.5, right, font=\footnotesize, black] {\(\boldsymbol{b}\)};
511 \end{scope}
512 \end{tikzpicture}
513
514 \begin{align*}
515 \text{e.g. Prove that}\\
516 \overrightharp{AC}=m\overrightharp{AB} \iff \boldsymbol{c}&=(1-m)\boldsymbol{a}+m\boldsymbol{b}\\
517 \implies \boldsymbol{c} &= \overrightharp{OA} + \overrightharp{AC}\\
518 &= \overrightharp{OA} + m\overrightharp{AB}\\
519 &=\boldsymbol{a}+m(\boldsymbol{b}-\boldsymbol{a})\\
520 &=\boldsymbol{a}+m\boldsymbol{b}-m\boldsymbol{a}\\
521 &=(1-m)\boldsymbol{a}+m{b}
522 \end{align*}
523 \begin{align*}
524 \text{Also, } \implies \overrightharp{OC} &= \lambda \vec{OA} + \mu \overrightharp{OB} \\
525 \text{where } \lambda + \mu &= 1\\
526 \text{If } C \text{ lies along } \overrightharp{AB}, & \implies 0 < \mu < 1
527 \end{align*}
528
529
530 \subsubsection*{Parallelograms}
531
532 \begin{center}\begin{tikzpicture}
533 \coordinate (O) at (0,0) node [below left] {\(O\)};
534 \coordinate (A) at (4,0);
535 \coordinate (B) at (6,2);
536 \coordinate (C) at (2,2);
537 \coordinate (D) at (6,0);
538
539 \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (O)--(A) node [below left] {\(A\)};
540 \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (A)--(B) node [above right] {\(B\)};
541 \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (B)--(C) node [above left] {\(C\)};
542 \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (C)--(O);
543
544 \draw [gray, dashed] (O) -- (B) node [pos=0.75] {\(\diagdown\diagdown\)} node [pos=0.25] {\(\diagdown\diagdown\)};
545 \draw [gray, dashed] (A) -- (C) node [pos=0.75] {\(\diagup\)} node [pos=0.25] {\(\diagup\)};
546 \begin{scope}
547 \path[clip] (C) -- (A) -- (O);
548 \fill[orange, opacity=0.5, draw=black] (0,0) circle (4mm);
549 \node at ($(0,0)+(20:8mm)$) {\(\theta\)};
550 \end{scope}
551 \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] {\(|\boldsymbol{c}|\cos\theta\)};
552 \draw pic [draw,thick,red,angle radius=2mm] {right angle=O--D--B};
553 \end{tikzpicture}\end{center}
554
555 \begin{itemize}
556 \item
557 Diagonals \(\overrightharp{OB}, \overrightharp{AC}\) bisect each other
558 \item
559 If diagonals are equal length, it is a rectangle
560 \item
561 \(|\overrightharp{OB}|^2 + |\overrightharp{CA}|^2 = |\overrightharp{OA}|^2 + |\overrightharp{AB}|^2 + |\overrightharp{CB}|^2 + |\overrightharp{OC}|^2\)
562 \item
563 Area \(=\boldsymbol{c} \cdot \boldsymbol{a}\)
564 \end{itemize}
565
566 \subsubsection*{Useful vector properties}
567
568 \begin{itemize}
569 \item
570 \(\boldsymbol{a} \parallel \boldsymbol{b} \implies \boldsymbol{b}=k\boldsymbol{a}\) for some
571 \(k \in \mathbb{R} \setminus \{0\}\)
572 \item
573 If \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are parallel with at
574 least one point in common, then they lie on the same straight line
575 \item
576 \(\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b}=0\)
577 \item
578 \(\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2\)
579 \end{itemize}
580
581 \subsection*{Linear dependence}
582
583 \(\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\) are linearly dependent if they are \(\nparallel\) and:
584 \begin{align*}
585 0&=k\boldsymbol{a}+l\boldsymbol{b}+m\boldsymbol{c}\\
586 \therefore \boldsymbol{c} &= m\boldsymbol{a} + n\boldsymbol{b} \quad \text{(simultaneous)}
587 \end{align*}
588
589 \noindent \(\boldsymbol{a}, \boldsymbol{b},\) and \(\boldsymbol{c}\) are linearly
590 independent if no vector in the set is expressible as a linear
591 combination of other vectors in set, or if they are parallel.
592
593 \subsection*{Three-dimensional vectors}
594
595 Right-hand rule for axes: \(z\) is up or out of page.
596
597 \tdplotsetmaincoords{60}{120}
598 \begin{center}\begin{tikzpicture} [scale=3, tdplot_main_coords, axis/.style={->,thick},
599 vector/.style={-stealth,red,very thick},
600 vector guide/.style={dashed,gray,thick}]
601
602 %standard tikz coordinate definition using x, y, z coords
603 \coordinate (O) at (0,0,0);
604
605 %tikz-3dplot coordinate definition using x, y, z coords
606
607 \pgfmathsetmacro{\ax}{1}
608 \pgfmathsetmacro{\ay}{1}
609 \pgfmathsetmacro{\az}{1}
610
611 \coordinate (P) at (\ax,\ay,\az);
612
613 %draw axes
614 \draw[axis] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$};
615 \draw[axis] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$};
616 \draw[axis] (0,0,0) -- (0,0,1) node[anchor=south]{$z$};
617
618 %draw a vector from O to P
619 \draw[vector] (O) -- (P);
620
621 %draw guide lines to components
622 \draw[vector guide] (O) -- (\ax,\ay,0);
623 \draw[vector guide] (\ax,\ay,0) -- (P);
624 \draw[vector guide] (P) -- (0,0,\az);
625 \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
626 \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
627 \draw[vector guide] (\ax,\ay,0) -- (\ax,0,0);
628 \node[tdplot_main_coords,above right]
629 at (\ax,\ay,\az){(\ax, \ay, \az)};
630 \end{tikzpicture}\end{center}
631
632 \subsection*{Parametric vectors}
633
634 Parametric equation of line through point \((x_0, y_0, z_0)\) and
635 parallel to \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) is:
636
637 \[\begin{cases}x = x_o + a \cdot t \\ y = y_0 + b \cdot t \\ z = z_0 + c \cdot t\end{cases}\]
638
639 \section{Circular functions}
640
641 \(\sin(bx)\) or \(\cos(bx)\): period \(=\frac{2\pi}{b}\)
642
643 \noindent \(\tan(nx)\): period \(=\frac{\pi}{n}\)\\
644 \indent\indent\indent asymptotes at \(x=\frac{(2k+1)\pi}{2n} \> \vert \> k \in \mathbb{Z}\)
645
646 \subsection*{Reciprocal functions}
647
648 \subsubsection*{Cosecant}
649
650 \[\operatorname{cosec} \theta = \frac{1}{\sin \theta} \> \vert \> \sin \theta \ne 0\]
651
652 \begin{itemize}
653 \item
654 \textbf{Domain} \(= \mathbb{R} \setminus {n\pi : n \in \mathbb{Z}}\)
655 \item
656 \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
657 \item
658 \textbf{Turning points} at
659 \(\theta = \frac{(2n + 1)\pi}{2} \> \vert \> n \in \mathbb{Z}\)
660 \item
661 \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
662 \end{itemize}
663
664 \subsubsection*{Secant}
665
666\begin{tikzpicture}
667 \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={\(-2\pi\), \(-\pi\), \(\pi\), \(2\pi\)}]
668% \addplot[blue, domain=-6.2830:6.2830,unbounded coords=jump,samples=80] {sec(deg(x))};
669 \addplot[blue, restrict y to domain=-10:10, domain=-7:7,samples=100] {sec(deg(x))} node [pos=0.93, black, right] {\(\operatorname{sec} x\)};
670 \addplot[red, dashed, domain=-7:7,samples=100] {cos(deg(x))};
671 \draw [gray, dotted, thick] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{rel axis cs:0,1});
672 \draw [gray, dotted, thick] ({axis cs:4.71239,0}|-{rel axis cs:0,0}) -- ({axis cs:4.71239,0}|-{rel axis cs:0,1});
673 \draw [gray, dotted, thick] ({axis cs:-4.71239,0}|-{rel axis cs:0,0}) -- ({axis cs:-4.71239,0}|-{rel axis cs:0,1});
674 \draw [gray, dotted, thick] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1});
675\end{axis}
676 \node [black] at (7,3.5) {\(\cos x\)};
677\end{tikzpicture}
678
679 \[\operatorname{sec} \theta = \frac{1}{\cos \theta} \> \vert \> \cos \theta \ne 0\]
680
681 \begin{itemize}
682
683 \item
684 \textbf{Domain}
685 \(= \mathbb{R} \setminus \frac{(2n + 1) \pi}{2} : n \in \mathbb{Z}\}\)
686 \item
687 \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
688 \item
689 \textbf{Turning points} at
690 \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
691 \item
692 \textbf{Asymptotes} at
693 \(\theta = \frac{(2n + 1) \pi}{2} \> \vert \> n \in \mathbb{Z}\)
694 \end{itemize}
695
696 \subsubsection*{Cotangent}
697
698\begin{tikzpicture}
699 \begin{axis}[xmin=-3,xmax=3,ymin=-1.5,ymax=1.5,enlargelimits=true, xtick={-3.1415, -1.5708, 1.5708, 3.1415},xticklabels={\(-\pi\), \(-\frac{\pi}{2}\), \(\frac{\pi}{2}\), \(\pi\)}]
700 \addplot[blue, smooth, domain=-3:-0.1,unbounded coords=jump,samples=105] {cot(deg(x))} node [pos=0.3, left] {\(\operatorname{cot} x\)};
701\addplot[blue, smooth, domain=0.1:3,unbounded coords=jump,samples=105] {cot(deg(x))};
702\addplot[red, smooth, dashed] gnuplot [domain=-1.5:1.5,unbounded coords=jump,samples=105] {tan(x)};
703\addplot[red, smooth, dashed] gnuplot [domain=-3.5:-1.8,unbounded coords=jump,samples=105] {tan(x)} node [pos=0.5, right] {\(\tan x\)};
704\addplot[red, smooth, dashed] gnuplot [domain=1.8:3.5,unbounded coords=jump,samples=105] {tan(x)};
705 \draw [thick, red, dotted] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1});
706 \draw [thick, blue, dotted] ({axis cs:-3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:-3.1415,0}|-{rel axis cs:0,1});
707 \draw [thick, blue, dotted] ({axis cs:0,0}|-{rel axis cs:0,0}) -- ({axis cs:0,0}|-{rel axis cs:0,1});
708 \draw [thick, blue, dotted] ({axis cs:3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:3.1415,0}|-{rel axis cs:0,1});
709 \draw [thick, red, dotted] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{rel axis cs:0,1});
710\end{axis}
711\end{tikzpicture}
712
713 \[\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0\]
714
715 \begin{itemize}
716
717 \item
718 \textbf{Domain} \(= \mathbb{R} \setminus \{n \pi: n \in \mathbb{Z}\}\)
719 \item
720 \textbf{Range} \(= \mathbb{R}\)
721 \item
722 \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
723 \end{itemize}
724
725 \subsubsection*{Symmetry properties}
726
727 \[\begin{split}
728 \operatorname{sec} (\pi \pm x) & = -\operatorname{sec} x \\
729 \operatorname{sec} (-x) & = \operatorname{sec} x \\
730 \operatorname{cosec} (\pi \pm x) & = \mp \operatorname{cosec} x \\
731 \operatorname{cosec} (-x) & = - \operatorname{cosec} x \\
732 \operatorname{cot} (\pi \pm x) & = \pm \operatorname{cot} x \\
733 \operatorname{cot} (-x) & = - \operatorname{cot} x
734 \end{split}\]
735
736 \subsubsection*{Complementary properties}
737
738 \[\begin{split}
739 \operatorname{sec} \left({\pi \over 2} - x\right) & = \operatorname{cosec} x \\
740 \operatorname{cosec} \left({\pi \over 2} - x\right) & = \operatorname{sec} x \\
741 \operatorname{cot} \left({\pi \over 2} - x\right) & = \tan x \\
742 \tan \left({\pi \over 2} - x\right) & = \operatorname{cot} x
743 \end{split}\]
744
745 \subsubsection*{Pythagorean identities}
746
747 \[\begin{split}
748 1 + \operatorname{cot}^2 x & = \operatorname{cosec}^2 x, \quad \text{where } \sin x \ne 0 \\
749 1 + \tan^2 x & = \operatorname{sec}^2 x, \quad \text{where } \cos x \ne 0
750 \end{split}\]
751
752 \subsection*{Compound angle formulas}
753
754 \[\cos(x \pm y) = \cos x + \cos y \mp \sin x \sin y\]
755 \[\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y\]
756 \[\tan(x \pm y) = {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}\]
757
758 \subsection*{Double angle formulas}
759
760 \[\begin{split}
761 \cos 2x &= \cos^2 x - \sin^2 x \\
762 & = 1 - 2\sin^2 x \\
763 & = 2 \cos^2 x -1
764 \end{split}\]
765
766 \[\sin 2x = 2 \sin x \cos x\]
767
768 \[\tan 2x = {{2 \tan x} \over {1 - \tan^2 x}}\]
769
770 \subsection*{Inverse circular functions}
771
772 \begin{tikzpicture}
773 \begin{axis}[ymin=-2, ymax=4, xmin=-1.1, xmax=1.1, ytick={-1.5708, 1.5708, 3.14159},yticklabels={$-\frac{\pi}{2}$, $\frac{\pi}{2}$, $\pi$}]
774 \addplot[color=red, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {asin(x)} node [pos=0.25, below right] {\(\sin^{-1}x\)};
775 \addplot[color=blue, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {acos(x)} node [pos=0.25, below left] {\(\cos^{-1}x\)};
776 \addplot[mark=*, red] coordinates {(-1,-1.5708)} node[right, font=\footnotesize]{\((-1,-\frac{\pi}{2})\)} ;
777 \addplot[mark=*, red] coordinates {(1,1.5708)} node[left, font=\footnotesize]{\((1,\frac{\pi}{2})\)} ;
778 \addplot[mark=*, blue] coordinates {(1,0)};
779 \addplot[mark=*, blue] coordinates {(-1,3.1415)} node[right, font=\footnotesize]{\((-1,\pi)\)} ;
780 \end{axis}
781 \end{tikzpicture}\\
782
783 Inverse functions: \(f(f^{-1}(x)) = x\) (restrict domain)
784
785 \[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y\]
786 \hfill where \(\sin y = x, \> y \in [{-\pi \over 2}, {\pi \over 2}]\)
787
788 \[\cos^{-1}: [-1,1] \rightarrow \mathbb{R}, \quad \cos^{-1} x = y\]
789 \hfill where \(\cos y = x, \> y \in [0, \pi]\)
790
791 \[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\]
792 \hfill where \(\tan y = x, \> y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\)
793
794 \begin{tikzpicture}
795 \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={\(-\frac{\pi}{2}\),\(\frac{\pi}{2}\)}]
796 \addplot[color=orange, smooth] gnuplot [domain=-35:35, unbounded coords=jump,samples=350] {atan(x)} node [pos=0.5, above left] {\(\tan^{-1}x\)};
797 \addplot[gray, dotted, thick, domain=-35:35] {1.5708} node [black, font=\footnotesize, below right, pos=0] {\(y=\frac{\pi}{2}\)};
798 \addplot[gray, dotted, thick, domain=-35:35] {-1.5708} node [black, font=\footnotesize, above left, pos=1] {\(y=-\frac{\pi}{2}\)};
799 \end{axis}
800 \end{tikzpicture}
801\columnbreak
802 \section{Differential calculus}
803
804 \subsection*{Limits}
805
806 \[\lim_{x \rightarrow a}f(x)\]
807 \(L^-,\quad L^+\) \qquad limit from below/above\\
808 \(\lim_{x \to a} f(x)\) \quad limit of a point\\
809
810 \noindent For solving \(x\rightarrow\infty\), put all \(x\) terms in denominators\\
811 e.g. \[\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2\]
812
813 \subsubsection*{Limit theorems}
814
815 \begin{enumerate}
816 \item
817 For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\)
818 \item
819 \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\)
820 \item
821 \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\)
822 \item
823 \(\therefore \lim_{x \rightarrow a} c \times f(x)=cF\) where \(c=\) constant
824 \item
825 \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\)
826 \item
827 \(f(x)\) is continuous \(\iff L^-=L^+=f(x) \> \forall x\)
828 \end{enumerate}
829
830 \subsection*{Gradients of secants and tangents}
831
832 \textbf{Secant (chord)} - line joining two points on curve\\
833 \textbf{Tangent} - line that intersects curve at one point
834
835 \subsection*{First principles derivative}
836
837 \[f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={\frac{dy}{dx}}\]
838
839 \subsubsection*{Logarithmic identities}
840
841 \(\log_b (xy)=\log_b x + \log_b y\)\\
842 \(\log_b x^n = n \log_b x\)\\
843 \(\log_b y^{x^n} = x^n \log_b y\)
844
845 \subsubsection*{Index identities}
846
847 \(b^{m+n}=b^m \cdot b^n\)\\
848 \((b^m)^n=b^{m \cdot n}\)\\
849 \((b \cdot c)^n = b^n \cdot c^n\)\\
850 \({a^m \div a^n} = {a^{m-n}}\)
851
852 \subsection*{Derivative rules}
853
854 \renewcommand{\arraystretch}{1.4}
855 \begin{tabularx}{\columnwidth}{rX}
856 \hline
857 \(f(x)\) & \(f^\prime(x)\)\\
858 \hline
859 \(\sin x\) & \(\cos x\)\\
860 \(\sin ax\) & \(a\cos ax\)\\
861 \(\cos x\) & \(-\sin x\)\\
862 \(\cos ax\) & \(-a \sin ax\)\\
863 \(\tan f(x)\) & \(f^2(x) \sec^2f(x)\)\\
864 \(e^x\) & \(e^x\)\\
865 \(e^{ax}\) & \(ae^{ax}\)\\
866 \(ax^{nx}\) & \(an \cdot e^{nx}\)\\
867 \(\log_e x\) & \(\dfrac{1}{x}\)\\
868 \(\log_e {ax}\) & \(\dfrac{1}{x}\)\\
869 \(\log_e f(x)\) & \(\dfrac{f^\prime (x)}{f(x)}\)\\
870 \(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\\
871 \(\sin^{-1} x\) & \(\dfrac{1}{\sqrt{1-x^2}}\)\\
872 \(\cos^{-1} x\) & \(\dfrac{-1}{sqrt{1-x^2}}\)\\
873 \(\tan^{-1} x\) & \(\dfrac{1}{1 + x^2}\)\\
874 \(\frac{d}{dy}f(y)\) & \(\dfrac{1}{\frac{dx}{dy}}\) (reciprocal)\\
875 \(uv\) & \(u \frac{dv}{dx}+v\frac{du}{dx} (product rule)\)\\
876 \(\dfrac{u}{v}\) & \(\dfrac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\) (quotient rule)\\
877 \(f(g(x))\) & \(f^\prime(g(x))\cdot g^\prime(x)\)\\
878 \hline
879 \end{tabularx}
880
881 \subsection*{Reciprocal derivatives}
882
883 \[\frac{1}{\frac{dy}{dx}} = \frac{dx}{dy}\]
884
885 \subsection*{Differentiating \(x=f(y)\)}
886 \begin{align*}
887 \text{Find }& \frac{dx}{dy}\\
888 \text{Then, }\frac{dx}{dy} &= \frac{1}{\frac{dy}{dx}} \\
889 \implies {\frac{dy}{dx}} &= \frac{1}{\frac{dx}{dy}}\\
890 \therefore {\frac{dy}{dx}} &= \frac{1}{\frac{dx}{dy}}
891 \end{align*}
892
893 \subsection*{Second derivative}
894 \begin{align*}f(x) \longrightarrow &f^\prime (x) \longrightarrow f^{\prime\prime}(x)\\
895 \implies y \longrightarrow &\frac{dy}{dx} \longrightarrow \frac{d^2 y}{dx^2}\end{align*}
896
897 \noindent Order of polynomial \(n\)th derivative decrements each time the derivative is taken
898
899 \subsubsection*{Points of Inflection}
900
901 \emph{Stationary point} - i.e.
902 \(f^\prime(x)=0\)\\
903 \emph{Point of inflection} - max \(|\)gradient\(|\) (i.e.
904 \(f^{\prime\prime} = 0\))
905
906
907 \begin{table*}[ht]
908 \centering
909 \begin{tabularx}{\textwidth}{rXXX}
910 \hline
911 \rowcolor{shade2}
912 & \centering\(\dfrac{d^2 y}{dx^2} > 0\) & \centering \(\dfrac{d^2y}{dx^2}<0\) & \(\dfrac{d^2y}{dx^2}=0\) (inflection) \\
913 \hline
914 \(\dfrac{dy}{dx}>0\) &
915 \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)}&
916 \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)}&
917 \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}\\
918 \hline
919 \(\dfrac{dy}{dx}<0\) &
920 \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)}&
921 \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)}&
922 \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}\\
923 \hline
924 \(\dfrac{dy}{dx}=0\)&
925 \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}&
926 \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}\\
927 \hline
928 \end{tabularx}
929 \end{table*}
930 \begin{itemize}
931 \item
932 if \(f^\prime (a) = 0\) and \(f^{\prime\prime}(a) > 0\), then point
933 \((a, f(a))\) is a local min (curve is concave up)
934 \item
935 if \(f^\prime (a) = 0\) and \(f^{\prime\prime} (a) < 0\), then point
936 \((a, f(a))\) is local max (curve is concave down)
937 \item
938 if \(f^{\prime\prime}(a) = 0\), then point \((a, f(a))\) is a point of
939 inflection
940 \item
941 if also \(f^\prime(a)=0\), then it is a stationary point of inflection
942 \end{itemize}
943
944 \subsection*{Implicit Differentiation}
945
946 \noindent Used for differentiating circles etc.
947
948 If \(p\) and \(q\) are expressions in \(x\) and \(y\) such that \(p=q\),
949 for all \(x\) and \(y\), then:
950
951 \[{\frac{dp}{dx}} = {\frac{dq}{dx}} \quad \text{and} \quad {\frac{dp}{dy}} = {\frac{dq}{dy}}\]
952
953 \noindent \colorbox{cas}{\textbf{On CAS:}}\\
954 Action \(\rightarrow\) Calculation \(\rightarrow\) \texttt{impDiff(y\^{}2+ax=5,\ x,\ y)}\\
955 Returns \(y^\prime= \dots\).
956
957 \subsection*{Integration}
958
959 \[\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)\]
960
961 \subsection*{Integral laws}
962
963 \renewcommand{\arraystretch}{1.4}
964 \begin{tabularx}{\columnwidth}{rX}
965 \hline
966 \(f(x)\) & \(\int f(x) \cdot dx\) \\
967 \hline
968 \(k\) (constant) & \(kx + c\)\\
969 \(x^n\) & \(\dfrac{1}{n+1} x^{n+1}\) \\
970 \(a x^{-n}\) &\(a \cdot \log_e |x| + c\)\\
971 \(\dfrac{1}{ax+b}\) &\(\dfrac{1}{a} \log_e (ax+b) + c\)\\
972 \((ax+b)^n\) & \(\dfrac{1}{a(n+1)}(ax+b)^{n-1} + c\>|\>n\ne 1\)\\
973 \((ax+b)^{-1}\) & \(\dfrac{1}{a}\log_e |ax+b|+c\)\\
974 \(e^{kx}\) & \(\dfrac{1}{k} e^{kx} + c\)\\
975 \(e^k\) & \(e^kx + c\)\\
976 \(\sin kx\) & \(\dfrac{-1}{k} \cos (kx) + c\)\\
977 \(\cos kx\) & \(\dfrac{1}{k} \sin (kx) + c\)\\
978 \(\sec^2 kx\) & \(\dfrac{1}{k} \tan(kx) + c\)\\
979 \(\dfrac{1}{\sqrt{a^2-x^2}}\) & \(\sin^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
980 \(\dfrac{-1}{\sqrt{a^2-x^2}}\) & \(\cos^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
981 \(\frac{a}{a^2-x^2}\) & \(\tan^{-1} \frac{x}{a} + c\)\\
982 \(\frac{f^\prime (x)}{f(x)}\) & \(\log_e f(x) + c\)\\
983 \(\int f(u) \cdot \frac{du}{dx} \cdot dx\) & \(\int f(u) \cdot du\) (substitution)\\
984 \(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\
985 \hline
986 \end{tabularx}
987
988 Note \(\sin^{-1} {x \over a} + \cos^{-1} {x \over a}\) is constant \(\forall x \in (-a, a)\)
989
990 \subsection*{Definite integrals}
991
992 \[\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)\]
993
994 \begin{itemize}
995
996 \item
997 Signed area enclosed by\\
998 \(\> y=f(x), \quad y=0, \quad x=a, \quad x=b\).
999 \item
1000 \emph{Integrand} is \(f\).
1001 \end{itemize}
1002
1003 \subsubsection*{Properties}
1004
1005 \[\int^b_a f(x) \> dx = \int^c_a f(x) \> dx + \int^b_c f(x) \> dx\]
1006
1007 \[\int^a_a f(x) \> dx = 0\]
1008
1009 \[\int^b_a k \cdot f(x) \> dx = k \int^b_a f(x) \> dx\]
1010
1011 \[\int^b_a f(x) \pm g(x) \> dx = \int^b_a f(x) \> dx \pm \int^b_a g(x) \> dx\]
1012
1013 \[\int^b_a f(x) \> dx = - \int^a_b f(x) \> dx\]
1014
1015 \subsection*{Integration by substitution}
1016
1017 \[\int f(u) {\frac{du}{dx}} \cdot dx = \int f(u) \cdot du\]
1018
1019 \noindent Note \(f(u)\) must be 1:1 \(\implies\) one \(x\) for each \(y\)
1020 \begin{align*}\text{e.g. for } y&=\int(2x+1)\sqrt{x+4} \cdot dx\\
1021 \text{let } u&=x+4\\
1022 \implies& {\frac{du}{dx}} = 1\\
1023 \implies& x = u - 4\\
1024 \text{then } &y=\int (2(u-4)+1)u^{\frac{1}{2}} \cdot du\\
1025 &\text{(solve as normal integral)}
1026 \end{align*}
1027
1028 \subsubsection*{Definite integrals by substitution}
1029
1030 For \(\int^b_a f(x) {\frac{du}{dx}} \cdot dx\), evaluate new \(a\) and
1031 \(b\) for \(f(u) \cdot du\).
1032
1033 \subsubsection*{Trigonometric integration}
1034
1035 \[\sin^m x \cos^n x \cdot dx\]
1036
1037 \paragraph{\textbf{\(m\) is odd:}}
1038 \(m=2k+1\) where \(k \in \mathbb{Z}\)\\
1039 \(\implies \sin^{2k+1} x = (\sin^2 z)^k \sin x = (1 - \cos^2 x)^k \sin x\)\\
1040 Substitute \(u=\cos x\)
1041
1042 \paragraph{\textbf{\(n\) is odd:}}
1043 \(n=2k+1\) where \(k \in \mathbb{Z}\)\\
1044 \(\implies \cos^{2k+1} x = (\cos^2 x)^k \cos x = (1-\sin^2 x)^k \cos x\)\\
1045 Substitute \(u=\sin x\)
1046
1047 \paragraph{\textbf{\(m\) and \(n\) are even:}}
1048 use identities...
1049
1050 \begin{itemize}
1051
1052 \item
1053 \(\sin^2x={1 \over 2}(1-\cos 2x)\)
1054 \item
1055 \(\cos^2x={1 \over 2}(1+\cos 2x)\)
1056 \item
1057 \(\sin 2x = 2 \sin x \cos x\)
1058 \end{itemize}
1059
1060 \subsection*{Partial fractions}
1061
1062 \colorbox{cas}{On CAS:}\\
1063 \indent Action \(\rightarrow\) Transformation \(\rightarrow\)
1064 \texttt{expand/combine}\\
1065 \indent Interactive \(\rightarrow\) Transformation \(\rightarrow\)
1066 Expand \(\rightarrow\) Partial
1067
1068 \subsection*{Graphing integrals on CAS}
1069
1070 \colorbox{cas}{In main:} Interactive \(\rightarrow\) Calculation \(\rightarrow\)
1071 \(\int\) (\(\rightarrow\) Definite)\\
1072 Restrictions: \texttt{Define\ f(x)=..} then \texttt{f(x)\textbar{}x\textgreater{}..}
1073
1074 \subsection*{Applications of antidifferentiation}
1075
1076 \begin{itemize}
1077
1078 \item
1079 \(x\)-intercepts of \(y=f(x)\) identify \(x\)-coordinates of
1080 stationary points on \(y=F(x)\)
1081 \item
1082 nature of stationary points is determined by sign of \(y=f(x)\) on
1083 either side of its \(x\)-intercepts
1084 \item
1085 if \(f(x)\) is a polynomial of degree \(n\), then \(F(x)\) has degree
1086 \(n+1\)
1087 \end{itemize}
1088
1089 To find stationary points of a function, substitute \(x\) value of given
1090 point into derivative. Solve for \({\frac{dy}{dx}}=0\). Integrate to find
1091 original function.
1092
1093 \subsection*{Solids of revolution}
1094
1095 Approximate as sum of infinitesimally-thick cylinders
1096
1097 \subsubsection*{Rotation about \(x\)-axis}
1098
1099 \begin{align*}
1100 V &= \int^{x=b}_{x-a} \pi y^2 \> dx \\
1101 &= \pi \int^b_a (f(x))^2 \> dx
1102 \end{align*}
1103
1104 \subsubsection*{Rotation about \(y\)-axis}
1105
1106 \begin{align*}
1107 V &= \int^{y=b}_{y=a} \pi x^2 \> dy \\
1108 &= \pi \int^b_a (f(y))^2 \> dy
1109 \end{align*}
1110
1111 \subsubsection*{Regions not bound by \(y=0\)}
1112
1113 \[V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx\]
1114 \hfill where \(f(x) > g(x)\)
1115
1116 \subsection*{Length of a curve}
1117
1118 \[L = \int^b_a \sqrt{1 + ({\frac{dy}{dx}})^2} \> dx \quad \text{(Cartesian)}\]
1119
1120 \[L = \int^b_a \sqrt{{\frac{dx}{dt}} + ({\frac{dy}{dt}})^2} \> dt \quad \text{(parametric)}\]
1121
1122 \noindent \colorbox{cas}{On CAS:}\\
1123 \indent Evaluate formula,\\
1124 \indent or Interactive \(\rightarrow\) Calculation
1125 \(\rightarrow\) Line \(\rightarrow\) \texttt{arcLen}
1126
1127 \subsection*{Rates}
1128
1129 \subsubsection*{Gradient at a point on parametric curve}
1130
1131 \[{\frac{dy}{dx}} = {{\frac{dy}{dt}} \div {\frac{dx}{dt}}} \> \vert \> {\frac{dx}{dt}} \ne 0 \text{ (chain rule)}\]
1132
1133 \[\frac{d^2}{dx^2} = \frac{d(y^\prime)}{dx} = {\frac{dy^\prime}{dt} \div {\frac{dx}{dt}}} \> \vert \> y^\prime = {\frac{dy}{dx}}\]
1134
1135 \subsection*{Rational functions}
1136
1137 \[f(x) = \frac{P(x)}{Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}\]
1138
1139 \subsubsection*{Addition of ordinates}
1140
1141 \begin{itemize}
1142
1143 \item
1144 when two graphs have the same ordinate, \(y\)-coordinate is double the
1145 ordinate
1146 \item
1147 when two graphs have opposite ordinates, \(y\)-coordinate is 0 i.e.
1148 (\(x\)-intercept)
1149 \item
1150 when one of the ordinates is 0, the resulting ordinate is equal to the
1151 other ordinate
1152 \end{itemize}
1153
1154 \subsection*{Fundamental theorem of calculus}
1155
1156 If \(f\) is continuous on \([a, b]\), then
1157
1158 \[\int^b_a f(x) \> dx = F(b) - F(a)\]
1159 \hfill where \(F = \int f \> dx\)
1160
1161 \subsection*{Differential equations}
1162
1163 \noindent\textbf{Order} - highest power inside derivative\\
1164 \textbf{Degree} - highest power of highest derivative\\
1165 e.g. \({\left(\dfrac{dy^2}{d^2} x\right)}^3\) \qquad order 2, degree 3
1166
1167 \subsubsection*{Verifying solutions}
1168
1169 Start with \(y=\dots\), and differentiate. Substitute into original
1170 equation.
1171
1172 \subsubsection*{Function of the dependent
1173 variable}
1174
1175 If \({\frac{dy}{dx}}=g(y)\), then
1176 \(\frac{dx}{dy} = 1 \div \frac{dy}{dx} = \frac{1}{g(y)}\). Integrate both sides to solve equation. Only add \(c\) on one side. Express
1177 \(e^c\) as \(A\).
1178
1179
1180
1181 \subsubsection*{Mixing problems}
1182
1183 \[\left(\frac{dm}{dt}\right)_\Sigma = \left(\frac{dm}{dt}\right)_{\text{in}} - \left(\frac{dm}{dt}_{\text{out}}\right)\]
1184
1185 \subsubsection*{Separation of variables}
1186
1187 If \({\frac{dy}{dx}}=f(x)g(y)\), then:
1188
1189 \[\int f(x) \> dx = \int \frac{1}{g(y)} \> dy\]
1190
1191 \subsubsection*{Euler's method for solving DEs}
1192
1193 \[\frac{f(x+h) - f(x)}{h} \approx f^\prime (x) \quad \text{for small } h\]
1194
1195 \[\implies f(x+h) \approx f(x) + hf^\prime(x)\]
1196
1197
1198 \section{Kinematics \& Mechanics}
1199
1200 \subsection*{Constant acceleration}
1201
1202 \begin{itemize}
1203 \item \textbf{Position} - relative to origin
1204 \item \textbf{Displacement} - relative to starting point
1205 \end{itemize}
1206
1207 \subsubsection*{Velocity-time graphs}
1208
1209 \begin{itemize}
1210 \item Displacement: \textit{signed} area between graph and \(t\) axis
1211 \item Distance travelled: \textit{total} area between graph and \(t\) axis
1212 \end{itemize}
1213
1214 \[ \text{acceleration} = \frac{d^2x}{dt^2} = \frac{dv}{dt} = v\frac{dv}{dx} = \frac{d}{dx}\left(\frac{1}{2}v^2\right) \]
1215
1216 \begin{center}
1217 \renewcommand{\arraystretch}{1}
1218 \begin{tabular}{ l r }
1219 \hline & no \\ \hline
1220 \(v=u+at\) & \(x\) \\
1221 \(v^2 = u^2+2as\) & \(t\) \\
1222 \(s = \frac{1}{2} (v+u)t\) & \(a\) \\
1223 \(s = ut + \frac{1}{2} at^2\) & \(v\) \\
1224 \(s = vt- \frac{1}{2} at^2\) & \(u\) \\ \hline
1225 \end{tabular}
1226 \end{center}
1227
1228 \[ v_{\text{avg}} = \frac{\Delta\text{position}}{\Delta t} \]
1229 \begin{align*}
1230 \text{speed} &= |{\text{velocity}}| \\
1231 &= \sqrt{v_x^2 + v_y^2 + v_z^2}
1232 \end{align*}
1233
1234 \noindent \textbf{Distance travelled between \(t=a \rightarrow t=b\):}
1235 \[= \int^b_a \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \cdot dt \]
1236
1237 \noindent \textbf{Shortest distance between \(\boldsymbol{r}(t_0)\) and \(\boldsymbol{r}(t_1)\):}
1238 \[ = |\boldsymbol{r}(t_1) - \boldsymbol{r}(t_2)| \]
1239
1240 \subsection*{Vector functions}
1241
1242 \[ \boldsymbol{r}(t) = x \boldsymbol{i} + y \boldsymbol{j} + z \boldsymbol{k} \]
1243
1244 \begin{itemize}
1245 \item If \(\boldsymbol{r}(t) \equiv\) position with time, then the graph of endpoints of \(\boldsymbol{r}(t) \equiv\) Cartesian path
1246 \item Domain of \(\boldsymbol{r}(t)\) is the range of \(x(t)\)
1247 \item Range of \(\boldsymbol{r}(t)\) is the range of \(y(t)\)
1248 \end{itemize}
1249
1250 \subsection*{Vector calculus}
1251
1252 \subsubsection*{Derivative}
1253
1254 Let \(\boldsymbol{r}(t)=x(t)\boldsymbol{i} + y(t)\boldsymbol(j)\). If both \(x(t)\) and \(y(t)\) are differentiable, then:
1255 \[ \boldsymbol{r}(t)=x(t)\boldsymbol{i}+y(t)\boldsymbol{j} \]
1256
1257 \subfile{dynamics}
1258 \subfile{statistics}
1259 \end{multicols}
1260\end{document}