94d61876d683d0dd35373e447df8d2f1ff0f4207
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71\pagestyle{fancy}
72\fancyhead[LO,LE]{Year 12 Specialist}
73\fancyhead[CO,CE]{Andrew Lorimer}
74
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99
100\begin{document}
101
102\begin{multicols}{2}
103
104 \section{Complex numbers}
105
106 \[\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}\]
107
108 \begin{align*}
109 \text{Cartesian form: } & a+bi\\
110 \text{Polar form: } & r\operatorname{cis}\theta
111 \end{align*}
112
113 \subsection*{Operations}
114
115 \definecolor{shade1}{HTML}{ffffff}
116 \definecolor{shade2}{HTML}{e6f2ff}
117 \definecolor{shade3}{HTML}{cce2ff}
118 \begin{tabularx}{\columnwidth}{r|X|X}
119 & \textbf{Cartesian} & \textbf{Polar} \\
120 \hline
121 \(z_1 \pm z_2\) & \((a \pm c)(b \pm d)i\) & convert to \(a+bi\)\\
122 \hline
123 \(+k \times z\) & \multirow{2}{*}{\(ka \pm kbi\)} & \(kr\operatorname{cis} \theta\)\\
124 \cline{1-1}\cline{3-3}
125 \(-k \times z\) & & \(kr \operatorname{cis}(\theta\pm \pi)\)\\
126 \hline
127 \(z_1 \cdot z_2\) & \(ac-bd+(ad+bc)i\) & \(r_1r_2 \operatorname{cis}(\theta_1 + \theta_2)\)\\
128 \hline
129 \(z_1 \div z_2\) & \((z_1 \overline{z_2}) \div |z_2|^2\) & \(\left(\frac{r_1}{r_2}\right) \operatorname{cis}(\theta_1 - \theta_2)\)
130 \end{tabularx}
131
132 \subsubsection*{Scalar multiplication in polar form}
133
134 For \(k \in \mathbb{R}^+\):
135 \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\theta\]
136
137 \noindent For \(k \in \mathbb{R}^-\):
138 \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\left(\begin{cases}\theta - \pi & |0<\operatorname{Arg}(z)\le \pi \\ \theta + \pi & |-\pi<\operatorname{Arg}(z)\le 0\end{cases}\right)\]
139
140 \subsection*{Conjugate}
141
142 \begin{align*}
143 \overline{z} &= a \mp bi\\
144 &= r \operatorname{cis}(-\theta)
145 \end{align*}
146
147 \noindent \colorbox{cas}{On CAS: \texttt{conjg(a+bi)}}
148
149 \subsubsection*{Properties}
150
151 \begin{align*}
152 \overline{z_1 \pm z_2} &= \overline{z_1}\pm\overline{z_2}\\
153 \overline{z_1 \cdot z_2} &= \overline{z_1}\cdot\overline{z_2}\\
154 \overline{kz} &= k\overline{z} \quad | \quad k \in \mathbb{R}\\
155 z\overline{z} &= (a+bi)(a-bi)\\
156 &= a^2 + b^2\\
157 &= |z|^2
158 \end{align*}
159
160 \subsection*{Modulus}
161
162 \[|z|=|\vec{Oz}|=\sqrt{a^2 + b^2}\]
163
164 \subsubsection*{Properties}
165
166 \begin{align*}
167 |z_1z_2|&=|z_1||z_2|\\
168 \left|\frac{z_1}{z_2}\right|&=\frac{|z_1|}{|z_2|}\\
169 |z_1+z_2|&\le|z_1|+|z_2|
170 \end{align*}
171
172 \subsection*{Multiplicative inverse}
173
174 \begin{align*}
175 z^{-1}&=\frac{a-bi}{a^2+b^2}\\
176 &=\frac{\overline{z}}{|z|^2}a\\
177 &=r \operatorname{cis}(-\theta)
178 \end{align*}
179
180 \subsection*{Dividing over \(\mathbb{C}\)}
181
182 \begin{align*}
183 \frac{z_1}{z_2}&=z_1z_2^{-1}\\
184 &=\frac{z_1\overline{z_2}}{|z_2|^2}\\
185 &=\frac{(a+bi)(c-di)}{c^2+d^2}\\
186 & \qquad \text{(rationalise denominator)}
187 \end{align*}
188
189 \subsection*{Polar form}
190
191 \begin{align*}
192 z&=r\operatorname{cis}\theta\\
193 &=r(\cos \theta + i \sin \theta)
194 \end{align*}
195
196 \begin{itemize}
197 \item{\(r=|z|=\sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}\)}
198 \item{\(\theta = \operatorname{arg}(z)\) \quad \colorbox{cas}{On CAS: \texttt{arg(a+bi)}}}
199 \item{\(\operatorname{Arg}(z) \in (-\pi,\pi)\) \quad \bf{(principal argument)}}
200 \item{\colorbox{cas}{Convert on CAS:}\\ \verb|compToTrig(a+bi)| \(\iff\) \verb|cExpand{r·cisX}|}
201 \item{Multiple representations:\\\(r\operatorname{cis}\theta=r\operatorname{cis}(\theta+2n\pi)\) with \(n \in \mathbb{Z}\) revolutions}
202 \item{\(\operatorname{cis}\pi=-1,\qquad \operatorname{cis}0=1\)}
203 \end{itemize}
204
205 \subsection*{de Moivres' theorem}
206
207 \[(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}\]
208
209 \subsection*{Complex polynomials}
210
211 Include \(\pm\) for all solutions, incl. imaginary
212
213 \begin{tabularx}{\columnwidth}{ R{0.55} X }
214 \hline
215 Sum of squares & \(\begin{aligned}
216 z^2 + a^2 &= z^2-(ai)^2\\
217 &= (z+ai)(z-ai) \end{aligned}\) \\
218 \hline
219 Sum of cubes & \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)\\
220 \hline
221 Division & \(P(z)=D(z)Q(z)+R(z)\) \\
222 \hline
223 Remainder theorem & Let \(\alpha \in \mathbb{C}\). Remainder of \(P(z) \div (z-\alpha)\) is \(P(\alpha)\)\\
224 \hline
225 Factor theorem & \(z-\alpha\) is a factor of \(P(z) \iff P(\alpha)=0\) for \(\alpha \in \mathbb{C}\)\\
226 \hline
227 Conjugate root theorem & \(P(z)=0 \text{ at } z=a\pm bi\) (\(\implies\) both \(z_1\) and \(\overline{z_1}\) are solutions)\\
228 \hline
229 \end{tabularx}
230
231 \subsection*{\(n\)th roots}
232
233 \(n\)th roots of \(z=r\operatorname{cis}\theta\) are:
234
235 \[z = r^{\frac{1}{n}} \operatorname{cis}\left(\frac{\theta+2k\pi}{n}\right)\]
236
237 \begin{itemize}
238
239 \item{Same modulus for all solutions}
240 \item{Arguments separated by \(\frac{2\pi}{n} \therefore\) there are \(n\) roots}
241 \item{If one square root is \(a+bi\), the other is \(-a-bi\)}
242 \item{Give one implicit \(n\)th root \(z_1\), function is \(z=z_1^n\)}
243 \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 \(\frac{2\pi}{n}\))}
244 \end{itemize}
245
246 \noindent For \(0=az^2+bz+c\), use quadratic formula:
247
248 \[z=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]
249
250 \subsection*{Fundamental theorem of algebra}
251
252 A polynomial of degree \(n\) can be factorised into \(n\) linear factors in \(\mathbb{C}\):
253
254 \[\implies P(z)=a_n(z-\alpha_1)(z-\alpha_2)(z-\alpha_3)\dots(z-\alpha_n)\]
255 \[\text{ where } \alpha_1,\alpha_2,\alpha_3,\dots,\alpha_n \in \mathbb{C}\]
256
257 \subsection*{Argand planes}
258
259 \begin{center}\begin{tikzpicture}[scale=2]
260 \draw [->] (-0.2,0) -- (1.5,0) node [right] {$\operatorname{Re}(z)$};
261 \draw [->] (0,-0.2) -- (0,1.5) node [above] {$\operatorname{Im}(z)$};
262 \coordinate (P) at (1,1);
263 \coordinate (a) at (1,0);
264 \coordinate (b) at (0,1);
265 \coordinate (O) at (0,0);
266 \draw (0,0) -- (P) node[pos=0.5, above left]{\(r\)} node[pos=1, right]{\(\begin{aligned}z&=a+bi\\&=r\operatorname{cis}\theta\end{aligned}\)};
267 \draw [gray, dashed] (1,1) -- (1,0) node[black, pos=1, below]{\(a\)};
268 \draw [gray, dashed] (1,1) -- (0,1) node[black, pos=1, left]{\(b\)};
269 \begin{scope}
270 \path[clip] (O) -- (P) -- (a);
271 \fill[red, opacity=0.5, draw=black] (O) circle (2mm);
272 \node at ($(O)+(20:3mm)$) {$\theta$};
273 \end{scope}
274 \filldraw (P) circle (0.5pt);
275 \end{tikzpicture}\end{center}
276
277 \begin{itemize}
278 \item{Multiplication by \(i \implies\) CCW rotation of \(\frac{\pi}{2}\)}
279 \item{Addition: \(z_1 + z_2 \equiv\) \overrightharp{\(Oz_1\)} + \overrightharp{\(Oz_2\)}}
280 \end{itemize}
281
282 \subsection*{Sketching complex graphs}
283
284 \subsubsection*{Linear}
285
286 \begin{itemize}
287 \item{\(\operatorname{Re}(z)=c\) or \(\operatorname{Im}(z)=c\) (perpendicular bisector)}
288 \item{\(\operatorname{Im}(z)=m\operatorname{Re}(z)\)}
289 \item{\(|z+a|=|z+b| \implies 2(a-b)x=b^2-a^2\)\\Geometric: equidistant from \(a,b\)}
290 \end{itemize}
291
292 \subsubsection*{Circles}
293
294 \begin{itemize}
295 \item \(|z-z_1|^2=c^2|z_2+2|^2\)
296 \item \(|z-(a+bi)|=c \implies (x-a)^2+_(y-b)^2=c^2\)
297 \end{itemize}
298
299 \noindent \textbf{Loci} \qquad \(\operatorname{Arg}(z)<\theta\)
300
301 \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
302 \draw [->] (0,0) -- (1,0) node [right] {$\operatorname{Re}(z)$};
303 \draw [->] (0,-0.5) -- (0,1) node [above] {$\operatorname{Im}(z)$};
304 \draw [<-, dashed, thick, blue] (-1,0) -- (0,0);
305 \draw [->, thick, blue] (0,0) -- (1,1);
306 \fill [gray, opacity=0.2, domain=-1:1, variable=\x] (-1,-0.5) -- (-1,0) -- (0, 0) -- (1,1) -- (1,-0.5) -- cycle;
307 \begin{scope}
308 \path[clip] (0,0) -- (1,1) -- (1,0);
309 \fill[red, opacity=0.5, draw=black] (0,0) circle (2mm);
310 \node at ($(0,0)+(20:3mm)$) {$\frac{\pi}{4}$};
311 \end{scope}
312 \node [font=\footnotesize] at (0.5,-0.25) {\(\operatorname{Arg}(z)\le\frac{\pi}{4}\)};
313 \node [blue, mydot] {};
314 \end{tikzpicture}\end{center}
315
316 \noindent \textbf{Rays} \qquad \(\operatorname{Arg}(z-b)=\theta\)
317
318 \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
319 \draw [->] (-0.75,0) -- (1.5,0) node [right] {$\operatorname{Re}(z)$};
320 \draw [->] (0,-1) -- (0,1) node [above] {$\operatorname{Im}(z)$};
321 \draw [->, thick, brown] (-0.25,0) -- (-0.75,-1);
322 \node [above, font=\footnotesize] at (-0.25,0) {\(\frac{1}{4}\)};
323 \begin{scope}
324 \path[clip] (-0.25,0) -- (-0.75,-1) -- (0,0);
325 \fill[orange, opacity=0.5, draw=black] (-0.25,0) circle (2mm);
326 \end{scope}
327 \node at (-0.08,-0.3) {\(\frac{\pi}{8}\)};
328 \node [font=\footnotesize, left] at (-0.75,-1) {\(\operatorname{Arg}(z+\frac{1}{4})=\frac{\pi}{8}\)};
329 \node [brown, mydot] at (-0.25,0) {};
330 \draw [<->, thick, green] (0,-1) -- (1.5,0.5) node [pos=0.25, black, font=\footnotesize, right] {\(|z-2|=|z-(1+i)|\)};
331 \node [left, font=\footnotesize] at (0,-1) {\(-1\)};
332 \node [below, font=\footnotesize] at (1,0) {\(1\)};
333 \end{tikzpicture}\end{center}
334
335 \section{Vectors}
336 \begin{center}\begin{tikzpicture}
337 \draw [->] (-0.5,0) -- (3,0) node [right] {\(x\)};
338 \draw [->] (0,-0.5) -- (0,3) node [above] {\(y\)};
339 \draw [orange, ->, thick] (0.5,0.5) -- (2.5,2.5) node [pos=0.5, above] {\(\vec{u}\)};
340 \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
341 \draw [gray, dashed, thick] (0.5,0.5) -- (2.5,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, below]{\(x\vec{i}\)};
342 \draw [gray, dashed, thick] (2.5,0.5) -- (2.5,2.5) node [pos=0.5] {\midarrow};
343 \end{scope}
344 \node[black, right] at (2.5,1.5) {\(y\vec{j}\)};
345 \end{tikzpicture}\end{center}
346 \subsection*{Column notation}
347
348 \[\begin{bmatrix}x\\ y \end{bmatrix} \iff x\boldsymbol{i} + y\boldsymbol{j}\]
349 \(\begin{bmatrix}x_2-x_1\\ y_2-y_1 \end{bmatrix}\) \quad between \(A(x_1,y_1), \> B(x_2,y_2)\)
350
351 \subsection*{Scalar multiplication}
352
353 \[k\cdot (x\boldsymbol{i}+y\boldsymbol{j})=kx\boldsymbol{i}+ky\boldsymbol{j}\]
354
355 \noindent For \(k \in \mathbb{R}^-\), direction is reversed
356
357 \subsection*{Vector addition}
358 \begin{center}\begin{tikzpicture}[scale=1]
359 \coordinate (A) at (0,0);
360 \coordinate (B) at (2,2);
361 \draw [->, thick, red] (0,0) -- (2,2) node [pos=0.5, below right] {\(\vec{u}=2\vec{i}+2\vec{j}\)};
362 \draw [->, thick, blue] (2,2) -- (1,4) node [pos=0.5, above right] {\(\vec{v}=-\vec{i}+2\vec{j}\)};
363 \draw [->, thick, orange] (0,0) -- (1,4) node [pos=0.5, left] {\(\vec{u}+\vec{v}=\vec{i}+4\vec{j}\)};
364 \end{tikzpicture}\end{center}
365
366 \[(x\boldsymbol{i}+y\boldsymbol{j}) \pm (a\boldsymbol{i}+b\boldsymbol{j})=(x \pm a)\boldsymbol{i}+(y \pm b)\boldsymbol{j}\]
367
368 \begin{itemize}
369 \item Draw each vector head to tail then join lines
370 \item Addition is commutative (parallelogram)
371 \item \(\boldsymbol{u}-\boldsymbol{v}=\boldsymbol{u}+(-\boldsymbol{v}) \implies \overrightharp{AB}=\boldsymbol{b}-\boldsymbol{a}\)
372 \end{itemize}
373
374 \subsection*{Magnitude}
375
376 \[|(x\boldsymbol{i} + y\boldsymbol{j})|=\sqrt{x^2+y^2}\]
377
378 \subsection*{Parallel vectors}
379
380 \[\boldsymbol{u} || \boldsymbol{v} \iff \boldsymbol{u} = k \boldsymbol{v} \text{ where } k \in \mathbb{R} \setminus \{0\}\]
381
382 For parallel vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\):\\
383 \[\boldsymbol{a \cdot b}=\begin{cases}
384 |\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction}\\
385 -|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions}
386 \end{cases}\]
387 %\includegraphics[width=0.2,height=\textheight]{graphics/parallelogram-vectors.jpg}
388 %\includegraphics[width=1]{graphics/vector-subtraction.jpg}
389
390 \subsection*{Perpendicular vectors}
391
392 \[\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b} = 0\ \quad \text{(since \(\cos 90 = 0\))}\]
393
394 \subsection*{Unit vector \(|\hat{\boldsymbol{a}}|=1\)}
395 \[\begin{split}\hat{\boldsymbol{a}} & = {\frac{1}{|\boldsymbol{a}|}}\boldsymbol{a} \\ & = \boldsymbol{a} \cdot {|\boldsymbol{a}|}\end{split}\]
396
397 \subsection*{Scalar product \(\boldsymbol{a} \cdot \boldsymbol{b}\)}
398
399
400 \begin{center}\begin{tikzpicture}[scale=2]
401 \draw [->] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{b}\)};
402 \draw [->] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{a}\)};
403 \begin{scope}
404 \path[clip] (1,0.5) -- (1,0) -- (0,0);
405 \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
406 \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
407 \end{scope}
408 \end{tikzpicture}\end{center}
409 \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*}
410 \noindent\colorbox{cas}{On CAS: \texttt{dotP({[}a\ b\ c{]},\ {[}d\ e\ f{]})}}
411
412 \subsubsection*{Properties}
413
414 \begin{enumerate}
415 \item
416 \(k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k\boldsymbol{b})\)
417 \item
418 \(\boldsymbol{a \cdot 0}=0\)
419 \item
420 \(\boldsymbol{a} \cdot (\boldsymbol{b} + \boldsymbol{c})=\boldsymbol{a} \cdot \boldsymbol{b} + \boldsymbol{a} \cdot \boldsymbol{c}\)
421 \item
422 \(\boldsymbol{i \cdot i} = \boldsymbol{j \cdot j} = \boldsymbol{k \cdot k}= 1\)
423 \item
424 \(\boldsymbol{a} \cdot \boldsymbol{b} = 0 \quad \implies \quad \boldsymbol{a} \perp \boldsymbol{b}\)
425 \item
426 \(\boldsymbol{a \cdot a} = |\boldsymbol{a}|^2 = a^2\)
427 \end{enumerate}
428
429 \subsection*{Angle between vectors}
430
431 \[\cos \theta = \frac{\boldsymbol{a} \cdot \boldsymbol{b}}{|\boldsymbol{a}| |\boldsymbol{b}|} = \frac{a_1 b_1 + a_2 b_2}{|\boldsymbol{a}| |\boldsymbol{b}|}\]
432
433 \noindent \colorbox{cas}{On CAS:} \texttt{angle([a b c], [a b c])}
434
435 (Action \(\rightarrow\) Vector \(\rightarrow\)Angle)
436
437 \subsection*{Angle between vector and axis}
438
439 \noindent For\(\boldsymbol{a} = a_1 \boldsymbol{i} + a_2 \boldsymbol{j} + a_3 \boldsymbol{k}\)
440 which makes angles \(\alpha, \beta, \gamma\) with positive side of
441 \(x, y, z\) axes:
442 \[\cos \alpha = \frac{a_1}{|\boldsymbol{a}|}, \quad \cos \beta = \frac{a_2}{|\boldsymbol{a}|}, \quad \cos \gamma = \frac{a_3}{|\boldsymbol{a}|}\]
443
444 \noindent \colorbox{cas}{On CAS:} \texttt{angle({[}a\ b\ c{]},\ {[}1\ 0\ 0{]})}\\for angle
445 between \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) and
446 \(x\)-axis
447
448 \subsection*{Projections \& resolutes}
449
450 \begin{tikzpicture}[scale=3]
451 \draw [->, purple] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{a}\)};
452 \draw [->, orange] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{u}\)};
453 \draw [->, blue] (1,0) -- (2,0) node [pos=0.5, below] {\(\boldsymbol{b}\)};
454 \begin{scope}
455 \path[clip] (1,0.5) -- (1,0) -- (0,0);
456 \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
457 \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
458 \end{scope}
459 \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
460 \draw [gray, dashed, thick] (1,0) -- (1,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, right, rotate=-90]{\(\boldsymbol{w}\)};
461 \end{scope}
462 \draw (0,0) coordinate (O)
463 (1,0) coordinate (A)
464 (1,0.5) coordinate (B)
465 pic [draw,red,angle radius=2mm] {right angle = O--A--B};
466 \end{tikzpicture}
467
468 \subsubsection*{\(\parallel\boldsymbol{b}\) (vector projection/resolute)}
469
470 \begin{align*}
471 \boldsymbol{u} & = \frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|^2}\boldsymbol{b} \\
472 & = \left(\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\right)\left(\frac{\boldsymbol{b}}{|\boldsymbol{b}|}\right) \\
473 & = (\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}
474 \end{align*}
475
476 \subsubsection*{\(\perp\boldsymbol{b}\) (perpendicular projection)}
477 \[\boldsymbol{w} = \boldsymbol{a} - \boldsymbol{u}\]
478
479 \subsubsection*{\(|\boldsymbol{u}|\) (scalar projection/resolute)}
480 \begin{align*}
481 s &= |\boldsymbol{u}|\\
482 &= \boldsymbol{a} \cdot \hat{\boldsymbol{b}}\\
483 &=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\\
484 &= |\boldsymbol{a}| \cos \theta
485 \end{align*}
486
487 \subsubsection*{Rectangular (\(\parallel,\perp\)) components}
488
489 \[\boldsymbol{a}=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}+\left(\boldsymbol{a}-\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}\right)\]
490
491
492 \subsection*{Vector proofs}
493
494 \textbf{Concurrent:} intersection of \(\ge\) 3 lines
495
496 \begin{tikzpicture}
497 \draw [blue] (0,0) -- (1,1);
498 \draw [red] (1,0) -- (0,1);
499 \draw [brown] (0.4,0) -- (0.6,1);
500 \filldraw (0.5,0.5) circle (2pt);
501 \end{tikzpicture}
502
503 \subsubsection*{Collinear points}
504
505 \(\ge\) 3 points lie on the same line
506
507 \begin{tikzpicture}
508 \draw [purple] (0,0) -- (4,1);
509 \filldraw (2,0.5) circle (2pt) node [above] {\(C\)};
510 \filldraw (1,0.25) circle (2pt) node [above] {\(A\)};
511 \filldraw (3,0.75) circle (2pt) node [above] {\(B\)};
512 \coordinate (O) at (2.8,-0.2);
513 \node at (O) [below] {\(O\)};
514 \begin{scope}[->, orange, thick]
515 \draw (O) -- (2,0.5) node [pos=0.5, above, font=\footnotesize, black] {\(\boldsymbol{c}\)};
516 \draw (O) -- (1,0.25) node [pos=0.5, below, font=\footnotesize, black] {\(\boldsymbol{a}\)};
517 \draw (O) -- (3,0.75) node [pos=0.5, right, font=\footnotesize, black] {\(\boldsymbol{b}\)};
518 \end{scope}
519 \end{tikzpicture}
520
521 \begin{align*}
522 \text{e.g. Prove that}\\
523 \overrightharp{AC}=m\overrightharp{AB} \iff \boldsymbol{c}&=(1-m)\boldsymbol{a}+m\boldsymbol{b}\\
524 \implies \boldsymbol{c} &= \overrightharp{OA} + \overrightharp{AC}\\
525 &= \overrightharp{OA} + m\overrightharp{AB}\\
526 &=\boldsymbol{a}+m(\boldsymbol{b}-\boldsymbol{a})\\
527 &=\boldsymbol{a}+m\boldsymbol{b}-m\boldsymbol{a}\\
528 &=(1-m)\boldsymbol{a}+m{b}
529 \end{align*}
530 \begin{align*}
531 \text{Also, } \implies \overrightharp{OC} &= \lambda \vec{OA} + \mu \overrightharp{OB} \\
532 \text{where } \lambda + \mu &= 1\\
533 \text{If } C \text{ lies along } \overrightharp{AB}, & \implies 0 < \mu < 1
534 \end{align*}
535
536
537 \subsubsection*{Parallelograms}
538
539 \begin{center}\begin{tikzpicture}
540 \coordinate (O) at (0,0) node [below left] {\(O\)};
541 \coordinate (A) at (4,0);
542 \coordinate (B) at (6,2);
543 \coordinate (C) at (2,2);
544 \coordinate (D) at (6,0);
545
546 \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (O)--(A) node [below left] {\(A\)};
547 \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (A)--(B) node [above right] {\(B\)};
548 \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (B)--(C) node [above left] {\(C\)};
549 \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (C)--(O);
550
551 \draw [gray, dashed] (O) -- (B) node [pos=0.75] {\(\diagdown\diagdown\)} node [pos=0.25] {\(\diagdown\diagdown\)};
552 \draw [gray, dashed] (A) -- (C) node [pos=0.75] {\(\diagup\)} node [pos=0.25] {\(\diagup\)};
553 \begin{scope}
554 \path[clip] (C) -- (A) -- (O);
555 \fill[orange, opacity=0.5, draw=black] (0,0) circle (4mm);
556 \node at ($(0,0)+(20:8mm)$) {\(\theta\)};
557 \end{scope}
558 \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] {\(|\boldsymbol{c}|\cos\theta\)};
559 \draw pic [draw,thick,red,angle radius=2mm] {right angle=O--D--B};
560 \end{tikzpicture}\end{center}
561
562 \begin{itemize}
563 \item
564 Diagonals \(\overrightharp{OB}, \overrightharp{AC}\) bisect each other
565 \item
566 If diagonals are equal length, it is a rectangle
567 \item
568 \(|\overrightharp{OB}|^2 + |\overrightharp{CA}|^2 = |\overrightharp{OA}|^2 + |\overrightharp{AB}|^2 + |\overrightharp{CB}|^2 + |\overrightharp{OC}|^2\)
569 \item
570 Area \(=\boldsymbol{c} \cdot \boldsymbol{a}\)
571 \end{itemize}
572
573 \subsubsection*{Useful vector properties}
574
575 \begin{itemize}
576 \item
577 \(\boldsymbol{a} \parallel \boldsymbol{b} \implies \boldsymbol{b}=k\boldsymbol{a}\) for some
578 \(k \in \mathbb{R} \setminus \{0\}\)
579 \item
580 If \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are parallel with at
581 least one point in common, then they lie on the same straight line
582 \item
583 \(\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b}=0\)
584 \item
585 \(\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2\)
586 \end{itemize}
587
588 \subsection*{Linear dependence}
589
590 \(\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\) are linearly dependent if they are \(\nparallel\) and:
591 \begin{align*}
592 0&=k\boldsymbol{a}+l\boldsymbol{b}+m\boldsymbol{c}\\
593 \therefore \boldsymbol{c} &= m\boldsymbol{a} + n\boldsymbol{b} \quad \text{(simultaneous)}
594 \end{align*}
595
596 \noindent \(\boldsymbol{a}, \boldsymbol{b},\) and \(\boldsymbol{c}\) are linearly
597 independent if no vector in the set is expressible as a linear
598 combination of other vectors in set, or if they are parallel.
599
600 \subsection*{Three-dimensional vectors}
601
602 Right-hand rule for axes: \(z\) is up or out of page.
603
604 \tdplotsetmaincoords{60}{120}
605 \begin{center}\begin{tikzpicture} [scale=3, tdplot_main_coords, axis/.style={->,thick},
606 vector/.style={-stealth,red,very thick},
607 vector guide/.style={dashed,gray,thick}]
608
609 %standard tikz coordinate definition using x, y, z coords
610 \coordinate (O) at (0,0,0);
611
612 %tikz-3dplot coordinate definition using x, y, z coords
613
614 \pgfmathsetmacro{\ax}{1}
615 \pgfmathsetmacro{\ay}{1}
616 \pgfmathsetmacro{\az}{1}
617
618 \coordinate (P) at (\ax,\ay,\az);
619
620 %draw axes
621 \draw[axis] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$};
622 \draw[axis] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$};
623 \draw[axis] (0,0,0) -- (0,0,1) node[anchor=south]{$z$};
624
625 %draw a vector from O to P
626 \draw[vector] (O) -- (P);
627
628 %draw guide lines to components
629 \draw[vector guide] (O) -- (\ax,\ay,0);
630 \draw[vector guide] (\ax,\ay,0) -- (P);
631 \draw[vector guide] (P) -- (0,0,\az);
632 \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
633 \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
634 \draw[vector guide] (\ax,\ay,0) -- (\ax,0,0);
635 \node[tdplot_main_coords,above right]
636 at (\ax,\ay,\az){(\ax, \ay, \az)};
637 \end{tikzpicture}\end{center}
638
639 \subsection*{Parametric vectors}
640
641 Parametric equation of line through point \((x_0, y_0, z_0)\) and
642 parallel to \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) is:
643
644 \[\begin{cases}x = x_o + a \cdot t \\ y = y_0 + b \cdot t \\ z = z_0 + c \cdot t\end{cases}\]
645
646 \section{Circular functions}
647
648 \(\sin(bx)\) or \(\cos(bx)\): period \(=\frac{2\pi}{b}\)
649
650 \noindent \(\tan(nx)\): period \(=\frac{\pi}{n}\)\\
651 \indent\indent\indent asymptotes at \(x=\frac{(2k+1)\pi}{2n} \> \vert \> k \in \mathbb{Z}\)
652
653 \subsection*{Reciprocal functions}
654
655 \subsubsection*{Cosecant}
656
657 \[\operatorname{cosec} \theta = \frac{1}{\sin \theta} \> \vert \> \sin \theta \ne 0\]
658
659 \begin{itemize}
660 \item
661 \textbf{Domain} \(= \mathbb{R} \setminus {n\pi : n \in \mathbb{Z}}\)
662 \item
663 \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
664 \item
665 \textbf{Turning points} at
666 \(\theta = \frac{(2n + 1)\pi}{2} \> \vert \> n \in \mathbb{Z}\)
667 \item
668 \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
669 \end{itemize}
670
671 \subsubsection*{Secant}
672
673\begin{tikzpicture}
674 \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={\(-2\pi\), \(-\pi\), \(\pi\), \(2\pi\)}]
675% \addplot[blue, domain=-6.2830:6.2830,unbounded coords=jump,samples=80] {sec(deg(x))};
676 \addplot[blue, restrict y to domain=-10:10, domain=-7:7,samples=100] {sec(deg(x))} node [pos=0.93, black, right] {\(\operatorname{sec} x\)};
677 \addplot[red, dashed, domain=-7:7,samples=100] {cos(deg(x))};
678 \draw [gray, dotted, thick] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{rel axis cs:0,1});
679 \draw [gray, dotted, thick] ({axis cs:4.71239,0}|-{rel axis cs:0,0}) -- ({axis cs:4.71239,0}|-{rel axis cs:0,1});
680 \draw [gray, dotted, thick] ({axis cs:-4.71239,0}|-{rel axis cs:0,0}) -- ({axis cs:-4.71239,0}|-{rel axis cs:0,1});
681 \draw [gray, dotted, thick] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1});
682\end{axis}
683 \node [black] at (7,3.5) {\(\cos x\)};
684\end{tikzpicture}
685
686 \[\operatorname{sec} \theta = \frac{1}{\cos \theta} \> \vert \> \cos \theta \ne 0\]
687
688 \begin{itemize}
689
690 \item
691 \textbf{Domain}
692 \(= \mathbb{R} \setminus \frac{(2n + 1) \pi}{2} : n \in \mathbb{Z}\}\)
693 \item
694 \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
695 \item
696 \textbf{Turning points} at
697 \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
698 \item
699 \textbf{Asymptotes} at
700 \(\theta = \frac{(2n + 1) \pi}{2} \> \vert \> n \in \mathbb{Z}\)
701 \end{itemize}
702
703 \subsubsection*{Cotangent}
704
705\begin{tikzpicture}
706 \begin{axis}[xmin=-3,xmax=3,ymin=-1.5,ymax=1.5,enlargelimits=true, xtick={-3.1415, -1.5708, 1.5708, 3.1415},xticklabels={\(-\pi\), \(-\frac{\pi}{2}\), \(\frac{\pi}{2}\), \(\pi\)}]
707 \addplot[blue, smooth, domain=-3:-0.1,unbounded coords=jump,samples=105] {cot(deg(x))} node [pos=0.3, left] {\(\operatorname{cot} x\)};
708\addplot[blue, smooth, domain=0.1:3,unbounded coords=jump,samples=105] {cot(deg(x))};
709\addplot[red, smooth, dashed] gnuplot [domain=-1.5:1.5,unbounded coords=jump,samples=105] {tan(x)};
710\addplot[red, smooth, dashed] gnuplot [domain=-3.5:-1.8,unbounded coords=jump,samples=105] {tan(x)} node [pos=0.5, right] {\(\tan x\)};
711\addplot[red, smooth, dashed] gnuplot [domain=1.8:3.5,unbounded coords=jump,samples=105] {tan(x)};
712 \draw [thick, red, dotted] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1});
713 \draw [thick, blue, dotted] ({axis cs:-3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:-3.1415,0}|-{rel axis cs:0,1});
714 \draw [thick, blue, dotted] ({axis cs:0,0}|-{rel axis cs:0,0}) -- ({axis cs:0,0}|-{rel axis cs:0,1});
715 \draw [thick, blue, dotted] ({axis cs:3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:3.1415,0}|-{rel axis cs:0,1});
716 \draw [thick, red, dotted] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{rel axis cs:0,1});
717\end{axis}
718\end{tikzpicture}
719
720 \[\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0\]
721
722 \begin{itemize}
723
724 \item
725 \textbf{Domain} \(= \mathbb{R} \setminus \{n \pi: n \in \mathbb{Z}\}\)
726 \item
727 \textbf{Range} \(= \mathbb{R}\)
728 \item
729 \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
730 \end{itemize}
731
732 \subsubsection*{Symmetry properties}
733
734 \[\begin{split}
735 \operatorname{sec} (\pi \pm x) & = -\operatorname{sec} x \\
736 \operatorname{sec} (-x) & = \operatorname{sec} x \\
737 \operatorname{cosec} (\pi \pm x) & = \mp \operatorname{cosec} x \\
738 \operatorname{cosec} (-x) & = - \operatorname{cosec} x \\
739 \operatorname{cot} (\pi \pm x) & = \pm \operatorname{cot} x \\
740 \operatorname{cot} (-x) & = - \operatorname{cot} x
741 \end{split}\]
742
743 \subsubsection*{Complementary properties}
744
745 \[\begin{split}
746 \operatorname{sec} \left({\pi \over 2} - x\right) & = \operatorname{cosec} x \\
747 \operatorname{cosec} \left({\pi \over 2} - x\right) & = \operatorname{sec} x \\
748 \operatorname{cot} \left({\pi \over 2} - x\right) & = \tan x \\
749 \tan \left({\pi \over 2} - x\right) & = \operatorname{cot} x
750 \end{split}\]
751
752 \subsubsection*{Pythagorean identities}
753
754 \[\begin{split}
755 1 + \operatorname{cot}^2 x & = \operatorname{cosec}^2 x, \quad \text{where } \sin x \ne 0 \\
756 1 + \tan^2 x & = \operatorname{sec}^2 x, \quad \text{where } \cos x \ne 0
757 \end{split}\]
758
759 \subsection*{Compound angle formulas}
760
761 \[\cos(x \pm y) = \cos x + \cos y \mp \sin x \sin y\]
762 \[\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y\]
763 \[\tan(x \pm y) = {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}\]
764
765 \subsection*{Double angle formulas}
766
767 \[\begin{split}
768 \cos 2x &= \cos^2 x - \sin^2 x \\
769 & = 1 - 2\sin^2 x \\
770 & = 2 \cos^2 x -1
771 \end{split}\]
772
773 \[\sin 2x = 2 \sin x \cos x\]
774
775 \[\tan 2x = {{2 \tan x} \over {1 - \tan^2 x}}\]
776
777 \subsection*{Inverse circular functions}
778
779 \begin{tikzpicture}
780 \begin{axis}[ymin=-2, ymax=4, xmin=-1.1, xmax=1.1, ytick={-1.5708, 1.5708, 3.14159},yticklabels={$-\frac{\pi}{2}$, $\frac{\pi}{2}$, $\pi$}]
781 \addplot[color=red, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {asin(x)} node [pos=0.25, below right] {\(\sin^{-1}x\)};
782 \addplot[color=blue, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {acos(x)} node [pos=0.25, below left] {\(\cos^{-1}x\)};
783 \addplot[mark=*, red] coordinates {(-1,-1.5708)} node[right, font=\footnotesize]{\((-1,-\frac{\pi}{2})\)} ;
784 \addplot[mark=*, red] coordinates {(1,1.5708)} node[left, font=\footnotesize]{\((1,\frac{\pi}{2})\)} ;
785 \addplot[mark=*, blue] coordinates {(1,0)};
786 \addplot[mark=*, blue] coordinates {(-1,3.1415)} node[right, font=\footnotesize]{\((-1,\pi)\)} ;
787 \end{axis}
788 \end{tikzpicture}\\
789
790 Inverse functions: \(f(f^{-1}(x)) = x\) (restrict domain)
791
792 \[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y\]
793 \hfill where \(\sin y = x, \> y \in [{-\pi \over 2}, {\pi \over 2}]\)
794
795 \[\cos^{-1}: [-1,1] \rightarrow \mathbb{R}, \quad \cos^{-1} x = y\]
796 \hfill where \(\cos y = x, \> y \in [0, \pi]\)
797
798 \[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\]
799 \hfill where \(\tan y = x, \> y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\)
800
801 \begin{tikzpicture}
802 \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={\(-\frac{\pi}{2}\),\(\frac{\pi}{2}\)}]
803 \addplot[color=orange, smooth] gnuplot [domain=-35:35, unbounded coords=jump,samples=350] {atan(x)} node [pos=0.5, above left] {\(\tan^{-1}x\)};
804 \addplot[gray, dotted, thick, domain=-35:35] {1.5708} node [black, font=\footnotesize, below right, pos=0] {\(y=\frac{\pi}{2}\)};
805 \addplot[gray, dotted, thick, domain=-35:35] {-1.5708} node [black, font=\footnotesize, above left, pos=1] {\(y=-\frac{\pi}{2}\)};
806 \end{axis}
807 \end{tikzpicture}
808\columnbreak
809 \section{Differential calculus}
810
811 \subsection*{Limits}
812
813 \[\lim_{x \rightarrow a}f(x)\]
814 \(L^-,\quad L^+\) \qquad limit from below/above\\
815 \(\lim_{x \to a} f(x)\) \quad limit of a point\\
816
817 \noindent For solving \(x\rightarrow\infty\), put all \(x\) terms in denominators\\
818 e.g. \[\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2\]
819
820 \subsubsection*{Limit theorems}
821
822 \begin{enumerate}
823 \item
824 For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\)
825 \item
826 \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\)
827 \item
828 \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\)
829 \item
830 \(\therefore \lim_{x \rightarrow a} c \times f(x)=cF\) where \(c=\) constant
831 \item
832 \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\)
833 \item
834 \(f(x)\) is continuous \(\iff L^-=L^+=f(x) \> \forall x\)
835 \end{enumerate}
836
837 \subsection*{Gradients of secants and tangents}
838
839 \textbf{Secant (chord)} - line joining two points on curve\\
840 \textbf{Tangent} - line that intersects curve at one point
841
842 \subsection*{First principles derivative}
843
844 \[f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={\frac{dy}{dx}}\]
845
846 \subsubsection*{Logarithmic identities}
847
848 \(\log_b (xy)=\log_b x + \log_b y\)\\
849 \(\log_b x^n = n \log_b x\)\\
850 \(\log_b y^{x^n} = x^n \log_b y\)
851
852 \subsubsection*{Index identities}
853
854 \(b^{m+n}=b^m \cdot b^n\)\\
855 \((b^m)^n=b^{m \cdot n}\)\\
856 \((b \cdot c)^n = b^n \cdot c^n\)\\
857 \({a^m \div a^n} = {a^{m-n}}\)
858
859 \subsection*{Reciprocal derivatives}
860
861 \[\frac{1}{\frac{dy}{dx}} = \frac{dx}{dy}\]
862
863 \subsection*{Differentiating \(x=f(y)\)}
864 Find \(\dfrac{dx}{dy}\), then \(\dfrac{dy}{dx} = \dfrac{1}{\left(\dfrac{dx}{dy}\right)}\)
865
866 \subsection*{Second derivative}
867 \begin{align*}f(x) \longrightarrow &f^\prime (x) \longrightarrow f^{\prime\prime}(x)\\
868 \implies y \longrightarrow &\frac{dy}{dx} \longrightarrow \frac{d^2 y}{dx^2}\end{align*}
869
870 \noindent Order of polynomial \(n\)th derivative decrements each time the derivative is taken
871
872 \subsubsection*{Points of Inflection}
873
874 \emph{Stationary point} - i.e.
875 \(f^\prime(x)=0\)\\
876 \emph{Point of inflection} - max \(|\)gradient\(|\) (i.e.
877 \(f^{\prime\prime} = 0\))
878
879
880 \begin{table*}[ht]
881 \centering
882 \begin{tabularx}{\textwidth}{rXXX}
883 \hline
884 \rowcolor{shade2}
885 & \centering\(\dfrac{d^2 y}{dx^2} > 0\) & \centering \(\dfrac{d^2y}{dx^2}<0\) & \(\dfrac{d^2y}{dx^2}=0\) (inflection) \\[1.5em]
886 \hline
887 \(\dfrac{dy}{dx}>0\) &
888 \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)}&
889 \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)}&
890 \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}\\
891 \hline
892 \(\dfrac{dy}{dx}<0\) &
893 \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)}&
894 \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)}&
895 \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}\\
896 \hline
897 \(\dfrac{dy}{dx}=0\)&
898 \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}&
899 \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}\\
900 \hline
901 \end{tabularx}
902 \end{table*}
903 \begin{itemize}
904 \item
905 if \(f^\prime (a) = 0\) and \(f^{\prime\prime}(a) > 0\), then point
906 \((a, f(a))\) is a local min (curve is concave up)
907 \item
908 if \(f^\prime (a) = 0\) and \(f^{\prime\prime} (a) < 0\), then point
909 \((a, f(a))\) is local max (curve is concave down)
910 \item
911 if \(f^{\prime\prime}(a) = 0\), then point \((a, f(a))\) is a point of
912 inflection
913 \item
914 if also \(f^\prime(a)=0\), then it is a stationary point of inflection
915 \end{itemize}
916
917 \subsection*{Implicit Differentiation}
918
919 \noindent Used for differentiating circles etc.
920
921 If \(p\) and \(q\) are expressions in \(x\) and \(y\) such that \(p=q\),
922 for all \(x\) and \(y\), then:
923
924 \[{\frac{dp}{dx}} = {\frac{dq}{dx}} \quad \text{and} \quad {\frac{dp}{dy}} = {\frac{dq}{dy}}\]
925
926 \begin{cas}
927 Action \(\rightarrow\) Calculation \\
928 \hspace{1em}\texttt{impDiff(y\^{}2+ax=5,\ x,\ y)} \hfill(returns \(y^\prime= \dots\))
929 \end{cas}
930
931 \subsection*{Slope fields}
932
933 \begin{tikzpicture}[declare function={diff(\x,\y) = \x+\y;}]
934 \begin{axis}[axis equal, ymin=-4, ymax=4, xmin=-4, xmax=4, ticks=none, enlargelimits=true, ]
935 \addplot[thick, orange, domain=-4:2] {e^(x)-x-1};
936 \pgfplotsinvokeforeach{-4,...,4}{%
937 \draw[gray] ( {#1 -0.1}, {4 - diff(#1, 4) *0.1}) -- ( {#1 +0.1}, {4 + diff(#1, 4) *0.1});
938 \draw[gray] ( {#1 -0.1}, {3 - diff(#1, 3) *0.1}) -- ( {#1 +0.1}, {3 + diff(#1, 3) *0.1});
939 \draw[gray] ( {#1 -0.1}, {2 - diff(#1, 2) *0.1}) -- ( {#1 +0.1}, {2 + diff(#1, 2) *0.1});
940 \draw[gray] ( {#1 -0.1}, {1 - diff(#1, 1) *0.1}) -- ( {#1 +0.1}, {1 + diff(#1, 1) *0.1});
941 \draw[gray] ( {#1 -0.1}, {0 - diff(#1, 0) *0.1}) -- ( {#1 +0.1}, {0 + diff(#1, 0) *0.1});
942 \draw[gray] ( {#1 -0.1}, {-1 - diff(#1, -1) *0.1}) -- ( {#1 +0.1}, {-1 + diff(#1, -1) *0.1});
943 \draw[gray] ( {#1 -0.1}, {-2 - diff(#1, -2) *0.1}) -- ( {#1 +0.1}, {-2 + diff(#1, -2) *0.1});
944 \draw[gray] ( {#1 -0.1}, {-3 - diff(#1, -3) *0.1}) -- ( {#1 +0.1}, {-3 + diff(#1, -3) *0.1});
945 \draw[gray] ( {#1 -0.1}, {-4 - diff(#1, -4) *0.1}) -- ( {#1 +0.1}, {-4 + diff(#1, -4) *0.1});
946 }
947 \end{axis}
948 \end{tikzpicture}
949
950 \subsection*{Parametric equations}
951
952 For each point on \(\left( f(t), g(t) \right)\):
953
954 \begin{align*}
955 \dfrac{dy}{dt} &= \dfrac{dy}{dx} \cdot \dfrac{dx}{dt} \\
956 \therefore \dfrac{dy}{dx} &= \dfrac{\left(\dfrac{dy}{dt}\right)}{\left(\dfrac{dx}{dt}\right)} \text{ provided } \dfrac{dx}{dt} \ne 0 \\
957 \text{Also...} \\
958 \dfrac{d^2y}{dx^2} &= \dfrac{\left(\dfrac{dy^\prime}{dt}\right)}{\left(\dfrac{dx}{dt}\right)} \text{ where } y^\prime = \dfrac{dy}{dx}
959 \end{align*}
960
961 \subsection*{Integration}
962
963 \[\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)\]
964
965 \subsubsection*{Definite integrals}
966
967 \[\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)\]
968
969 \begin{itemize}
970
971 \item
972 Signed area enclosed by\\
973 \(\> y=f(x), \quad y=0, \quad x=a, \quad x=b\).
974 \item
975 \emph{Integrand} is \(f\).
976 \end{itemize}
977
978 \subsubsection*{Properties}
979
980 \begin{align*}
981 \int^b_a f(x) \> dx &= \int^c_a f(x) \> dx + \int^b_c f(x) \> dx \\
982 \int^a_a f(x) \> dx &= 0 \\
983 \int^b_a k \cdot f(x) \> dx &= k \int^b_a f(x) \> dx \\
984 \int^b_a f(x) \pm g(x) \> dx &= \int^b_a f(x) \> dx \pm \int^b_a g(x) \> dx \\
985 \int^b_a f(x) \> dx &= - \int^a_b f(x) \> dx \\
986 \end{align*}
987
988 \subsection*{Integration by substitution}
989
990 \[\int f(u) {\frac{du}{dx}} \cdot dx = \int f(u) \cdot du\]
991
992 \begin{warning}
993 \(\boldsymbol{f(u)}\) must be 1:1 \(\boldsymbol{\implies}\) one \(\boldsymbol{x}\) for each \(\boldsymbol{y}\)
994 \end{warning}
995 \begin{align*}\text{e.g. for } y&=\int(2x+1)\sqrt{x+4} \cdot dx\\
996 \text{let } u&=x+4\\
997 \implies& {\frac{du}{dx}} = 1\\
998 \implies& x = u - 4\\
999 \text{then } &y=\int (2(u-4)+1)u^{\frac{1}{2}} \cdot du\\
1000 &\text{(solve as normal integral)}
1001 \end{align*}
1002
1003 \subsubsection*{Definite integrals by substitution}
1004
1005 For \(\int^b_a f(x) {\frac{du}{dx}} \cdot dx\), evaluate new \(a\) and
1006 \(b\) for \(f(u) \cdot du\).
1007
1008 \subsubsection*{Trigonometric integration}
1009
1010 \[\sin^m x \cos^n x \cdot dx\]
1011
1012 \paragraph{\textbf{\(m\) is odd:}}
1013 \(m=2k+1\) where \(k \in \mathbb{Z}\)\\
1014 \(\implies \sin^{2k+1} x = (\sin^2 z)^k \sin x = (1 - \cos^2 x)^k \sin x\)\\
1015 Substitute \(u=\cos x\)
1016
1017 \paragraph{\textbf{\(n\) is odd:}}
1018 \(n=2k+1\) where \(k \in \mathbb{Z}\)\\
1019 \(\implies \cos^{2k+1} x = (\cos^2 x)^k \cos x = (1-\sin^2 x)^k \cos x\)\\
1020 Substitute \(u=\sin x\)
1021
1022 \paragraph{\textbf{\(m\) and \(n\) are even:}}
1023 use identities...
1024
1025 \begin{itemize}
1026
1027 \item
1028 \(\sin^2x={1 \over 2}(1-\cos 2x)\)
1029 \item
1030 \(\cos^2x={1 \over 2}(1+\cos 2x)\)
1031 \item
1032 \(\sin 2x = 2 \sin x \cos x\)
1033 \end{itemize}
1034
1035 \subsection*{Partial fractions}
1036
1037 To factorise \(f(x) = \frac{\delta}{\alpha \cdot \beta}\):
1038 \begin{align*}
1039 \dfrac{\delta}{\alpha \cdot \beta \cdot \gamma} &= \dfrac{A}{\alpha} + \dfrac{B}{\beta} + \dfrac{C}{\gamma} \tag{1} \\
1040 \text{Multiply by } & (\alpha \cdot \beta \cdot \gamma) \text{:} \\
1041 \delta &= \beta\gamma A + \alpha\gamma B +\alpha\beta C \tag{2} \\
1042 \text{Substitute } x &= \{\alpha, \beta, \gamma\} \text{ into (2) to find denominators}
1043 \end{align*}
1044
1045 \subsubsection*{Repeated linear factors}
1046
1047 \[ \dfrac{p(x)}{(x-a)^n} = \dfrac{A_1}{(x-a)} + \dfrac{A_2}{(x-a)^2} + \dots + \dfrac{A_n}{(x-a)^n} \]
1048
1049 \subsubsection*{Irreducible quadratic factors}
1050
1051 \[ \text{e.g. } \dfrac{3x-4}{(2x-3)(x^2+5)} = \dfrac{A}{2x-3} + \dfrac{Bx+C}{x^2+5} \]
1052
1053 \begin{cas}
1054 Action \(\rightarrow\) Transformation:\\
1055 \hspace{1em} \texttt{expand(..., x)}
1056
1057 To reverse, use \texttt{combine(...)}
1058 \end{cas}
1059
1060 \subsection*{Graphing integrals on CAS}
1061
1062 \begin{cas}
1063 \textbf{In main:} Interactive \(\rightarrow\) Calculation \(\rightarrow\) \(\int\)\\
1064 Restrictions: \texttt{Define\ f(x)=..} then \texttt{f(x)\textbar{}x\textgreater{}..}
1065 \end{cas}
1066
1067 \subsection*{Applications of antidifferentiation}
1068
1069 \begin{itemize}
1070
1071 \item
1072 \(x\)-intercepts of \(y=f(x)\) identify \(x\)-coordinates of
1073 stationary points on \(y=F(x)\)
1074 \item
1075 nature of stationary points is determined by sign of \(y=f(x)\) on
1076 either side of its \(x\)-intercepts
1077 \item
1078 if \(f(x)\) is a polynomial of degree \(n\), then \(F(x)\) has degree
1079 \(n+1\)
1080 \end{itemize}
1081
1082 To find stationary points of a function, substitute \(x\) value of given
1083 point into derivative. Solve for \({\frac{dy}{dx}}=0\). Integrate to find
1084 original function.
1085
1086 \subsection*{Solids of revolution}
1087
1088 Approximate as sum of infinitesimally-thick cylinders
1089
1090 \subsubsection*{Rotation about \(\boldsymbol{x}\)-axis}
1091
1092 \[ V = \pi\int^{x=b}_{x=a} f(x)^2 \> dx \]
1093
1094 \subsubsection*{Rotation about \(\boldsymbol{y}\)-axis}
1095
1096 \begin{align*}
1097 V &= \pi \int^{y=b}_{y=a} x^2 \> dy \\
1098 &= \pi \int^{y=b}_{y=a} (f(y))^2 \> dy
1099 \end{align*}
1100
1101 \subsubsection*{Regions not bound by \(\boldsymbol{y=0}\)}
1102
1103 \[V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx\]
1104 \hfill where \(f(x) > g(x)\)
1105
1106 \subsection*{Length of a curve}
1107
1108 \[L = \int^b_a \sqrt{1 + ({\frac{dy}{dx}})^2} \> dx \quad \text{(Cartesian)}\]
1109
1110 \[L = \int^b_a \sqrt{{\frac{dx}{dt}} + ({\frac{dy}{dt}})^2} \> dt \quad \text{(parametric)}\]
1111
1112 \begin{cas}
1113 \begin{enumerate}[label=\alph*), leftmargin=5mm]
1114 \item Evaluate formula
1115 \item Interactive \(\rightarrow\) Calculation \(\rightarrow\) Line \(\rightarrow\) \texttt{arcLen}
1116 \end{enumerate}
1117 \end{cas}
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 \include{calculus-rules}
1190
1191 \section{Kinematics \& Mechanics}
1192
1193 \subsection*{Constant acceleration}
1194
1195 \begin{itemize}
1196 \item \textbf{Position} - relative to origin
1197 \item \textbf{Displacement} - relative to starting point
1198 \end{itemize}
1199
1200 \subsubsection*{Velocity-time graphs}
1201
1202 \begin{itemize}
1203 \item Displacement: \textit{signed} area between graph and \(t\) axis
1204 \item Distance travelled: \textit{total} area between graph and \(t\) axis
1205 \end{itemize}
1206
1207 \[ \text{acceleration} = \frac{d^2x}{dt^2} = \frac{dv}{dt} = v\frac{dv}{dx} = \frac{d}{dx}\left(\frac{1}{2}v^2\right) \]
1208
1209 \begin{center}
1210 \renewcommand{\arraystretch}{1}
1211 \begin{tabular}{ l r }
1212 \hline & no \\ \hline
1213 \(v=u+at\) & \(x\) \\
1214 \(v^2 = u^2+2as\) & \(t\) \\
1215 \(s = \frac{1}{2} (v+u)t\) & \(a\) \\
1216 \(s = ut + \frac{1}{2} at^2\) & \(v\) \\
1217 \(s = vt- \frac{1}{2} at^2\) & \(u\) \\ \hline
1218 \end{tabular}
1219 \end{center}
1220
1221 \[ v_{\text{avg}} = \frac{\Delta\text{position}}{\Delta t} \]
1222 \begin{align*}
1223 \text{speed} &= |{\text{velocity}}| \\
1224 &= \sqrt{v_x^2 + v_y^2 + v_z^2}
1225 \end{align*}
1226
1227 \noindent \textbf{Distance travelled between \(t=a \rightarrow t=b\):}
1228 \[= \int^b_a \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \cdot dt \]
1229
1230 \noindent \textbf{Shortest distance between \(\boldsymbol{r}(t_0)\) and \(\boldsymbol{r}(t_1)\):}
1231 \[ = |\boldsymbol{r}(t_1) - \boldsymbol{r}(t_2)| \]
1232
1233 \subsection*{Vector functions}
1234
1235 \[ \boldsymbol{r}(t) = x \boldsymbol{i} + y \boldsymbol{j} + z \boldsymbol{k} \]
1236
1237 \begin{itemize}
1238 \item If \(\boldsymbol{r}(t) \equiv\) position with time, then the graph of endpoints of \(\boldsymbol{r}(t) \equiv\) Cartesian path
1239 \item Domain of \(\boldsymbol{r}(t)\) is the range of \(x(t)\)
1240 \item Range of \(\boldsymbol{r}(t)\) is the range of \(y(t)\)
1241 \end{itemize}
1242
1243 \subsection*{Vector calculus}
1244
1245 \subsubsection*{Derivative}
1246
1247 Let \(\boldsymbol{r}(t)=x(t)\boldsymbol{i} + y(t)\boldsymbol(j)\). If both \(x(t)\) and \(y(t)\) are differentiable, then:
1248 \[ \boldsymbol{r}(t)=x(t)\boldsymbol{i}+y(t)\boldsymbol{j} \]
1249
1250 \subfile{dynamics}
1251 \subfile{statistics}
1252 \end{multicols}
1253\end{document}