785a7fdcb6da91ca2a2b82ef125294daf4e1b9a4
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73\fancypagestyle{plain}{\fancyhead[LO,LE]{} \fancyhead[CO,CE]{}} % rm title & author for first page
74\fancyhead[LO,LE]{Year 12 Specialist}
75\fancyhead[CO,CE]{Andrew Lorimer}
76
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104\begin{document}
105
106\title{\vspace{-22mm}Year 12 Specialist\vspace{-4mm}}
107\author{Andrew Lorimer}
108\date{}
109\maketitle
110\vspace{-9mm}
111\begin{multicols}{2}
112
113 \section{Complex numbers}
114
115 \[\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}\]
116 \begin{align*}
117 \text{Cartesian form: } & a+bi\\
118 \text{Polar form: } & r\operatorname{cis}\theta
119 \end{align*}
120
121 \subsection*{Operations}
122
123 \begin{tabularx}{\columnwidth}{|r|X|X|}
124 \hline
125 \rowcolor{cas}
126 & \textbf{Cartesian} & \textbf{Polar} \\
127 \hline
128 \(z_1 \pm z_2\) & \((a \pm c)(b \pm d)i\) & convert to \(a+bi\)\\
129 \hline
130 \(+k \times z\) & \multirow{2}{*}{\(ka \pm kbi\)} & \(kr\operatorname{cis} \theta\)\\
131 \cline{1-1}\cline{3-3}
132 \(-k \times z\) & & \(kr \operatorname{cis}(\theta\pm \pi)\)\\
133 \hline
134 \(z_1 \cdot z_2\) & \(ac-bd+(ad+bc)i\) & \(r_1r_2 \operatorname{cis}(\theta_1 + \theta_2)\)\\
135 \hline
136 \(z_1 \div z_2\) & \((z_1 \overline{z_2}) \div |z_2|^2\) & \(\left(\frac{r_1}{r_2}\right) \operatorname{cis}(\theta_1 - \theta_2)\) \\
137 \hline
138 \end{tabularx}
139
140 \subsubsection*{Scalar multiplication in polar form}
141
142 For \(k \in \mathbb{R}^+\):
143 \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\theta\]
144
145 \noindent For \(k \in \mathbb{R}^-\):
146 \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\left(\begin{cases}\theta - \pi & |0<\operatorname{Arg}(z)\le \pi \\ \theta + \pi & |-\pi<\operatorname{Arg}(z)\le 0\end{cases}\right)\]
147
148 \subsection*{Conjugate}
149 \vspace{-7mm} \hfill \colorbox{cas}{\texttt{conjg(a+bi)}}
150 \begin{align*}
151 \overline{z} &= a \mp bi\\
152 &= r \operatorname{cis}(-\theta)
153 \end{align*}
154
155 \subsubsection*{Properties}
156
157 \begin{align*}
158 \overline{z_1 \pm z_2} &= \overline{z_1}\pm\overline{z_2}\\
159 \overline{z_1 \cdot z_2} &= \overline{z_1}\cdot\overline{z_2}\\
160 \overline{kz} &= k\overline{z} \> \forall \> k \in \mathbb{R}\\
161 z\overline{z} &= (a+bi)(a-bi)\\
162 &= a^2 + b^2\\
163 &= |z|^2
164 \end{align*}
165
166 \subsection*{Modulus}
167
168 \[|z|=|\vec{Oz}|=\sqrt{a^2 + b^2}\]
169
170 \subsubsection*{Properties}
171
172 \begin{align*}
173 |z_1z_2|&=|z_1||z_2|\\
174 \left|\frac{z_1}{z_2}\right|&=\frac{|z_1|}{|z_2|}\\
175 |z_1+z_2|&\le|z_1|+|z_2|
176 \end{align*}
177
178 \subsection*{Multiplicative inverse}
179
180 \begin{align*}
181 z^{-1}&=\frac{a-bi}{a^2+b^2}\\
182 &=\frac{\overline{z}}{|z|^2}a\\
183 &=r \operatorname{cis}(-\theta)
184 \end{align*}
185
186 \subsection*{Dividing over \(\mathbb{C}\)}
187
188 \begin{align*}
189 \frac{z_1}{z_2}&=z_1z_2^{-1}\\
190 &=\frac{z_1\overline{z_2}}{|z_2|^2}\\
191 &=\frac{(a+bi)(c-di)}{c^2+d^2}\\
192 & \text{then rationalise denominator}
193 \end{align*}
194
195 \subsection*{Polar form}
196
197 \[ r \operatorname{cis} \theta = r\left( \cos \theta + i \sin \theta \right) \]
198
199 \begin{itemize}
200 \item{\(r=|z|=\sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}\)}
201 \item{\(\theta = \operatorname{arg}(z)\) \hfill \colorbox{cas}{\texttt{arg(a+bi)}}}
202 \item{\(\operatorname{Arg}(z) \in (-\pi,\pi)\) \quad \bf{(principal argument)}}
203 \item{Multiple representations:\\\(r\operatorname{cis}\theta=r\operatorname{cis}(\theta+2n\pi)\) with \(n \in \mathbb{Z}\) revolutions}
204 \item{\(\operatorname{cis}\pi=-1,\qquad \operatorname{cis}0=1\)}
205 \end{itemize}
206
207 \begin{cas}
208 \-\hspace{1em}\verb|compToTrig(a+bi)| \(\iff\) \verb|cExpand{r·cisX}|
209 \end{cas}
210
211 \subsection*{de Moivres' theorem}
212
213 \begin{theorembox}{}
214 \[(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}\]
215 \end{theorembox}
216
217 \subsection*{Complex polynomials}
218
219 Include \(\pm\) for all solutions, incl. imaginary
220
221 \begin{tabularx}{\columnwidth}{ R{0.55} X }
222 \hline
223 Sum of squares & \(\begin{aligned}
224 z^2 + a^2 &= z^2-(ai)^2\\
225 &= (z+ai)(z-ai) \end{aligned}\) \\
226 \hline
227 Sum of cubes & \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)\\
228 \hline
229 Division & \(P(z)=D(z)Q(z)+R(z)\) \\
230 \hline
231 Remainder theorem & Let \(\alpha \in \mathbb{C}\). Remainder of \(P(z) \div (z-\alpha)\) is \(P(\alpha)\)\\
232 \hline
233 Factor theorem & \(z-\alpha\) is a factor of \(P(z) \iff P(\alpha)=0\) for \(\alpha \in \mathbb{C}\)\\
234 \hline
235 Conjugate root theorem & \(P(z)=0 \text{ at } z=a\pm bi\) (\(\implies\) both \(z_1\) and \(\overline{z_1}\) are solutions)\\
236 \hline
237 \end{tabularx}
238
239 \subsection*{\(n\)th roots}
240
241 \(n\)th roots of \(z=r\operatorname{cis}\theta\) are:
242
243 \[z = r^{\frac{1}{n}} \operatorname{cis}\left(\frac{\theta+2k\pi}{n}\right)\]
244
245 \begin{itemize}
246
247 \item{Same modulus for all solutions}
248 \item{Arguments separated by \(\frac{2\pi}{n} \therefore\) there are \(n\) roots}
249 \item{If one square root is \(a+bi\), the other is \(-a-bi\)}
250 \item{Give one implicit \(n\)th root \(z_1\), function is \(z=z_1^n\)}
251 \item{Solutions of \(z^n=a\) where \(a \in \mathbb{C}\) lie on the circle \(x^2+y^2=\left(|a|^{\frac{1}{n}}\right)^2\) \quad (intervals of \(\frac{2\pi}{n}\))}
252 \end{itemize}
253
254 \noindent For \(0=az^2+bz+c\), use quadratic formula:
255
256 \[z=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]
257
258 \subsection*{Fundamental theorem of algebra}
259
260 A polynomial of degree \(n\) can be factorised into \(n\) linear factors in \(\mathbb{C}\):
261
262 \[\implies P(z)=a_n(z-\alpha_1)(z-\alpha_2)(z-\alpha_3)\dots(z-\alpha_n)\]
263 \[\text{ where } \alpha_1,\alpha_2,\alpha_3,\dots,\alpha_n \in \mathbb{C}\]
264
265 \subsection*{Argand planes}
266
267 \begin{center}\begin{tikzpicture}[scale=2]
268 \draw [->] (-0.2,0) -- (1.5,0) node [right] {$\operatorname{Re}(z)$};
269 \draw [->] (0,-0.2) -- (0,1.5) node [above] {$\operatorname{Im}(z)$};
270 \coordinate (P) at (1,1);
271 \coordinate (a) at (1,0);
272 \coordinate (b) at (0,1);
273 \coordinate (O) at (0,0);
274 \draw (0,0) -- (P) node[pos=0.5, above left]{\(r\)} node[pos=1, right]{\(\begin{aligned}z&=a+bi\\&=r\operatorname{cis}\theta\end{aligned}\)};
275 \draw [gray, dashed] (1,1) -- (1,0) node[black, pos=1, below]{\(a\)};
276 \draw [gray, dashed] (1,1) -- (0,1) node[black, pos=1, left]{\(b\)};
277 \begin{scope}
278 \path[clip] (O) -- (P) -- (a);
279 \fill[red, opacity=0.5, draw=black] (O) circle (2mm);
280 \node at ($(O)+(20:3mm)$) {$\theta$};
281 \end{scope}
282 \filldraw (P) circle (0.5pt);
283 \end{tikzpicture}\end{center}
284
285 \begin{itemize}
286 \item{Multiplication by \(i \implies\) CCW rotation of \(\frac{\pi}{2}\)}
287 \item{Addition: \(z_1 + z_2 \equiv\) \overrightharp{\(Oz_1\)} + \overrightharp{\(Oz_2\)}}
288 \end{itemize}
289
290 \subsection*{Sketching complex graphs}
291
292 \subsubsection*{Linear}
293
294 \begin{itemize}
295 \item{\(\operatorname{Re}(z)=c\) or \(\operatorname{Im}(z)=c\) (perpendicular bisector)}
296 \item{\(\operatorname{Im}(z)=m\operatorname{Re}(z)\)}
297 \item{\(|z+a|=|z+b| \implies 2(a-b)x=b^2-a^2\)\\Geometric: equidistant from \(a,b\)}
298 \end{itemize}
299
300 \subsubsection*{Circles}
301
302 \begin{itemize}
303 \item \(|z-z_1|^2=c^2|z_2+2|^2\)
304 \item \(|z-(a+bi)|=c \implies (x-a)^2+_(y-b)^2=c^2\)
305 \end{itemize}
306
307 \noindent \textbf{Loci} \qquad \(\operatorname{Arg}(z)<\theta\)
308
309 \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
310 \draw [->] (0,0) -- (1,0) node [right] {$\operatorname{Re}(z)$};
311 \draw [->] (0,-0.5) -- (0,1) node [above] {$\operatorname{Im}(z)$};
312 \draw [<-, dashed, thick, blue] (-1,0) -- (0,0);
313 \draw [->, thick, blue] (0,0) -- (1,1);
314 \fill [gray, opacity=0.2, domain=-1:1, variable=\x] (-1,-0.5) -- (-1,0) -- (0, 0) -- (1,1) -- (1,-0.5) -- cycle;
315 \begin{scope}
316 \path[clip] (0,0) -- (1,1) -- (1,0);
317 \fill[red, opacity=0.5, draw=black] (0,0) circle (2mm);
318 \node at ($(0,0)+(20:3mm)$) {$\frac{\pi}{4}$};
319 \end{scope}
320 \node [font=\footnotesize] at (0.5,-0.25) {\(\operatorname{Arg}(z)\le\frac{\pi}{4}\)};
321 \node [blue, mydot] {};
322 \end{tikzpicture}\end{center}
323
324 \noindent \textbf{Rays} \qquad \(\operatorname{Arg}(z-b)=\theta\)
325
326 \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
327 \draw [->] (-0.75,0) -- (1.5,0) node [right] {$\operatorname{Re}(z)$};
328 \draw [->] (0,-1) -- (0,1) node [above] {$\operatorname{Im}(z)$};
329 \draw [->, thick, brown] (-0.25,0) -- (-0.75,-1);
330 \node [above, font=\footnotesize] at (-0.25,0) {\(\frac{1}{4}\)};
331 \begin{scope}
332 \path[clip] (-0.25,0) -- (-0.75,-1) -- (0,0);
333 \fill[orange, opacity=0.5, draw=black] (-0.25,0) circle (2mm);
334 \end{scope}
335 \node at (-0.08,-0.3) {\(\frac{\pi}{8}\)};
336 \node [font=\footnotesize, left] at (-0.75,-1) {\(\operatorname{Arg}(z+\frac{1}{4})=\frac{\pi}{8}\)};
337 \node [brown, mydot] at (-0.25,0) {};
338 \draw [<->, thick, green] (0,-1) -- (1.5,0.5) node [pos=0.25, black, font=\footnotesize, right] {\(|z-2|=|z-(1+i)|\)};
339 \node [left, font=\footnotesize] at (0,-1) {\(-1\)};
340 \node [below, font=\footnotesize] at (1,0) {\(1\)};
341 \end{tikzpicture}\end{center}
342
343 \section{Vectors}
344 \begin{center}\begin{tikzpicture}
345 \draw [->] (-0.5,0) -- (3,0) node [right] {\(x\)};
346 \draw [->] (0,-0.5) -- (0,3) node [above] {\(y\)};
347 \draw [orange, ->, thick] (0.5,0.5) -- (2.5,2.5) node [pos=0.5, above] {\(\vec{u}\)};
348 \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
349 \draw [gray, dashed, thick] (0.5,0.5) -- (2.5,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, below]{\(x\vec{i}\)};
350 \draw [gray, dashed, thick] (2.5,0.5) -- (2.5,2.5) node [pos=0.5] {\midarrow};
351 \end{scope}
352 \node[black, right] at (2.5,1.5) {\(y\vec{j}\)};
353 \end{tikzpicture}\end{center}
354 \subsection*{Column notation}
355
356 \[\begin{bmatrix}x\\ y \end{bmatrix} \iff x\boldsymbol{i} + y\boldsymbol{j}\]
357 \(\begin{bmatrix}x_2-x_1\\ y_2-y_1 \end{bmatrix}\) \quad between \(A(x_1,y_1), \> B(x_2,y_2)\)
358
359 \subsection*{Scalar multiplication}
360
361 \[k\cdot (x\boldsymbol{i}+y\boldsymbol{j})=kx\boldsymbol{i}+ky\boldsymbol{j}\]
362
363 \noindent For \(k \in \mathbb{R}^-\), direction is reversed
364
365 \subsection*{Vector addition}
366 \begin{center}\begin{tikzpicture}[scale=1]
367 \coordinate (A) at (0,0);
368 \coordinate (B) at (2,2);
369 \draw [->, thick, red] (0,0) -- (2,2) node [pos=0.5, below right] {\(\vec{u}=2\vec{i}+2\vec{j}\)};
370 \draw [->, thick, blue] (2,2) -- (1,4) node [pos=0.5, above right] {\(\vec{v}=-\vec{i}+2\vec{j}\)};
371 \draw [->, thick, orange] (0,0) -- (1,4) node [pos=0.5, left] {\(\vec{u}+\vec{v}=\vec{i}+4\vec{j}\)};
372 \end{tikzpicture}\end{center}
373
374 \[(x\boldsymbol{i}+y\boldsymbol{j}) \pm (a\boldsymbol{i}+b\boldsymbol{j})=(x \pm a)\boldsymbol{i}+(y \pm b)\boldsymbol{j}\]
375
376 \begin{itemize}
377 \item Draw each vector head to tail then join lines
378 \item Addition is commutative (parallelogram)
379 \item \(\boldsymbol{u}-\boldsymbol{v}=\boldsymbol{u}+(-\boldsymbol{v}) \implies \overrightharp{AB}=\boldsymbol{b}-\boldsymbol{a}\)
380 \end{itemize}
381
382 \subsection*{Magnitude}
383
384 \[|(x\boldsymbol{i} + y\boldsymbol{j})|=\sqrt{x^2+y^2}\]
385
386 \subsection*{Parallel vectors}
387
388 \[\boldsymbol{u} || \boldsymbol{v} \iff \boldsymbol{u} = k \boldsymbol{v} \text{ where } k \in \mathbb{R} \setminus \{0\}\]
389
390 For parallel vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\):\\
391 \[\boldsymbol{a \cdot b}=\begin{cases}
392 |\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction}\\
393 -|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions}
394 \end{cases}\]
395 %\includegraphics[width=0.2,height=\textheight]{graphics/parallelogram-vectors.jpg}
396 %\includegraphics[width=1]{graphics/vector-subtraction.jpg}
397
398 \subsection*{Perpendicular vectors}
399
400 \[\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b} = 0\ \quad \text{(since \(\cos 90 = 0\))}\]
401
402 \subsection*{Unit vector \(|\hat{\boldsymbol{a}}|=1\)}
403 \[\begin{split}\hat{\boldsymbol{a}} & = {\frac{1}{|\boldsymbol{a}|}}\boldsymbol{a} \\ & = \boldsymbol{a} \cdot {|\boldsymbol{a}|}\end{split}\]
404
405 \subsection*{Scalar product \(\boldsymbol{a} \cdot \boldsymbol{b}\)}
406
407
408 \begin{center}\begin{tikzpicture}[scale=2]
409 \draw [->] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{b}\)};
410 \draw [->] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{a}\)};
411 \begin{scope}
412 \path[clip] (1,0.5) -- (1,0) -- (0,0);
413 \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
414 \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
415 \end{scope}
416 \end{tikzpicture}\end{center}
417 \begin{align*}\boldsymbol{a} \cdot \boldsymbol{b} &= a_1 b_1 + a_2 b_2 \\ &= |\boldsymbol{a}| |\boldsymbol{b}| \cos \theta \\ &\quad (\> 0 \le \theta \le \pi) \text{ - from cosine rule}\end{align*}
418 \noindent\colorbox{cas}{On CAS: \texttt{dotP({[}a\ b\ c{]},\ {[}d\ e\ f{]})}}
419
420 \subsubsection*{Properties}
421
422 \begin{enumerate}
423 \item
424 \(k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k\boldsymbol{b})\)
425 \item
426 \(\boldsymbol{a \cdot 0}=0\)
427 \item
428 \(\boldsymbol{a} \cdot (\boldsymbol{b} + \boldsymbol{c})=\boldsymbol{a} \cdot \boldsymbol{b} + \boldsymbol{a} \cdot \boldsymbol{c}\)
429 \item
430 \(\boldsymbol{i \cdot i} = \boldsymbol{j \cdot j} = \boldsymbol{k \cdot k}= 1\)
431 \item
432 \(\boldsymbol{a} \cdot \boldsymbol{b} = 0 \quad \implies \quad \boldsymbol{a} \perp \boldsymbol{b}\)
433 \item
434 \(\boldsymbol{a \cdot a} = |\boldsymbol{a}|^2 = a^2\)
435 \end{enumerate}
436
437 \subsection*{Angle between vectors}
438
439 \[\cos \theta = \frac{\boldsymbol{a} \cdot \boldsymbol{b}}{|\boldsymbol{a}| |\boldsymbol{b}|} = \frac{a_1 b_1 + a_2 b_2}{|\boldsymbol{a}| |\boldsymbol{b}|}\]
440
441 \noindent \colorbox{cas}{On CAS:} \texttt{angle([a b c], [a b c])}
442
443 (Action \(\rightarrow\) Vector \(\rightarrow\)Angle)
444
445 \subsection*{Angle between vector and axis}
446
447 \noindent For\(\boldsymbol{a} = a_1 \boldsymbol{i} + a_2 \boldsymbol{j} + a_3 \boldsymbol{k}\)
448 which makes angles \(\alpha, \beta, \gamma\) with positive side of
449 \(x, y, z\) axes:
450 \[\cos \alpha = \frac{a_1}{|\boldsymbol{a}|}, \quad \cos \beta = \frac{a_2}{|\boldsymbol{a}|}, \quad \cos \gamma = \frac{a_3}{|\boldsymbol{a}|}\]
451
452 \noindent \colorbox{cas}{On CAS:} \texttt{angle({[}a\ b\ c{]},\ {[}1\ 0\ 0{]})}\\for angle
453 between \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) and
454 \(x\)-axis
455
456 \subsection*{Projections \& resolutes}
457
458 \begin{tikzpicture}[scale=3]
459 \draw [->, purple] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{a}\)};
460 \draw [->, orange] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{u}\)};
461 \draw [->, blue] (1,0) -- (2,0) node [pos=0.5, below] {\(\boldsymbol{b}\)};
462 \begin{scope}
463 \path[clip] (1,0.5) -- (1,0) -- (0,0);
464 \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
465 \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
466 \end{scope}
467 \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
468 \draw [gray, dashed, thick] (1,0) -- (1,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, right, rotate=-90]{\(\boldsymbol{w}\)};
469 \end{scope}
470 \draw (0,0) coordinate (O)
471 (1,0) coordinate (A)
472 (1,0.5) coordinate (B)
473 pic [draw,red,angle radius=2mm] {right angle = O--A--B};
474 \end{tikzpicture}
475
476 \subsubsection*{\(\parallel\boldsymbol{b}\) (vector projection/resolute)}
477
478 \begin{align*}
479 \boldsymbol{u} & = \frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|^2}\boldsymbol{b} \\
480 & = \left(\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\right)\left(\frac{\boldsymbol{b}}{|\boldsymbol{b}|}\right) \\
481 & = (\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}
482 \end{align*}
483
484 \subsubsection*{\(\perp\boldsymbol{b}\) (perpendicular projection)}
485 \[\boldsymbol{w} = \boldsymbol{a} - \boldsymbol{u}\]
486
487 \subsubsection*{\(|\boldsymbol{u}|\) (scalar projection/resolute)}
488 \begin{align*}
489 s &= |\boldsymbol{u}|\\
490 &= \boldsymbol{a} \cdot \hat{\boldsymbol{b}}\\
491 &=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\\
492 &= |\boldsymbol{a}| \cos \theta
493 \end{align*}
494
495 \subsubsection*{Rectangular (\(\parallel,\perp\)) components}
496
497 \[\boldsymbol{a}=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}+\left(\boldsymbol{a}-\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}\right)\]
498
499
500 \subsection*{Vector proofs}
501
502 \textbf{Concurrent:} intersection of \(\ge\) 3 lines
503
504 \begin{tikzpicture}
505 \draw [blue] (0,0) -- (1,1);
506 \draw [red] (1,0) -- (0,1);
507 \draw [brown] (0.4,0) -- (0.6,1);
508 \filldraw (0.5,0.5) circle (2pt);
509 \end{tikzpicture}
510
511 \subsubsection*{Collinear points}
512
513 \(\ge\) 3 points lie on the same line
514
515 \begin{tikzpicture}
516 \draw [purple] (0,0) -- (4,1);
517 \filldraw (2,0.5) circle (2pt) node [above] {\(C\)};
518 \filldraw (1,0.25) circle (2pt) node [above] {\(A\)};
519 \filldraw (3,0.75) circle (2pt) node [above] {\(B\)};
520 \coordinate (O) at (2.8,-0.2);
521 \node at (O) [below] {\(O\)};
522 \begin{scope}[->, orange, thick]
523 \draw (O) -- (2,0.5) node [pos=0.5, above, font=\footnotesize, black] {\(\boldsymbol{c}\)};
524 \draw (O) -- (1,0.25) node [pos=0.5, below, font=\footnotesize, black] {\(\boldsymbol{a}\)};
525 \draw (O) -- (3,0.75) node [pos=0.5, right, font=\footnotesize, black] {\(\boldsymbol{b}\)};
526 \end{scope}
527 \end{tikzpicture}
528
529 \begin{align*}
530 \text{e.g. Prove that}\\
531 \overrightharp{AC}=m\overrightharp{AB} \iff \boldsymbol{c}&=(1-m)\boldsymbol{a}+m\boldsymbol{b}\\
532 \implies \boldsymbol{c} &= \overrightharp{OA} + \overrightharp{AC}\\
533 &= \overrightharp{OA} + m\overrightharp{AB}\\
534 &=\boldsymbol{a}+m(\boldsymbol{b}-\boldsymbol{a})\\
535 &=\boldsymbol{a}+m\boldsymbol{b}-m\boldsymbol{a}\\
536 &=(1-m)\boldsymbol{a}+m{b}
537 \end{align*}
538 \begin{align*}
539 \text{Also, } \implies \overrightharp{OC} &= \lambda \vec{OA} + \mu \overrightharp{OB} \\
540 \text{where } \lambda + \mu &= 1\\
541 \text{If } C \text{ lies along } \overrightharp{AB}, & \implies 0 < \mu < 1
542 \end{align*}
543
544
545 \subsubsection*{Parallelograms}
546
547 \begin{center}\begin{tikzpicture}
548 \coordinate (O) at (0,0) node [below left] {\(O\)};
549 \coordinate (A) at (4,0);
550 \coordinate (B) at (6,2);
551 \coordinate (C) at (2,2);
552 \coordinate (D) at (6,0);
553
554 \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (O)--(A) node [below left] {\(A\)};
555 \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (A)--(B) node [above right] {\(B\)};
556 \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (B)--(C) node [above left] {\(C\)};
557 \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (C)--(O);
558
559 \draw [gray, dashed] (O) -- (B) node [pos=0.75] {\(\diagdown\diagdown\)} node [pos=0.25] {\(\diagdown\diagdown\)};
560 \draw [gray, dashed] (A) -- (C) node [pos=0.75] {\(\diagup\)} node [pos=0.25] {\(\diagup\)};
561 \begin{scope}
562 \path[clip] (C) -- (A) -- (O);
563 \fill[orange, opacity=0.5, draw=black] (0,0) circle (4mm);
564 \node at ($(0,0)+(20:8mm)$) {\(\theta\)};
565 \end{scope}
566 \draw [gray, thick, dotted] (B) -- (D) node [pos=0.5, right, black, font=\footnotesize] {\(|\boldsymbol{c}|\sin\theta\)} (A) -- (D) node [pos=0.5, below, black, font=\footnotesize] {\(|\boldsymbol{c}|\cos\theta\)};
567 \draw pic [draw,thick,red,angle radius=2mm] {right angle=O--D--B};
568 \end{tikzpicture}\end{center}
569
570 \begin{itemize}
571 \item
572 Diagonals \(\overrightharp{OB}, \overrightharp{AC}\) bisect each other
573 \item
574 If diagonals are equal length, it is a rectangle
575 \item
576 \(|\overrightharp{OB}|^2 + |\overrightharp{CA}|^2 = |\overrightharp{OA}|^2 + |\overrightharp{AB}|^2 + |\overrightharp{CB}|^2 + |\overrightharp{OC}|^2\)
577 \item
578 Area \(=\boldsymbol{c} \cdot \boldsymbol{a}\)
579 \end{itemize}
580
581 \subsubsection*{Useful vector properties}
582
583 \begin{itemize}
584 \item
585 \(\boldsymbol{a} \parallel \boldsymbol{b} \implies \boldsymbol{b}=k\boldsymbol{a}\) for some
586 \(k \in \mathbb{R} \setminus \{0\}\)
587 \item
588 If \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are parallel with at
589 least one point in common, then they lie on the same straight line
590 \item
591 \(\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b}=0\)
592 \item
593 \(\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2\)
594 \end{itemize}
595
596 \subsection*{Linear dependence}
597
598 \(\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\) are linearly dependent if they are \(\nparallel\) and:
599 \begin{align*}
600 0&=k\boldsymbol{a}+l\boldsymbol{b}+m\boldsymbol{c}\\
601 \therefore \boldsymbol{c} &= m\boldsymbol{a} + n\boldsymbol{b} \quad \text{(simultaneous)}
602 \end{align*}
603
604 \noindent \(\boldsymbol{a}, \boldsymbol{b},\) and \(\boldsymbol{c}\) are linearly
605 independent if no vector in the set is expressible as a linear
606 combination of other vectors in set, or if they are parallel.
607
608 \subsection*{Three-dimensional vectors}
609
610 Right-hand rule for axes: \(z\) is up or out of page.
611
612 \tdplotsetmaincoords{60}{120}
613 \begin{center}\begin{tikzpicture} [scale=3, tdplot_main_coords, axis/.style={->,thick},
614 vector/.style={-stealth,red,very thick},
615 vector guide/.style={dashed,gray,thick}]
616
617 %standard tikz coordinate definition using x, y, z coords
618 \coordinate (O) at (0,0,0);
619
620 %tikz-3dplot coordinate definition using x, y, z coords
621
622 \pgfmathsetmacro{\ax}{1}
623 \pgfmathsetmacro{\ay}{1}
624 \pgfmathsetmacro{\az}{1}
625
626 \coordinate (P) at (\ax,\ay,\az);
627
628 %draw axes
629 \draw[axis] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$};
630 \draw[axis] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$};
631 \draw[axis] (0,0,0) -- (0,0,1) node[anchor=south]{$z$};
632
633 %draw a vector from O to P
634 \draw[vector] (O) -- (P);
635
636 %draw guide lines to components
637 \draw[vector guide] (O) -- (\ax,\ay,0);
638 \draw[vector guide] (\ax,\ay,0) -- (P);
639 \draw[vector guide] (P) -- (0,0,\az);
640 \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
641 \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
642 \draw[vector guide] (\ax,\ay,0) -- (\ax,0,0);
643 \node[tdplot_main_coords,above right]
644 at (\ax,\ay,\az){(\ax, \ay, \az)};
645 \end{tikzpicture}\end{center}
646
647 \subsection*{Parametric vectors}
648
649 Parametric equation of line through point \((x_0, y_0, z_0)\) and
650 parallel to \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) is:
651
652 \[\begin{cases}x = x_o + a \cdot t \\ y = y_0 + b \cdot t \\ z = z_0 + c \cdot t\end{cases}\]
653
654 \section{Circular functions}
655
656 \(\sin(bx)\) or \(\cos(bx)\): period \(=\frac{2\pi}{b}\)
657
658 \noindent \(\tan(nx)\): period \(=\frac{\pi}{n}\)\\
659 \indent\indent\indent asymptotes at \(x=\frac{(2k+1)\pi}{2n} \> \vert \> k \in \mathbb{Z}\)
660
661 \subsection*{Reciprocal functions}
662
663 \subsubsection*{Cosecant}
664
665 \[\operatorname{cosec} \theta = \frac{1}{\sin \theta} \> \vert \> \sin \theta \ne 0\]
666
667 \begin{itemize}
668 \item
669 \textbf{Domain} \(= \mathbb{R} \setminus {n\pi : n \in \mathbb{Z}}\)
670 \item
671 \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
672 \item
673 \textbf{Turning points} at
674 \(\theta = \frac{(2n + 1)\pi}{2} \> \vert \> n \in \mathbb{Z}\)
675 \item
676 \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
677 \end{itemize}
678
679 \subsubsection*{Secant}
680
681\begin{tikzpicture}
682 \begin{axis}[ytick={-1,1}, yticklabels={\(-1\), \(1\)}, xmin=-7,xmax=7,ymin=-3,ymax=3,enlargelimits=true, xtick={-6.2830, -3.1415, 3.1415, 6.2830},xticklabels={\(-2\pi\), \(-\pi\), \(\pi\), \(2\pi\)}]
683% \addplot[blue, domain=-6.2830:6.2830,unbounded coords=jump,samples=80] {sec(deg(x))};
684 \addplot[blue, restrict y to domain=-10:10, domain=-7:7,samples=100] {sec(deg(x))} node [pos=0.93, black, right] {\(\operatorname{sec} x\)};
685 \addplot[red, dashed, domain=-7:7,samples=100] {cos(deg(x))};
686 \draw [gray, dotted, thick] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{rel axis cs:0,1});
687 \draw [gray, dotted, thick] ({axis cs:4.71239,0}|-{rel axis cs:0,0}) -- ({axis cs:4.71239,0}|-{rel axis cs:0,1});
688 \draw [gray, dotted, thick] ({axis cs:-4.71239,0}|-{rel axis cs:0,0}) -- ({axis cs:-4.71239,0}|-{rel axis cs:0,1});
689 \draw [gray, dotted, thick] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1});
690\end{axis}
691 \node [black] at (7,3.5) {\(\cos x\)};
692\end{tikzpicture}
693
694 \[\operatorname{sec} \theta = \frac{1}{\cos \theta} \> \vert \> \cos \theta \ne 0\]
695
696 \begin{itemize}
697
698 \item
699 \textbf{Domain}
700 \(= \mathbb{R} \setminus \frac{(2n + 1) \pi}{2} : n \in \mathbb{Z}\}\)
701 \item
702 \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
703 \item
704 \textbf{Turning points} at
705 \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
706 \item
707 \textbf{Asymptotes} at
708 \(\theta = \frac{(2n + 1) \pi}{2} \> \vert \> n \in \mathbb{Z}\)
709 \end{itemize}
710
711 \subsubsection*{Cotangent}
712
713\begin{tikzpicture}
714 \begin{axis}[xmin=-3,xmax=3,ymin=-1.5,ymax=1.5,enlargelimits=true, xtick={-3.1415, -1.5708, 1.5708, 3.1415},xticklabels={\(-\pi\), \(-\frac{\pi}{2}\), \(\frac{\pi}{2}\), \(\pi\)}]
715 \addplot[blue, smooth, domain=-3:-0.1,unbounded coords=jump,samples=105] {cot(deg(x))} node [pos=0.3, left] {\(\operatorname{cot} x\)};
716\addplot[blue, smooth, domain=0.1:3,unbounded coords=jump,samples=105] {cot(deg(x))};
717\addplot[red, smooth, dashed] gnuplot [domain=-1.5:1.5,unbounded coords=jump,samples=105] {tan(x)};
718\addplot[red, smooth, dashed] gnuplot [domain=-3.5:-1.8,unbounded coords=jump,samples=105] {tan(x)} node [pos=0.5, right] {\(\tan x\)};
719\addplot[red, smooth, dashed] gnuplot [domain=1.8:3.5,unbounded coords=jump,samples=105] {tan(x)};
720 \draw [thick, red, dotted] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1});
721 \draw [thick, blue, dotted] ({axis cs:-3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:-3.1415,0}|-{rel axis cs:0,1});
722 \draw [thick, blue, dotted] ({axis cs:0,0}|-{rel axis cs:0,0}) -- ({axis cs:0,0}|-{rel axis cs:0,1});
723 \draw [thick, blue, dotted] ({axis cs:3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:3.1415,0}|-{rel axis cs:0,1});
724 \draw [thick, red, dotted] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{rel axis cs:0,1});
725\end{axis}
726\end{tikzpicture}
727
728 \[\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0\]
729
730 \begin{itemize}
731
732 \item
733 \textbf{Domain} \(= \mathbb{R} \setminus \{n \pi: n \in \mathbb{Z}\}\)
734 \item
735 \textbf{Range} \(= \mathbb{R}\)
736 \item
737 \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
738 \end{itemize}
739
740 \subsubsection*{Symmetry properties}
741
742 \[\begin{split}
743 \operatorname{sec} (\pi \pm x) & = -\operatorname{sec} x \\
744 \operatorname{sec} (-x) & = \operatorname{sec} x \\
745 \operatorname{cosec} (\pi \pm x) & = \mp \operatorname{cosec} x \\
746 \operatorname{cosec} (-x) & = - \operatorname{cosec} x \\
747 \operatorname{cot} (\pi \pm x) & = \pm \operatorname{cot} x \\
748 \operatorname{cot} (-x) & = - \operatorname{cot} x
749 \end{split}\]
750
751 \subsubsection*{Complementary properties}
752
753 \[\begin{split}
754 \operatorname{sec} \left({\pi \over 2} - x\right) & = \operatorname{cosec} x \\
755 \operatorname{cosec} \left({\pi \over 2} - x\right) & = \operatorname{sec} x \\
756 \operatorname{cot} \left({\pi \over 2} - x\right) & = \tan x \\
757 \tan \left({\pi \over 2} - x\right) & = \operatorname{cot} x
758 \end{split}\]
759
760 \subsubsection*{Pythagorean identities}
761
762 \[\begin{split}
763 1 + \operatorname{cot}^2 x & = \operatorname{cosec}^2 x, \quad \text{where } \sin x \ne 0 \\
764 1 + \tan^2 x & = \operatorname{sec}^2 x, \quad \text{where } \cos x \ne 0
765 \end{split}\]
766
767 \subsection*{Compound angle formulas}
768
769 \[\cos(x \pm y) = \cos x + \cos y \mp \sin x \sin y\]
770 \[\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y\]
771 \[\tan(x \pm y) = {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}\]
772
773 \subsection*{Double angle formulas}
774
775 \[\begin{split}
776 \cos 2x &= \cos^2 x - \sin^2 x \\
777 & = 1 - 2\sin^2 x \\
778 & = 2 \cos^2 x -1
779 \end{split}\]
780
781 \[\sin 2x = 2 \sin x \cos x\]
782
783 \[\tan 2x = {{2 \tan x} \over {1 - \tan^2 x}}\]
784
785 \subsection*{Inverse circular functions}
786
787 \begin{tikzpicture}
788 \begin{axis}[ymin=-2, ymax=4, xmin=-1.1, xmax=1.1, ytick={-1.5708, 1.5708, 3.14159},yticklabels={$-\frac{\pi}{2}$, $\frac{\pi}{2}$, $\pi$}]
789 \addplot[color=red, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {asin(x)} node [pos=0.25, below right] {\(\sin^{-1}x\)};
790 \addplot[color=blue, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {acos(x)} node [pos=0.25, below left] {\(\cos^{-1}x\)};
791 \addplot[mark=*, red] coordinates {(-1,-1.5708)} node[right, font=\footnotesize]{\((-1,-\frac{\pi}{2})\)} ;
792 \addplot[mark=*, red] coordinates {(1,1.5708)} node[left, font=\footnotesize]{\((1,\frac{\pi}{2})\)} ;
793 \addplot[mark=*, blue] coordinates {(1,0)};
794 \addplot[mark=*, blue] coordinates {(-1,3.1415)} node[right, font=\footnotesize]{\((-1,\pi)\)} ;
795 \end{axis}
796 \end{tikzpicture}\\
797
798 Inverse functions: \(f(f^{-1}(x)) = x\) (restrict domain)
799
800 \[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y\]
801 \hfill where \(\sin y = x, \> y \in [{-\pi \over 2}, {\pi \over 2}]\)
802
803 \[\cos^{-1}: [-1,1] \rightarrow \mathbb{R}, \quad \cos^{-1} x = y\]
804 \hfill where \(\cos y = x, \> y \in [0, \pi]\)
805
806 \[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\]
807 \hfill where \(\tan y = x, \> y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\)
808
809 \begin{tikzpicture}
810 \begin{axis}[yticklabel style={yshift=1.0pt, anchor=north east},x=0.1cm, y=1cm, ymax=2, ymin=-2, xticklabels={}, ytick={-1.5708,1.5708},yticklabels={\(-\frac{\pi}{2}\),\(\frac{\pi}{2}\)}]
811 \addplot[color=orange, smooth] gnuplot [domain=-35:35, unbounded coords=jump,samples=350] {atan(x)} node [pos=0.5, above left] {\(\tan^{-1}x\)};
812 \addplot[gray, dotted, thick, domain=-35:35] {1.5708} node [black, font=\footnotesize, below right, pos=0] {\(y=\frac{\pi}{2}\)};
813 \addplot[gray, dotted, thick, domain=-35:35] {-1.5708} node [black, font=\footnotesize, above left, pos=1] {\(y=-\frac{\pi}{2}\)};
814 \end{axis}
815 \end{tikzpicture}
816\columnbreak
817 \section{Differential calculus}
818
819 \subsection*{Limits}
820
821 \[\lim_{x \rightarrow a}f(x)\]
822 \(L^-,\quad L^+\) \qquad limit from below/above\\
823 \(\lim_{x \to a} f(x)\) \quad limit of a point\\
824
825 \noindent For solving \(x\rightarrow\infty\), put all \(x\) terms in denominators\\
826 e.g. \[\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2\]
827
828 \subsubsection*{Limit theorems}
829
830 \begin{enumerate}
831 \item
832 For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\)
833 \item
834 \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\)
835 \item
836 \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\)
837 \item
838 \(\therefore \lim_{x \rightarrow a} c \times f(x)=cF\) where \(c=\) constant
839 \item
840 \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\)
841 \item
842 \(f(x)\) is continuous \(\iff L^-=L^+=f(x) \> \forall x\)
843 \end{enumerate}
844
845 \subsection*{Gradients of secants and tangents}
846
847 \textbf{Secant (chord)} - line joining two points on curve\\
848 \textbf{Tangent} - line that intersects curve at one point
849
850 \subsection*{First principles derivative}
851
852 \[f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={\frac{dy}{dx}}\]
853
854 \subsubsection*{Logarithmic identities}
855
856 \(\log_b (xy)=\log_b x + \log_b y\)\\
857 \(\log_b x^n = n \log_b x\)\\
858 \(\log_b y^{x^n} = x^n \log_b y\)
859
860 \subsubsection*{Index identities}
861
862 \(b^{m+n}=b^m \cdot b^n\)\\
863 \((b^m)^n=b^{m \cdot n}\)\\
864 \((b \cdot c)^n = b^n \cdot c^n\)\\
865 \({a^m \div a^n} = {a^{m-n}}\)
866
867 \subsection*{Reciprocal derivatives}
868
869 \[\frac{1}{\frac{dy}{dx}} = \frac{dx}{dy}\]
870
871 \subsection*{Differentiating \(x=f(y)\)}
872 Find \(\dfrac{dx}{dy}\), then \(\dfrac{dy}{dx} = \dfrac{1}{\left(\dfrac{dx}{dy}\right)}\)
873
874 \subsection*{Second derivative}
875 \begin{align*}f(x) \longrightarrow &f^\prime (x) \longrightarrow f^{\prime\prime}(x)\\
876 \implies y \longrightarrow &\frac{dy}{dx} \longrightarrow \frac{d^2 y}{dx^2}\end{align*}
877
878 \noindent Order of polynomial \(n\)th derivative decrements each time the derivative is taken
879
880 \subsubsection*{Points of Inflection}
881
882 \emph{Stationary point} - i.e.
883 \(f^\prime(x)=0\)\\
884 \emph{Point of inflection} - max \(|\)gradient\(|\) (i.e.
885 \(f^{\prime\prime} = 0\))
886
887 \subsubsection*{Strictly increasing/decreasing}
888
889 For \(x_2\) and \(x_1\) where \(x_2 > x_1\):
890
891 \textbf{strictly increasing}\\
892 \-\hspace{1em}where \(f(x_2) > f(x_1)\) or \(f^\prime(x)>0\)
893
894 \textbf{strictly decreasing}\\
895 \hspace{1em}where \(f(x_2) < f(x_1)\) or \(f^\prime(x)<0\)
896 \begin{warning}
897 Endpoints are included, even where \(\boldsymbol{\frac{dy}{dx}=0}\)
898 \end{warning}
899
900
901 \begin{table*}[ht]
902 \centering
903 \begin{tabularx}{\textwidth}{|r|Y|Y|Y|}
904 \hline
905 \rowcolor{lblue}
906 & \adjustbox{margin=0 1ex, valign=m}{\centering\(\dfrac{d^2 y}{dx^2} > 0\)} & \adjustbox{margin=0 1ex, valign=m}{\centering \(\dfrac{d^2y}{dx^2}<0\)} & \adjustbox{margin=0 1ex, valign=m}{\(\dfrac{d^2y}{dx^2}=0\) (inflection)} \\
907 \hline
908 \(\dfrac{dy}{dx}>0\) &
909 \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-3, xmax=0.8, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(e^(x)}; \addplot[red] {x/2.5+0.75}; \end{axis}\end{tikzpicture} \\Rising (concave up)}&
910 \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=0.1, xmax=4, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(ln(x))}; \addplot[red] {x/1.5-0.56}; \end{axis}\end{tikzpicture} \\Rising (concave down)}&
911 \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1.5, xmax=1.5, scale=0.2, samples=100] \addplot[blue] {(sin((deg x)))}; \addplot[red] {x}; \end{axis}\end{tikzpicture} \\Rising inflection point}\\
912 \hline
913 \(\dfrac{dy}{dx}<0\) &
914 \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-.5, xmax=1, ymin=-.5, ymax=.5, scale=0.2, samples=100] \addplot[blue] {1/(x+1)-1}; \addplot[red] {-x}; \end{axis}\end{tikzpicture} \\Falling (concave up)}&
915 \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=0, xmax=1.5, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(2-x*x)^(1/2)}; \addplot[red] {-x+2}; \end{axis}\end{tikzpicture} \\Falling (concave down)}&
916 \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=1.5, xmax=4.5, scale=0.2, samples=100] \addplot[blue] {(sin((deg x)))}; \addplot[red] {-x+3.1415}; \end{axis}\end{tikzpicture} \\Falling inflection point}\\
917 \hline
918 \(\dfrac{dy}{dx}=0\)&
919 \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(x*x)}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \\Local minimum}& \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(-x*x)}; \addplot[red, very thick] {0}; \end{axis}\end{tikzpicture} \\Local maximum}&
920 \makecell{\\\begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(x*x*x)}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \(\>\) \begin{tikzpicture}\begin{axis}[axis x line=none, axis y line=none, xmin=-1, xmax=1, scale=0.2, samples=50, unbounded coords=jump] \addplot[blue] {(-x*x*x)}; \addplot[red, thick] {0}; \end{axis}\end{tikzpicture} \\Stationary inflection point}\\
921 \hline
922 \end{tabularx}
923 \end{table*}
924 \begin{itemize}
925 \item
926 \(f^\prime (a) = 0, \> f^{\prime\prime}(a) > 0\) \\
927 \textbf{local min} at \((a, f(a))\) (concave up)
928 \item
929 \(f^\prime (a) = 0, \> f^{\prime\prime} (a) < 0\) \\
930 \textbf{local max} at \((a, f(a))\) (concave down)
931 \item
932 \(f^{\prime\prime}(a) = 0\) \\
933 \textbf{point of inflection} at \((a, f(a))\)
934 \item
935 \(f^{\prime\prime}(a) = 0, \> f^\prime(a)=0\) \\
936 stationary point of inflection at \((a, f(a)\)
937 \end{itemize}
938
939 \subsection*{Implicit Differentiation}
940
941 \noindent Used for differentiating circles etc.
942
943 If \(p\) and \(q\) are expressions in \(x\) and \(y\) such that \(p=q\),
944 for all \(x\) and \(y\), then:
945
946 \[{\frac{dp}{dx}} = {\frac{dq}{dx}} \quad \text{and} \quad {\frac{dp}{dy}} = {\frac{dq}{dy}}\]
947
948 \begin{cas}
949 Action \(\rightarrow\) Calculation \\
950 \-\hspace{1em}\texttt{impDiff(y\^{}2+ax=5,\ x,\ y)}
951 \end{cas}
952
953 \subsection*{Slope fields}
954
955 \begin{tikzpicture}[declare function={diff(\x,\y) = \x+\y;}]
956 \begin{axis}[axis equal, ymin=-4, ymax=4, xmin=-4, xmax=4, ticks=none, enlargelimits=true, ]
957 \addplot[thick, orange, domain=-4:2] {e^(x)-x-1};
958 \pgfplotsinvokeforeach{-4,...,4}{%
959 \draw[gray] ( {#1 -0.1}, {4 - diff(#1, 4) *0.1}) -- ( {#1 +0.1}, {4 + diff(#1, 4) *0.1});
960 \draw[gray] ( {#1 -0.1}, {3 - diff(#1, 3) *0.1}) -- ( {#1 +0.1}, {3 + diff(#1, 3) *0.1});
961 \draw[gray] ( {#1 -0.1}, {2 - diff(#1, 2) *0.1}) -- ( {#1 +0.1}, {2 + diff(#1, 2) *0.1});
962 \draw[gray] ( {#1 -0.1}, {1 - diff(#1, 1) *0.1}) -- ( {#1 +0.1}, {1 + diff(#1, 1) *0.1});
963 \draw[gray] ( {#1 -0.1}, {0 - diff(#1, 0) *0.1}) -- ( {#1 +0.1}, {0 + diff(#1, 0) *0.1});
964 \draw[gray] ( {#1 -0.1}, {-1 - diff(#1, -1) *0.1}) -- ( {#1 +0.1}, {-1 + diff(#1, -1) *0.1});
965 \draw[gray] ( {#1 -0.1}, {-2 - diff(#1, -2) *0.1}) -- ( {#1 +0.1}, {-2 + diff(#1, -2) *0.1});
966 \draw[gray] ( {#1 -0.1}, {-3 - diff(#1, -3) *0.1}) -- ( {#1 +0.1}, {-3 + diff(#1, -3) *0.1});
967 \draw[gray] ( {#1 -0.1}, {-4 - diff(#1, -4) *0.1}) -- ( {#1 +0.1}, {-4 + diff(#1, -4) *0.1});
968 }
969 \end{axis}
970 \end{tikzpicture}
971
972 \subsection*{Parametric equations}
973
974
975 \begin{align*}
976 \dfrac{dy}{dt} &= \dfrac{dy}{dx} \cdot \dfrac{dx}{dt} \\
977 \therefore \dfrac{dy}{dx} &= \dfrac{\left(\dfrac{dy}{dt}\right)}{\left(\dfrac{dx}{dt}\right)} \text{ provided } \dfrac{dx}{dt} \ne 0 \\
978 \dfrac{d^2y}{dx^2} &= \dfrac{\left(\dfrac{dy^\prime}{dt}\right)}{\left(\dfrac{dx}{dt}\right)} \text{ where } y^\prime = \dfrac{dy}{dx}
979 \end{align*}
980
981 \subsection*{Integration}
982
983 \[\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)\]
984
985 \subsubsection*{Definite integrals}
986
987 \[\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)\]
988
989 \begin{itemize}
990
991 \item
992 Signed area enclosed by\\
993 \(\> y=f(x), \quad y=0, \quad x=a, \quad x=b\).
994 \item
995 \emph{Integrand} is \(f\).
996 \end{itemize}
997
998 \subsubsection*{Properties}
999
1000 \begin{align*}
1001 \int^b_a f(x) \> dx &= \int^c_a f(x) \> dx + \int^b_c f(x) \> dx \\
1002 \int^a_a f(x) \> dx &= 0 \\
1003 \int^b_a k \cdot f(x) \> dx &= k \int^b_a f(x) \> dx \\
1004 \int^b_a f(x) \pm g(x) \> dx &= \int^b_a f(x) \> dx \pm \int^b_a g(x) \> dx \\
1005 \int^b_a f(x) \> dx &= - \int^a_b f(x) \> dx \\
1006 \end{align*}
1007
1008 \subsection*{Integration by substitution}
1009
1010 \[\int f(u) {\frac{du}{dx}} \cdot dx = \int f(u) \cdot du\]
1011
1012 \begin{warning}
1013 \(\boldsymbol{f(u)}\) must be 1:1 \(\boldsymbol{\implies}\) one \(\boldsymbol{x}\) for each \(\boldsymbol{y}\)
1014 \end{warning}
1015 \begin{align*}\text{e.g. for } y&=\int(2x+1)\sqrt{x+4} \cdot dx\\
1016 \text{let } u&=x+4\\
1017 \implies& {\frac{du}{dx}} = 1\\
1018 \implies& x = u - 4\\
1019 \text{then } &y=\int (2(u-4)+1)u^{\frac{1}{2}} \cdot du\\
1020 &\text{(solve as normal integral)}
1021 \end{align*}
1022
1023 \subsubsection*{Definite integrals by substitution}
1024
1025 For \(\int^b_a f(x) {\frac{du}{dx}} \cdot dx\), evaluate new \(a\) and
1026 \(b\) for \(f(u) \cdot du\).
1027
1028 \subsubsection*{Trigonometric integration}
1029
1030 \[\sin^m x \cos^n x \cdot dx\]
1031
1032 \paragraph{\textbf{\(m\) is odd:}}
1033 \(m=2k+1\) where \(k \in \mathbb{Z}\)\\
1034 \(\implies \sin^{2k+1} x = (\sin^2 z)^k \sin x = (1 - \cos^2 x)^k \sin x\)\\
1035 Substitute \(u=\cos x\)
1036
1037 \paragraph{\textbf{\(n\) is odd:}}
1038 \(n=2k+1\) where \(k \in \mathbb{Z}\)\\
1039 \(\implies \cos^{2k+1} x = (\cos^2 x)^k \cos x = (1-\sin^2 x)^k \cos x\)\\
1040 Substitute \(u=\sin x\)
1041
1042 \paragraph{\textbf{\(m\) and \(n\) are even:}}
1043 use identities...
1044
1045 \begin{itemize}
1046
1047 \item
1048 \(\sin^2x={1 \over 2}(1-\cos 2x)\)
1049 \item
1050 \(\cos^2x={1 \over 2}(1+\cos 2x)\)
1051 \item
1052 \(\sin 2x = 2 \sin x \cos x\)
1053 \end{itemize}
1054
1055 \subsection*{Partial fractions}
1056
1057 To factorise \(f(x) = \frac{\delta}{\alpha \cdot \beta}\):
1058 \begin{align*}
1059 \dfrac{\delta}{\alpha \cdot \beta \cdot \gamma} &= \dfrac{A}{\alpha} + \dfrac{B}{\beta} + \dfrac{C}{\gamma} \tag{1} \\
1060 \text{Multiply by } & (\alpha \cdot \beta \cdot \gamma) \text{:} \\
1061 \delta &= \beta\gamma A + \alpha\gamma B +\alpha\beta C \tag{2} \\
1062 \text{Substitute } x &= \{\alpha, \beta, \gamma\} \text{ into (2) to find denominators}
1063 \end{align*}
1064
1065 \subsubsection*{Repeated linear factors}
1066
1067 \[ \dfrac{p(x)}{(x-a)^n} = \dfrac{A_1}{(x-a)} + \dfrac{A_2}{(x-a)^2} + \dots + \dfrac{A_n}{(x-a)^n} \]
1068
1069 \subsubsection*{Irreducible quadratic factors}
1070
1071 \[ \text{e.g. } \dfrac{3x-4}{(2x-3)(x^2+5)} = \dfrac{A}{2x-3} + \dfrac{Bx+C}{x^2+5} \]
1072
1073 \begin{cas}
1074 Action \(\rightarrow\) Transformation:\\
1075 \-\hspace{1em} \texttt{expand(..., x)}
1076
1077 To reverse, use \texttt{combine(...)}
1078 \end{cas}
1079
1080 \subsection*{Graphing integrals on CAS}
1081
1082 \begin{cas}
1083 \textbf{In main:} Interactive \(\rightarrow\) Calculation \(\rightarrow\) \(\int\)\\
1084 For restrictions, \texttt{Define\ f(x)=...} then \texttt{f(x)\textbar{}x\textgreater{}...}
1085 \end{cas}
1086
1087 \subsection*{Applications of antidifferentiation}
1088
1089 \begin{itemize}
1090
1091 \item
1092 \(x\)-intercepts of \(y=f(x)\) identify \(x\)-coordinates of
1093 stationary points on \(y=F(x)\)
1094 \item
1095 nature of stationary points is determined by sign of \(y=f(x)\) on
1096 either side of its \(x\)-intercepts
1097 \item
1098 if \(f(x)\) is a polynomial of degree \(n\), then \(F(x)\) has degree
1099 \(n+1\)
1100 \end{itemize}
1101
1102 To find stationary points of a function, substitute \(x\) value of given
1103 point into derivative. Solve for \({\frac{dy}{dx}}=0\). Integrate to find
1104 original function.
1105
1106 \subsection*{Solids of revolution}
1107
1108 Approximate as sum of infinitesimally-thick cylinders
1109
1110 \subsubsection*{Rotation about \(\boldsymbol{x}\)-axis}
1111
1112 \[ V = \pi\int^{x=b}_{x=a} f(x)^2 \> dx \]
1113
1114 \subsubsection*{Rotation about \(\boldsymbol{y}\)-axis}
1115
1116 \begin{align*}
1117 V &= \pi \int^{y=b}_{y=a} x^2 \> dy \\
1118 &= \pi \int^{y=b}_{y=a} (f(y))^2 \> dy
1119 \end{align*}
1120
1121 \subsubsection*{Regions not bound by \(\boldsymbol{y=0}\)}
1122
1123 \[V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx\]
1124 \hfill where \(f(x) > g(x)\)
1125
1126 \subsection*{Length of a curve}
1127
1128 For length of \(f(x)\) from \(x=a \rightarrow x=b\):
1129 \begin{align*}
1130 &\text{Cartesian} \> & L &= \int^b_a \sqrt{1 + \left(\dfrac{dy}{dx}\right)^2} \> dx \\
1131 &\text{Parametric} \> & L & = \int^b_a \sqrt{\left(\dfrac{dx}{dt}\right)^2 + \left(\dfrac{dy}{dt}\right)^2} \> dt
1132 \end{align*}
1133
1134 \begin{cas}
1135 \begin{enumerate}[label=\alph*), leftmargin=5mm]
1136 \item Evaluate formula
1137 \item Interactive \(\rightarrow\) Calculation \(\rightarrow\) Line \(\rightarrow\) \texttt{arcLen}
1138 \end{enumerate}
1139 \end{cas}
1140
1141 \subsection*{Rates}
1142
1143 \subsubsection*{Gradient at a point on parametric curve}
1144
1145 \[{\frac{dy}{dx}} = {{\frac{dy}{dt}} \div {\frac{dx}{dt}}} \> \vert \> {\frac{dx}{dt}} \ne 0 \text{ (chain rule)}\]
1146
1147 \[\frac{d^2}{dx^2} = \frac{d(y^\prime)}{dx} = {\frac{dy^\prime}{dt} \div {\frac{dx}{dt}}} \> \vert \> y^\prime = {\frac{dy}{dx}}\]
1148
1149 \subsection*{Rational functions}
1150
1151 \[f(x) = \frac{P(x)}{Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}\]
1152
1153 \subsection*{Fundamental theorem of calculus}
1154
1155 If \(f\) is continuous on \([a, b]\), then
1156
1157 \[\int^b_a f(x) \> dx = F(b) - F(a)\]
1158 \hfill where \(F = \int f \> dx\)
1159
1160 \subsection*{Differential equations}
1161
1162 \noindent\textbf{Order} - highest power inside derivative\\
1163 \textbf{Degree} - highest power of highest derivative\\
1164 e.g. \({\left(\dfrac{dy^2}{d^2} x\right)}^3\) \qquad order 2, degree 3
1165
1166 \begin{warning}
1167 To verify solutions, find \(\frac{dy}{dx}\) from \(y\) and substitute into original
1168 \end{warning}
1169
1170 \subsubsection*{Function of the dependent
1171 variable}
1172
1173 If \({\frac{dy}{dx}}=g(y)\), then
1174 \(\frac{dx}{dy} = 1 \div \frac{dy}{dx} = \frac{1}{g(y)}\). Integrate both sides to solve equation. Only add \(c\) on one side. Express
1175 \(e^c\) as \(A\).
1176
1177
1178
1179 \subsubsection*{Mixing problems}
1180
1181 \[\left(\frac{dm}{dt}\right)_\Sigma = \left(\frac{dm}{dt}\right)_{\text{in}} - \left(\frac{dm}{dt}_{\text{out}}\right)\]
1182
1183 \subsubsection*{Separation of variables}
1184
1185 If \({\frac{dy}{dx}}=f(x)g(y)\), then:
1186
1187 \[\int f(x) \> dx = \int \frac{1}{g(y)} \> dy\]
1188
1189 \subsubsection*{Euler's method for solving DEs}
1190
1191 \[\frac{f(x+h) - f(x)}{h} \approx f^\prime (x) \quad \text{for small } h\]
1192
1193 \[\implies f(x+h) \approx f(x) + hf^\prime(x)\]
1194
1195 \include{calculus-rules}
1196
1197 \section{Kinematics \& Mechanics}
1198
1199 \subsection*{Constant acceleration}
1200
1201 \begin{itemize}
1202 \item \textbf{Position} - relative to origin
1203 \item \textbf{Displacement} - relative to starting point
1204 \end{itemize}
1205
1206 \subsubsection*{Velocity-time graphs}
1207
1208 \begin{description}[nosep, labelindent=0.5cm, leftmargin=0.5\columnwidth]
1209 \item[Displacement:] \textit{signed} area
1210 \item[Distance travelled:] \textit{total} area
1211 \end{description}
1212
1213 \[ \text{acceleration} = \frac{d^2x}{dt^2} = \frac{dv}{dt} = v\frac{dv}{dx} = \frac{d}{dx}\left(\frac{1}{2}v^2\right) \]
1214
1215 \begin{center}
1216 \renewcommand{\arraystretch}{1}
1217 \begin{tabular}{ l r }
1218 \hline & no \\ \hline
1219 \(v=u+at\) & \(x\) \\
1220 \(v^2 = u^2+2as\) & \(t\) \\
1221 \(s = \frac{1}{2} (v+u)t\) & \(a\) \\
1222 \(s = ut + \frac{1}{2} at^2\) & \(v\) \\
1223 \(s = vt- \frac{1}{2} at^2\) & \(u\) \\ \hline
1224 \end{tabular}
1225 \end{center}
1226
1227 \[ v_{\text{avg}} = \frac{\Delta\text{position}}{\Delta t} \]
1228 \begin{align*}
1229 \text{speed} &= |{\text{velocity}}| \\
1230 &= \sqrt{v_x^2 + v_y^2 + v_z^2}
1231 \end{align*}
1232
1233 \noindent \textbf{Distance travelled between \(t=a \rightarrow t=b\):}
1234 \[= \int^b_a \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \cdot dt \]
1235
1236 \noindent \textbf{Shortest distance between \(\boldsymbol{r}(t_0)\) and \(\boldsymbol{r}(t_1)\):}
1237 \[ = |\boldsymbol{r}(t_1) - \boldsymbol{r}(t_2)| \]
1238
1239 \subsection*{Vector functions}
1240
1241 \[ \boldsymbol{r}(t) = x \boldsymbol{i} + y \boldsymbol{j} + z \boldsymbol{k} \]
1242
1243 \begin{itemize}
1244 \item If \(\boldsymbol{r}(t) \equiv\) position with time, then the graph of endpoints of \(\boldsymbol{r}(t) \equiv\) Cartesian path
1245 \item Domain of \(\boldsymbol{r}(t)\) is the range of \(x(t)\)
1246 \item Range of \(\boldsymbol{r}(t)\) is the range of \(y(t)\)
1247 \end{itemize}
1248
1249 \subsection*{Vector calculus}
1250
1251 \subsubsection*{Derivative}
1252
1253 Let \(\boldsymbol{r}(t)=x(t)\boldsymbol{i} + y(t)\boldsymbol(j)\). If both \(x(t)\) and \(y(t)\) are differentiable, then:
1254 \[ \boldsymbol{r}(t)=x(t)\boldsymbol{i}+y(t)\boldsymbol{j} \]
1255
1256 \subfile{dynamics}
1257 \subfile{statistics}
1258 \end{multicols}
1259\end{document}