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75\fancypagestyle{plain}{\fancyhead[LO,LE]{} \fancyhead[CO,CE]{}} % rm title & author for first page
76\fancyhead[LO,LE]{Year 12 Specialist}
77\fancyhead[CO,CE]{Andrew Lorimer}
78
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106\begin{document}
107
108\title{\vspace{-22mm}Year 12 Specialist\vspace{-4mm}}
109\author{Andrew Lorimer}
110\date{}
111\maketitle
112\vspace{-9mm}
113\begin{multicols}{2}
114
115 \section{Complex numbers}
116
117 \[\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}\]
118 \begin{align*}
119 \text{Cartesian form: } & a+bi\\
120 \text{Polar form: } & r\operatorname{cis}\theta
121 \end{align*}
122
123 \subsection*{Operations}
124
125 \begin{tabularx}{\columnwidth}{|r|X|X|}
126 \hline
127 \rowcolor{cas}
128 & \textbf{Cartesian} & \textbf{Polar} \\
129 \hline
130 \(z_1 \pm z_2\) & \((a \pm c)(b \pm d)i\) & convert to \(a+bi\)\\
131 \hline
132 \(+k \times z\) & \multirow{2}{*}{\(ka \pm kbi\)} & \(kr\operatorname{cis} \theta\)\\
133 \cline{1-1}\cline{3-3}
134 \(-k \times z\) & & \(kr \operatorname{cis}(\theta\pm \pi)\)\\
135 \hline
136 \(z_1 \cdot z_2\) & \(ac-bd+(ad+bc)i\) & \(r_1r_2 \operatorname{cis}(\theta_1 + \theta_2)\)\\
137 \hline
138 \(z_1 \div z_2\) & \((z_1 \overline{z_2}) \div |z_2|^2\) & \(\left(\frac{r_1}{r_2}\right) \operatorname{cis}(\theta_1 - \theta_2)\) \\
139 \hline
140 \end{tabularx}
141
142 \subsubsection*{Scalar multiplication in polar form}
143
144 For \(k \in \mathbb{R}^+\):
145 \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\theta\]
146
147 \noindent For \(k \in \mathbb{R}^-\):
148 \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\left(\begin{cases}\theta - \pi & |0<\operatorname{Arg}(z)\le \pi \\ \theta + \pi & |-\pi<\operatorname{Arg}(z)\le 0\end{cases}\right)\]
149
150 \subsection*{Conjugate}
151 \vspace{-7mm} \hfill \colorbox{cas}{\texttt{conjg(a+bi)}}
152 \begin{align*}
153 \overline{z} &= a \mp bi\\
154 &= r \operatorname{cis}(-\theta)
155 \end{align*}
156
157 \subsubsection*{Properties}
158
159 \begin{align*}
160 \overline{z_1 \pm z_2} &= \overline{z_1}\pm\overline{z_2}\\
161 \overline{z_1 \cdot z_2} &= \overline{z_1}\cdot\overline{z_2}\\
162 \overline{kz} &= k\overline{z} \> \forall \> k \in \mathbb{R}\\
163 z\overline{z} &= (a+bi)(a-bi)\\
164 &= a^2 + b^2\\
165 &= |z|^2
166 \end{align*}
167
168 \subsection*{Modulus}
169
170 \[|z|=|\vec{Oz}|=\sqrt{a^2 + b^2}\]
171
172 \subsubsection*{Properties}
173
174 \begin{align*}
175 |z_1z_2|&=|z_1||z_2|\\
176 \left|\frac{z_1}{z_2}\right|&=\frac{|z_1|}{|z_2|}\\
177 |z_1+z_2|&\le|z_1|+|z_2|
178 \end{align*}
179
180 \subsection*{Multiplicative inverse}
181
182 \begin{align*}
183 z^{-1}&=\frac{a-bi}{a^2+b^2}\\
184 &=\frac{\overline{z}}{|z|^2}a\\
185 &=r \operatorname{cis}(-\theta)
186 \end{align*}
187
188 \subsection*{Dividing over \(\mathbb{C}\)}
189
190 \begin{align*}
191 \frac{z_1}{z_2}&=z_1z_2^{-1}\\
192 &=\frac{z_1\overline{z_2}}{|z_2|^2}\\
193 &=\frac{(a+bi)(c-di)}{c^2+d^2}\\
194 & \text{then rationalise denominator}
195 \end{align*}
196
197 \subsection*{Polar form}
198
199 \[ r \operatorname{cis} \theta = r\left( \cos \theta + i \sin \theta \right) \]
200
201 \begin{itemize}
202 \item{\(r=|z|=\sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}\)}
203 \item{\(\theta = \operatorname{arg}(z)\) \hfill \colorbox{cas}{\texttt{arg(a+bi)}}}
204 \item{\(\operatorname{Arg}(z) \in (-\pi,\pi)\) \quad \bf{(principal argument)}}
205 \item{Multiple representations:\\\(r\operatorname{cis}\theta=r\operatorname{cis}(\theta+2n\pi)\) with \(n \in \mathbb{Z}\) revolutions}
206 \item{\(\operatorname{cis}\pi=-1,\qquad \operatorname{cis}0=1\)}
207 \end{itemize}
208
209 \begin{cas}
210 \-\hspace{1em}\verb|compToTrig(a+bi)| \(\iff\) \verb|cExpand{r·cisX}|
211 \end{cas}
212
213 \subsection*{de Moivres' theorem}
214
215 \begin{theorembox}{}
216 \[(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}\]
217 \end{theorembox}
218
219 \subsection*{Complex polynomials}
220
221 Include \(\pm\) for all solutions, incl. imaginary
222
223 \begin{tabularx}{\columnwidth}{ R{0.55} X }
224 \hline
225 Sum of squares & \(\begin{aligned}
226 z^2 + a^2 &= z^2-(ai)^2\\
227 &= (z+ai)(z-ai) \end{aligned}\) \\
228 \hline
229 Sum of cubes & \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)\\
230 \hline
231 Division & \(P(z)=D(z)Q(z)+R(z)\) \\
232 \hline
233 Remainder theorem & Let \(\alpha \in \mathbb{C}\). Remainder of \(P(z) \div (z-\alpha)\) is \(P(\alpha)\)\\
234 \hline
235 Factor theorem & \(z-\alpha\) is a factor of \(P(z) \iff P(\alpha)=0\) for \(\alpha \in \mathbb{C}\)\\
236 \hline
237 Conjugate root theorem & \(P(z)=0 \text{ at } z=a\pm bi\) (\(\implies\) both \(z_1\) and \(\overline{z_1}\) are solutions)\\
238 \hline
239 \end{tabularx}
240
241 \begin{theorembox}{title=Factor theorem}
242 If \(\beta z + \alpha\) is a factor of \(P(z)\), \\
243 \-\hspace{1em}then \(P(-\dfrac{\alpha}{\beta})=0\).
244 \end{theorembox}
245
246 \subsection*{\(n\)th roots}
247
248 \(n\)th roots of \(z=r\operatorname{cis}\theta\) are:
249
250 \[z = r^{\frac{1}{n}} \operatorname{cis}\left(\frac{\theta+2k\pi}{n}\right)\]
251
252 \begin{itemize}
253
254 \item{Same modulus for all solutions}
255 \item{Arguments separated by \(\frac{2\pi}{n} \therefore\) there are \(n\) roots}
256 \item{If one square root is \(a+bi\), the other is \(-a-bi\)}
257 \item{Give one implicit \(n\)th root \(z_1\), function is \(z=z_1^n\)}
258 \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 \(\frac{2\pi}{n}\))}
259 \end{itemize}
260
261 \noindent For \(0=az^2+bz+c\), use quadratic formula:
262
263 \[z=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]
264
265 \subsection*{Fundamental theorem of algebra}
266
267 A polynomial of degree \(n\) can be factorised into \(n\) linear factors in \(\mathbb{C}\):
268
269 \[\implies P(z)=a_n(z-\alpha_1)(z-\alpha_2)(z-\alpha_3)\dots(z-\alpha_n)\]
270 \[\text{ where } \alpha_1,\alpha_2,\alpha_3,\dots,\alpha_n \in \mathbb{C}\]
271
272 \subsection*{Argand planes}
273
274 \begin{center}\begin{tikzpicture}[scale=2]
275 \draw [->] (-0.2,0) -- (1.5,0) node [right] {$\operatorname{Re}(z)$};
276 \draw [->] (0,-0.2) -- (0,1.5) node [above] {$\operatorname{Im}(z)$};
277 \coordinate (P) at (1,1);
278 \coordinate (a) at (1,0);
279 \coordinate (b) at (0,1);
280 \coordinate (O) at (0,0);
281 \draw (0,0) -- (P) node[pos=0.5, above left]{\(r\)} node[pos=1, right]{\(\begin{aligned}z&=a+bi\\&=r\operatorname{cis}\theta\end{aligned}\)};
282 \draw [gray, dashed] (1,1) -- (1,0) node[black, pos=1, below]{\(a\)};
283 \draw [gray, dashed] (1,1) -- (0,1) node[black, pos=1, left]{\(b\)};
284 \begin{scope}
285 \path[clip] (O) -- (P) -- (a);
286 \fill[red, opacity=0.5, draw=black] (O) circle (2mm);
287 \node at ($(O)+(20:3mm)$) {$\theta$};
288 \end{scope}
289 \filldraw (P) circle (0.5pt);
290 \end{tikzpicture}\end{center}
291
292 \begin{itemize}
293 \item{Multiplication by \(i \implies\) CCW rotation of \(\frac{\pi}{2}\)}
294 \item{Addition: \(z_1 + z_2 \equiv\) \overrightharp{\(Oz_1\)} + \overrightharp{\(Oz_2\)}}
295 \end{itemize}
296
297 \subsection*{Sketching complex graphs}
298
299 \subsubsection*{Linear}
300
301 \begin{itemize}
302 \item{\(\operatorname{Re}(z)=c\) or \(\operatorname{Im}(z)=c\) (perpendicular bisector)}
303 \item{\(\operatorname{Im}(z)=m\operatorname{Re}(z)\)}
304 \item{\(|z+a|=|z+b| \implies 2(a-b)x=b^2-a^2\)\\Geometric: equidistant from \(a,b\)}
305 \end{itemize}
306
307 \subsubsection*{Circles}
308
309 \begin{itemize}
310 \item \(|z-z_1|^2=c^2|z_2+2|^2\)
311 \item \(|z-(a+bi)|=c \implies (x-a)^2+_(y-b)^2=c^2\)
312 \end{itemize}
313
314 \noindent \textbf{Loci} \qquad \(\operatorname{Arg}(z)<\theta\)
315
316 \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
317 \draw [->] (0,0) -- (1,0) node [right] {$\operatorname{Re}(z)$};
318 \draw [->] (0,-0.5) -- (0,1) node [above] {$\operatorname{Im}(z)$};
319 \draw [<-, dashed, thick, blue] (-1,0) -- (0,0);
320 \draw [->, thick, blue] (0,0) -- (1,1);
321 \fill [gray, opacity=0.2, domain=-1:1, variable=\x] (-1,-0.5) -- (-1,0) -- (0, 0) -- (1,1) -- (1,-0.5) -- cycle;
322 \begin{scope}
323 \path[clip] (0,0) -- (1,1) -- (1,0);
324 \fill[red, opacity=0.5, draw=black] (0,0) circle (2mm);
325 \node at ($(0,0)+(20:3mm)$) {$\frac{\pi}{4}$};
326 \end{scope}
327 \node [font=\footnotesize] at (0.5,-0.25) {\(\operatorname{Arg}(z)\le\frac{\pi}{4}\)};
328 \node [blue, mydot] {};
329 \end{tikzpicture}\end{center}
330
331 \noindent \textbf{Rays} \qquad \(\operatorname{Arg}(z-b)=\theta\)
332
333 \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
334 \draw [->] (-0.75,0) -- (1.5,0) node [right] {$\operatorname{Re}(z)$};
335 \draw [->] (0,-1) -- (0,1) node [above] {$\operatorname{Im}(z)$};
336 \draw [->, thick, brown] (-0.25,0) -- (-0.75,-1);
337 \node [above, font=\footnotesize] at (-0.25,0) {\(\frac{1}{4}\)};
338 \begin{scope}
339 \path[clip] (-0.25,0) -- (-0.75,-1) -- (0,0);
340 \fill[orange, opacity=0.5, draw=black] (-0.25,0) circle (2mm);
341 \end{scope}
342 \node at (-0.08,-0.3) {\(\frac{\pi}{8}\)};
343 \node [font=\footnotesize, left] at (-0.75,-1) {\(\operatorname{Arg}(z+\frac{1}{4})=\frac{\pi}{8}\)};
344 \node [brown, mydot] at (-0.25,0) {};
345 \draw [<->, thick, green] (0,-1) -- (1.5,0.5) node [pos=0.25, black, font=\footnotesize, right] {\(|z-2|=|z-(1+i)|\)};
346 \node [left, font=\footnotesize] at (0,-1) {\(-1\)};
347 \node [below, font=\footnotesize] at (1,0) {\(1\)};
348 \end{tikzpicture}\end{center}
349
350 \section{Vectors}
351 \begin{center}\begin{tikzpicture}
352 \draw [->] (-0.5,0) -- (3,0) node [right] {\(x\)};
353 \draw [->] (0,-0.5) -- (0,3) node [above] {\(y\)};
354 \draw [orange, ->, thick] (0.5,0.5) -- (2.5,2.5) node [pos=0.5, above] {\(\vec{u}\)};
355 \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
356 \draw [gray, dashed, thick] (0.5,0.5) -- (2.5,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, below]{\(x\vec{i}\)};
357 \draw [gray, dashed, thick] (2.5,0.5) -- (2.5,2.5) node [pos=0.5] {\midarrow};
358 \end{scope}
359 \node[black, right] at (2.5,1.5) {\(y\vec{j}\)};
360 \end{tikzpicture}\end{center}
361
362 \subsection*{Column notation}
363
364 \[\begin{bmatrix}x\\ y \end{bmatrix} \iff x\boldsymbol{i} + y\boldsymbol{j}\]
365 \(\begin{bmatrix}x_2-x_1\\ y_2-y_1 \end{bmatrix}\) \quad between \(A(x_1,y_1), \> B(x_2,y_2)\)
366
367 \subsection*{Scalar multiplication}
368
369 \[k\cdot (x\boldsymbol{i}+y\boldsymbol{j})=kx\boldsymbol{i}+ky\boldsymbol{j}\]
370
371 \noindent For \(k \in \mathbb{R}^-\), direction is reversed
372
373 \subsection*{Vector addition}
374 \begin{center}\begin{tikzpicture}[scale=1]
375 \coordinate (A) at (0,0);
376 \coordinate (B) at (2,2);
377 \draw [->, thick, red] (0,0) -- (2,2) node [pos=0.5, below right] {\(\vec{u}=2\vec{i}+2\vec{j}\)};
378 \draw [->, thick, blue] (2,2) -- (1,4) node [pos=0.5, above right] {\(\vec{v}=-\vec{i}+2\vec{j}\)};
379 \draw [->, thick, orange] (0,0) -- (1,4) node [pos=0.5, left] {\(\vec{u}+\vec{v}=\vec{i}+4\vec{j}\)};
380 \end{tikzpicture}\end{center}
381
382 \[(x\boldsymbol{i}+y\boldsymbol{j}) \pm (a\boldsymbol{i}+b\boldsymbol{j})=(x \pm a)\boldsymbol{i}+(y \pm b)\boldsymbol{j}\]
383
384 \begin{itemize}
385 \item Draw each vector head to tail then join lines
386 \item Addition is commutative (parallelogram)
387 \item \(\boldsymbol{u}-\boldsymbol{v}=\boldsymbol{u}+(-\boldsymbol{v}) \implies \overrightharp{AB}=\boldsymbol{b}-\boldsymbol{a}\)
388 \end{itemize}
389
390 \subsection*{Magnitude}
391
392 \[|(x\boldsymbol{i} + y\boldsymbol{j})|=\sqrt{x^2+y^2}\]
393
394 \subsection*{Parallel vectors}
395
396 \[\boldsymbol{u} || \boldsymbol{v} \iff \boldsymbol{u} = k \boldsymbol{v} \text{ where } k \in \mathbb{R} \setminus \{0\}\]
397
398 For parallel vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\):\\
399 \[\boldsymbol{a \cdot b}=\begin{cases}
400 |\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction}\\
401 -|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions}
402 \end{cases}\]
403 %\includegraphics[width=0.2,height=\textheight]{graphics/parallelogram-vectors.jpg}
404 %\includegraphics[width=1]{graphics/vector-subtraction.jpg}
405
406 \subsection*{Perpendicular vectors}
407
408 \[\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b} = 0\ \quad \text{(since \(\cos 90 = 0\))}\]
409
410 \subsection*{Unit vector \(|\hat{\boldsymbol{a}}|=1\)}
411 \[\begin{split}\hat{\boldsymbol{a}} & = {\frac{1}{|\boldsymbol{a}|}}\boldsymbol{a} \\ & = \boldsymbol{a} \cdot {|\boldsymbol{a}|}\end{split}\]
412
413 \subsection*{Scalar product \(\boldsymbol{a} \cdot \boldsymbol{b}\)}
414
415
416 \begin{center}\begin{tikzpicture}[scale=2]
417 \draw [->] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{b}\)};
418 \draw [->] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{a}\)};
419 \begin{scope}
420 \path[clip] (1,0.5) -- (1,0) -- (0,0);
421 \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
422 \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
423 \end{scope}
424 \end{tikzpicture}\end{center}
425 \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*}
426 \noindent\colorbox{cas}{On CAS:} \texttt{dotP({[}a\ b\ c{]},\ {[}d\ e\ f{]})}
427
428 \subsubsection*{Properties}
429
430 \begin{enumerate}
431 \item
432 \(k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k\boldsymbol{b})\)
433 \item
434 \(\boldsymbol{a \cdot 0}=0\)
435 \item
436 \(\boldsymbol{a} \cdot (\boldsymbol{b} + \boldsymbol{c})=\boldsymbol{a} \cdot \boldsymbol{b} + \boldsymbol{a} \cdot \boldsymbol{c}\)
437 \item
438 \(\boldsymbol{i \cdot i} = \boldsymbol{j \cdot j} = \boldsymbol{k \cdot k}= 1\)
439 \item
440 \(\boldsymbol{a} \cdot \boldsymbol{b} = 0 \quad \implies \quad \boldsymbol{a} \perp \boldsymbol{b}\)
441 \item
442 \(\boldsymbol{a \cdot a} = |\boldsymbol{a}|^2 = a^2\)
443 \end{enumerate}
444
445 \subsection*{Angle between vectors}
446
447 \[\cos \theta = \frac{\boldsymbol{a} \cdot \boldsymbol{b}}{|\boldsymbol{a}| |\boldsymbol{b}|} = \frac{a_1 b_1 + a_2 b_2}{|\boldsymbol{a}| |\boldsymbol{b}|}\]
448
449 \noindent \colorbox{cas}{On CAS:} \texttt{angle([a b c], [a b c])}
450
451 (Action \(\rightarrow\) Vector \(\rightarrow\)Angle)
452
453 \subsection*{Angle between vector and axis}
454
455 \noindent For\(\boldsymbol{a} = a_1 \boldsymbol{i} + a_2 \boldsymbol{j} + a_3 \boldsymbol{k}\)
456 which makes angles \(\alpha, \beta, \gamma\) with positive side of
457 \(x, y, z\) axes:
458 \[\cos \alpha = \frac{a_1}{|\boldsymbol{a}|}, \quad \cos \beta = \frac{a_2}{|\boldsymbol{a}|}, \quad \cos \gamma = \frac{a_3}{|\boldsymbol{a}|}\]
459
460 \noindent \colorbox{cas}{On CAS:} \texttt{angle({[}a\ b\ c{]},\ {[}1\ 0\ 0{]})}\\for angle
461 between \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) and
462 \(x\)-axis
463
464 \subsection*{Projections \& resolutes}
465
466 \begin{tikzpicture}[scale=3]
467 \draw [->, purple] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{a}\)};
468 \draw [->, orange] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{u}\)};
469 \draw [->, blue] (1,0) -- (2,0) node [pos=0.5, below] {\(\boldsymbol{b}\)};
470 \begin{scope}
471 \path[clip] (1,0.5) -- (1,0) -- (0,0);
472 \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
473 \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
474 \end{scope}
475 \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
476 \draw [gray, dashed, thick] (1,0) -- (1,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, right, rotate=-90]{\(\boldsymbol{w}\)};
477 \end{scope}
478 \draw (0,0) coordinate (O)
479 (1,0) coordinate (A)
480 (1,0.5) coordinate (B)
481 pic [draw,red,angle radius=2mm] {right angle = O--A--B};
482 \end{tikzpicture}
483
484 \subsubsection*{\(\parallel\boldsymbol{b}\) (vector projection/resolute)}
485
486 \begin{align*}
487 \boldsymbol{u} & = \frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|^2}\boldsymbol{b} \\
488 & = \left(\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\right)\left(\frac{\boldsymbol{b}}{|\boldsymbol{b}|}\right) \\
489 & = (\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}
490 \end{align*}
491
492 \subsubsection*{\(\perp\boldsymbol{b}\) (perpendicular projection)}
493 \[\boldsymbol{w} = \boldsymbol{a} - \boldsymbol{u}\]
494
495 \subsubsection*{\(|\boldsymbol{u}|\) (scalar projection/resolute)}
496 \begin{align*}
497 s &= |\boldsymbol{u}|\\
498 &= \boldsymbol{a} \cdot \hat{\boldsymbol{b}}\\
499 &=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\\
500 &= |\boldsymbol{a}| \cos \theta
501 \end{align*}
502
503 \subsubsection*{Rectangular (\(\parallel,\perp\)) components}
504
505 \[\boldsymbol{a}=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}+\left(\boldsymbol{a}-\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}\right)\]
506
507
508 \subsection*{Vector proofs}
509
510 \textbf{Concurrent:} intersection of \(\ge\) 3 lines
511
512 \begin{tikzpicture}
513 \draw [blue] (0,0) -- (1,1);
514 \draw [red] (1,0) -- (0,1);
515 \draw [brown] (0.4,0) -- (0.6,1);
516 \filldraw (0.5,0.5) circle (2pt);
517 \end{tikzpicture}
518
519 \subsubsection*{Collinear points}
520
521 \(\ge\) 3 points lie on the same line
522
523 \begin{tikzpicture}
524 \draw [purple] (0,0) -- (4,1);
525 \filldraw (2,0.5) circle (2pt) node [above] {\(C\)};
526 \filldraw (1,0.25) circle (2pt) node [above] {\(A\)};
527 \filldraw (3,0.75) circle (2pt) node [above] {\(B\)};
528 \coordinate (O) at (2.8,-0.2);
529 \node at (O) [below] {\(O\)};
530 \begin{scope}[->, orange, thick]
531 \draw (O) -- (2,0.5) node [pos=0.5, above, font=\footnotesize, black] {\(\boldsymbol{c}\)};
532 \draw (O) -- (1,0.25) node [pos=0.5, below, font=\footnotesize, black] {\(\boldsymbol{a}\)};
533 \draw (O) -- (3,0.75) node [pos=0.5, right, font=\footnotesize, black] {\(\boldsymbol{b}\)};
534 \end{scope}
535 \end{tikzpicture}
536
537 \begin{align*}
538 \text{e.g. Prove that}\\
539 \overrightharp{AC}=m\overrightharp{AB} \iff \boldsymbol{c}&=(1-m)\boldsymbol{a}+m\boldsymbol{b}\\
540 \implies \boldsymbol{c} &= \overrightharp{OA} + \overrightharp{AC}\\
541 &= \overrightharp{OA} + m\overrightharp{AB}\\
542 &=\boldsymbol{a}+m(\boldsymbol{b}-\boldsymbol{a})\\
543 &=\boldsymbol{a}+m\boldsymbol{b}-m\boldsymbol{a}\\
544 &=(1-m)\boldsymbol{a}+m{b}
545 \end{align*}
546 \begin{align*}
547 \text{Also, } \implies \overrightharp{OC} &= \lambda \vec{OA} + \mu \overrightharp{OB} \\
548 \text{where } \lambda + \mu &= 1\\
549 \text{If } C \text{ lies along } \overrightharp{AB}, & \implies 0 < \mu < 1
550 \end{align*}
551
552
553 \subsubsection*{Parallelograms}
554
555 \begin{center}\begin{tikzpicture}
556 \coordinate (O) at (0,0) node [below left] {\(O\)};
557 \coordinate (A) at (4,0);
558 \coordinate (B) at (6,2);
559 \coordinate (C) at (2,2);
560 \coordinate (D) at (6,0);
561
562 \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (O)--(A) node [below left] {\(A\)};
563 \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (A)--(B) node [above right] {\(B\)};
564 \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (B)--(C) node [above left] {\(C\)};
565 \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (C)--(O);
566
567 \draw [gray, dashed] (O) -- (B) node [pos=0.75] {\(\diagdown\diagdown\)} node [pos=0.25] {\(\diagdown\diagdown\)};
568 \draw [gray, dashed] (A) -- (C) node [pos=0.75] {\(\diagup\)} node [pos=0.25] {\(\diagup\)};
569 \begin{scope}
570 \path[clip] (C) -- (A) -- (O);
571 \fill[orange, opacity=0.5, draw=black] (0,0) circle (4mm);
572 \node at ($(0,0)+(20:8mm)$) {\(\theta\)};
573 \end{scope}
574 \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] {\(|\boldsymbol{c}|\cos\theta\)};
575 \draw pic [draw,thick,red,angle radius=2mm] {right angle=O--D--B};
576 \end{tikzpicture}\end{center}
577
578 \begin{itemize}
579 \item
580 Diagonals \(\overrightharp{OB}, \overrightharp{AC}\) bisect each other
581 \item
582 If diagonals are equal length, it is a rectangle
583 \item
584 \(|\overrightharp{OB}|^2 + |\overrightharp{CA}|^2 = |\overrightharp{OA}|^2 + |\overrightharp{AB}|^2 + |\overrightharp{CB}|^2 + |\overrightharp{OC}|^2\)
585 \item
586 Area \(=\boldsymbol{c} \cdot \boldsymbol{a}\)
587 \end{itemize}
588
589 \subsubsection*{Perpendicular bisectors of a triangle}
590
591\hspace{-1.5cm}\begin{tikzpicture}
592 [
593 scale=3,
594 >=stealth,
595 point/.style = {draw, circle, fill = black, inner sep = 1pt},
596 dot/.style = {draw, circle, fill = black, inner sep = .2pt},
597 thick
598 ]
599
600 \node at (-1,1) [text width=5cm, rounded corners, fill=lblue, inner sep=1ex]
601 {
602 \sffamily The three bisectors meet at the circumcenter \(Z\) where \(|\overrightharp{ZA}| = |\overrightharp{ZB}| = |\overrightharp{ZC}|\).
603 };
604
605 % the circle
606 \def\rad{1}
607 \node (origin) at (0,0) [point, label = {right: {\(Z\)}}]{};
608 \draw [thin] (origin) circle (\rad);
609
610 % triangle nodes: just points on the circle
611 \node (n1) at +(60:\rad) [point, label = above:\(A\)] {};
612 \node (n2) at +(-145:\rad) [point, label = below:\(B\)] {};
613 \node (n3) at +(-45:\rad) [point, label = {below right:\(C\)}] {};
614
615 % triangle edges: connect the vertices, and leave a node at the midpoint
616 \draw[orange] (n3) -- node (a) [label = {above right:\(D\)}] {} (n1);
617 \draw[blue] (n3) -- node (b) [label = {below right:\(F\)}] {} (n2);
618 \draw[red] (n1) -- node (c) [label = {left: \(E\)}] {} (n2);
619
620 % Bisectors
621 % start at the point lying on the line from (origin) to (a), at
622 % twice that distance, and then draw a path going to the point on
623 % the line lying on the line from (a) to the (origin), at 3 times
624 % that distance.
625 \draw[orange, dotted]
626 ($ (origin) ! 2 ! (a) $)
627 node [right] {\sffamily Bisector \(AC\)}
628 -- ($(a) ! 3 ! (origin)$ );
629
630 % similarly for origin and b
631 \draw[blue, dotted]
632 ($ (origin) ! 2 ! (b) $)
633 -- ($(b) ! 3 ! (origin)$ )
634 node [right] {\sffamily Bisector \(BC\)};
635
636 \draw[red, dotted]
637 ($ (origin) ! 5 ! (c) $)
638 -- ($(c) ! 7 ! (origin)$ )
639 node [right] {\sffamily Bisector \(AB\)};
640
641 \draw[gray, dashed, thin] (n1) -- (origin) -- (n2);
642 \draw[gray, dashed, thin] (origin) -- (n3);
643
644 % Right angle symbols
645 \def\ralen{.5ex} % length of the short segment
646 \foreach \inter/\first/\last in {a/n3/origin, b/n2/origin, c/n2/origin}
647 {
648 \draw [thin] let \p1 = ($(\inter)!\ralen!(\first)$), % point along first path
649 \p2 = ($(\inter)!\ralen!(\last)$), % point along second path
650 \p3 = ($(\p1)+(\p2)-(\inter)$) % corner point
651 in
652 (\p1) -- (\p3) -- (\p2); % path
653 }
654\end{tikzpicture}
655
656 \begin{theorembox}{title=Perpendicular bisector theorem}
657 If a point \(P\) lies on the perpendicular bisector of line \(\overrightharp{XY}\), then \(P\) is equidistant from the endpoints of the bisected segment
658 \[ \text{i.e. } |\overrightharp{PX}| = |\overrightharp{PY}| \]
659 \end{theorembox}
660
661 \subsubsection*{Useful vector properties}
662
663 \begin{itemize}
664 \item
665 \(\boldsymbol{a} \parallel \boldsymbol{b} \implies \boldsymbol{b}=k\boldsymbol{a}\) for some
666 \(k \in \mathbb{R} \setminus \{0\}\)
667 \item
668 If \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are parallel with at
669 least one point in common, then they lie on the same straight line
670 \item
671 \(\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b}=0\)
672 \item
673 \(\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2\)
674 \end{itemize}
675
676 \subsection*{Linear dependence}
677
678 \(\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\) are linearly dependent if they are \(\nparallel\) and:
679 \begin{align*}
680 0&=k\boldsymbol{a}+l\boldsymbol{b}+m\boldsymbol{c}\\
681 \therefore \boldsymbol{c} &= m\boldsymbol{a} + n\boldsymbol{b} \quad \text{(simultaneous)}
682 \end{align*}
683
684 \noindent \(\boldsymbol{a}, \boldsymbol{b},\) and \(\boldsymbol{c}\) are linearly
685 independent if no vector in the set is expressible as a linear
686 combination of other vectors in set, or if they are parallel.
687
688 \subsection*{Three-dimensional vectors}
689
690 Right-hand rule for axes: \(z\) is up or out of page.
691
692 \tdplotsetmaincoords{60}{120}
693 \begin{center}\begin{tikzpicture} [scale=3, tdplot_main_coords, axis/.style={->,thick},
694 vector/.style={-stealth,red,very thick},
695 vector guide/.style={dashed,gray,thick}]
696
697 %standard tikz coordinate definition using x, y, z coords
698 \coordinate (O) at (0,0,0);
699
700 %tikz-3dplot coordinate definition using x, y, z coords
701
702 \pgfmathsetmacro{\ax}{1}
703 \pgfmathsetmacro{\ay}{1}
704 \pgfmathsetmacro{\az}{1}
705
706 \coordinate (P) at (\ax,\ay,\az);
707
708 %draw axes
709 \draw[axis] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$};
710 \draw[axis] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$};
711 \draw[axis] (0,0,0) -- (0,0,1) node[anchor=south]{$z$};
712
713 %draw a vector from O to P
714 \draw[vector] (O) -- (P);
715
716 %draw guide lines to components
717 \draw[vector guide] (O) -- (\ax,\ay,0);
718 \draw[vector guide] (\ax,\ay,0) -- (P);
719 \draw[vector guide] (P) -- (0,0,\az);
720 \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
721 \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
722 \draw[vector guide] (\ax,\ay,0) -- (\ax,0,0);
723 \node[tdplot_main_coords,above right]
724 at (\ax,\ay,\az){(\ax, \ay, \az)};
725 \end{tikzpicture}\end{center}
726
727 \subsection*{Parametric vectors}
728
729 Parametric equation of line through point \((x_0, y_0, z_0)\) and
730 parallel to \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) is:
731
732 \[\begin{cases}x = x_o + a \cdot t \\ y = y_0 + b \cdot t \\ z = z_0 + c \cdot t\end{cases}\]
733
734 \section{Circular functions}
735
736 \(\sin(bx)\) or \(\cos(bx)\): period \(=\frac{2\pi}{b}\)
737
738 \noindent \(\tan(nx)\): period \(=\frac{\pi}{n}\)\\
739 \indent\indent\indent asymptotes at \(x=\frac{(2k+1)\pi}{2n} \> \vert \> k \in \mathbb{Z}\)
740
741 \subsection*{Reciprocal functions}
742
743 \subsubsection*{Cosecant}
744
745 \[\operatorname{cosec} \theta = \frac{1}{\sin \theta} \> \vert \> \sin \theta \ne 0\]
746
747 \begin{itemize}
748 \item
749 \textbf{Domain} \(= \mathbb{R} \setminus {n\pi : n \in \mathbb{Z}}\)
750 \item
751 \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
752 \item
753 \textbf{Turning points} at
754 \(\theta = \frac{(2n + 1)\pi}{2} \> \vert \> n \in \mathbb{Z}\)
755 \item
756 \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
757 \end{itemize}
758
759 \subsubsection*{Secant}
760
761\begin{tikzpicture}
762 \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={\(-2\pi\), \(-\pi\), \(\pi\), \(2\pi\)}]
763% \addplot[blue, domain=-6.2830:6.2830,unbounded coords=jump,samples=80] {sec(deg(x))};
764 \addplot[blue, restrict y to domain=-10:10, domain=-7:7,samples=100] {sec(deg(x))} node [pos=0.93, black, right] {\(\operatorname{sec} x\)};
765 \addplot[red, dashed, domain=-7:7,samples=100] {cos(deg(x))};
766 \draw [gray, dotted, thick] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{rel axis cs:0,1});
767 \draw [gray, dotted, thick] ({axis cs:4.71239,0}|-{rel axis cs:0,0}) -- ({axis cs:4.71239,0}|-{rel axis cs:0,1});
768 \draw [gray, dotted, thick] ({axis cs:-4.71239,0}|-{rel axis cs:0,0}) -- ({axis cs:-4.71239,0}|-{rel axis cs:0,1});
769 \draw [gray, dotted, thick] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1});
770\end{axis}
771 \node [black] at (7,3.5) {\(\cos x\)};
772\end{tikzpicture}
773
774 \[\operatorname{sec} \theta = \frac{1}{\cos \theta} \> \vert \> \cos \theta \ne 0\]
775
776 \begin{itemize}
777
778 \item
779 \textbf{Domain}
780 \(= \mathbb{R} \setminus \frac{(2n + 1) \pi}{2} : n \in \mathbb{Z}\}\)
781 \item
782 \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
783 \item
784 \textbf{Turning points} at
785 \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
786 \item
787 \textbf{Asymptotes} at
788 \(\theta = \frac{(2n + 1) \pi}{2} \> \vert \> n \in \mathbb{Z}\)
789 \end{itemize}
790
791 \subsubsection*{Cotangent}
792
793\begin{tikzpicture}
794 \begin{axis}[xmin=-3,xmax=3,ymin=-1.5,ymax=1.5,enlargelimits=true, xtick={-3.1415, -1.5708, 1.5708, 3.1415},xticklabels={\(-\pi\), \(-\frac{\pi}{2}\), \(\frac{\pi}{2}\), \(\pi\)}]
795 \addplot[blue, smooth, domain=-3:-0.1,unbounded coords=jump,samples=105] {cot(deg(x))} node [pos=0.3, left] {\(\operatorname{cot} x\)};
796\addplot[blue, smooth, domain=0.1:3,unbounded coords=jump,samples=105] {cot(deg(x))};
797\addplot[red, smooth, dashed] gnuplot [domain=-1.5:1.5,unbounded coords=jump,samples=105] {tan(x)};
798\addplot[red, smooth, dashed] gnuplot [domain=-3.5:-1.8,unbounded coords=jump,samples=105] {tan(x)} node [pos=0.5, right] {\(\tan x\)};
799\addplot[red, smooth, dashed] gnuplot [domain=1.8:3.5,unbounded coords=jump,samples=105] {tan(x)};
800 \draw [thick, red, dotted] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1});
801 \draw [thick, blue, dotted] ({axis cs:-3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:-3.1415,0}|-{rel axis cs:0,1});
802 \draw [thick, blue, dotted] ({axis cs:0,0}|-{rel axis cs:0,0}) -- ({axis cs:0,0}|-{rel axis cs:0,1});
803 \draw [thick, blue, dotted] ({axis cs:3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:3.1415,0}|-{rel axis cs:0,1});
804 \draw [thick, red, dotted] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{rel axis cs:0,1});
805\end{axis}
806\end{tikzpicture}
807
808 \[\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0\]
809
810 \begin{itemize}
811
812 \item
813 \textbf{Domain} \(= \mathbb{R} \setminus \{n \pi: n \in \mathbb{Z}\}\)
814 \item
815 \textbf{Range} \(= \mathbb{R}\)
816 \item
817 \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
818 \end{itemize}
819
820 \subsubsection*{Symmetry properties}
821
822 \[\begin{split}
823 \operatorname{sec} (\pi \pm x) & = -\operatorname{sec} x \\
824 \operatorname{sec} (-x) & = \operatorname{sec} x \\
825 \operatorname{cosec} (\pi \pm x) & = \mp \operatorname{cosec} x \\
826 \operatorname{cosec} (-x) & = - \operatorname{cosec} x \\
827 \operatorname{cot} (\pi \pm x) & = \pm \operatorname{cot} x \\
828 \operatorname{cot} (-x) & = - \operatorname{cot} x
829 \end{split}\]
830
831 \subsubsection*{Complementary properties}
832
833 \[\begin{split}
834 \operatorname{sec} \left({\pi \over 2} - x\right) & = \operatorname{cosec} x \\
835 \operatorname{cosec} \left({\pi \over 2} - x\right) & = \operatorname{sec} x \\
836 \operatorname{cot} \left({\pi \over 2} - x\right) & = \tan x \\
837 \tan \left({\pi \over 2} - x\right) & = \operatorname{cot} x
838 \end{split}\]
839
840 \subsubsection*{Pythagorean identities}
841
842 \[\begin{split}
843 1 + \operatorname{cot}^2 x & = \operatorname{cosec}^2 x, \quad \text{where } \sin x \ne 0 \\
844 1 + \tan^2 x & = \operatorname{sec}^2 x, \quad \text{where } \cos x \ne 0
845 \end{split}\]
846
847 \subsection*{Compound angle formulas}
848
849 \[\cos(x \pm y) = \cos x + \cos y \mp \sin x \sin y\]
850 \[\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y\]
851 \[\tan(x \pm y) = {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}\]
852
853 \subsection*{Double angle formulas}
854
855 \[\begin{split}
856 \cos 2x &= \cos^2 x - \sin^2 x \\
857 & = 1 - 2\sin^2 x \\
858 & = 2 \cos^2 x -1
859 \end{split}\]
860
861 \[\sin 2x = 2 \sin x \cos x\]
862
863 \[\tan 2x = {{2 \tan x} \over {1 - \tan^2 x}}\]
864
865 \subsection*{Inverse circular functions}
866
867 \begin{tikzpicture}
868 \begin{axis}[ymin=-2, ymax=4, xmin=-1.1, xmax=1.1, ytick={-1.5708, 1.5708, 3.14159},yticklabels={$-\frac{\pi}{2}$, $\frac{\pi}{2}$, $\pi$}]
869 \addplot[color=red, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {asin(x)} node [pos=0.25, below right] {\(\sin^{-1}x\)};
870 \addplot[color=blue, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {acos(x)} node [pos=0.25, below left] {\(\cos^{-1}x\)};
871 \addplot[mark=*, red] coordinates {(-1,-1.5708)} node[right, font=\footnotesize]{\((-1,-\frac{\pi}{2})\)} ;
872 \addplot[mark=*, red] coordinates {(1,1.5708)} node[left, font=\footnotesize]{\((1,\frac{\pi}{2})\)} ;
873 \addplot[mark=*, blue] coordinates {(1,0)};
874 \addplot[mark=*, blue] coordinates {(-1,3.1415)} node[right, font=\footnotesize]{\((-1,\pi)\)} ;
875 \end{axis}
876 \end{tikzpicture}\\
877
878 Inverse functions: \(f(f^{-1}(x)) = x\) (restrict domain)
879
880 \[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y\]
881 \hfill where \(\sin y = x, \> y \in [{-\pi \over 2}, {\pi \over 2}]\)
882
883 \[\cos^{-1}: [-1,1] \rightarrow \mathbb{R}, \quad \cos^{-1} x = y\]
884 \hfill where \(\cos y = x, \> y \in [0, \pi]\)
885
886 \[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\]
887 \hfill where \(\tan y = x, \> y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\)
888
889 \begin{tikzpicture}
890 \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={\(-\frac{\pi}{2}\),\(\frac{\pi}{2}\)}]
891 \addplot[color=orange, smooth] gnuplot [domain=-35:35, unbounded coords=jump,samples=350] {atan(x)} node [pos=0.5, above left] {\(\tan^{-1}x\)};
892 \addplot[gray, dotted, thick, domain=-35:35] {1.5708} node [black, font=\footnotesize, below right, pos=0] {\(y=\frac{\pi}{2}\)};
893 \addplot[gray, dotted, thick, domain=-35:35] {-1.5708} node [black, font=\footnotesize, above left, pos=1] {\(y=-\frac{\pi}{2}\)};
894 \end{axis}
895 \end{tikzpicture}
896
897 \subsection*{Mensuration}
898
899 \begin{tikzpicture}[draw=blue!70,thick]
900 \filldraw[fill=lblue] circle (2cm);
901 \filldraw[fill=white]
902 (320:2cm) node[right] {}
903 -- (220:2cm) node[left] {}
904 arc[start angle=220, end angle=320, radius=2cm]
905 -- cycle;
906 \node {Major Segment};
907 \node at (-90:1.5) {Minor Segment};
908
909 \begin{scope}[xshift=4.5cm]
910 \draw [fill=lblue] circle (2cm);
911 \filldraw[fill=white]
912 (320:2cm) node[right] {}
913 -- (0,0) node[above] {}
914 -- (220:2cm) node[left] {}
915 arc[start angle=220, end angle=320, radius=2cm]
916 -- cycle;
917 \node at (90:1cm) {Major Sector};
918 \node at (-90:1.5) {Minor Sector};
919 \end{scope}
920 \end{tikzpicture}
921
922
923 \begin{align*}
924 \textbf{Sectors: } A &= \pi r^2 \dfrac{\theta}{2\pi} \\
925 &= \dfrac{r^2 \theta}{2}
926 \end{align*}
927
928 \[ \textbf{Segments: } A = \dfrac{r^2}{2} \left(\theta - \sin \theta \right) \]
929
930 \begin{align*}
931 \textbf{Chords: } \operatorname{crd} \theta &= \sqrt{(1 - \cos\theta)^2 + \sin^2 \theta} \\
932 &= \sqrt{2 - 2\cos\theta} \\
933 &= 2 \sin \left(\dfrac{\theta}{2}\right)
934 \end{align*}
935
936 \section{Differential calculus}
937
938 \[f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={\frac{dy}{dx}}\]
939
940 \subsection*{Limits}
941
942 \[\lim_{x \rightarrow a}f(x)\]
943 \(L^-,\quad L^+\) \qquad limit from below/above\\
944 \(\lim_{x \to a} f(x)\) \quad limit of a point\\
945
946 \noindent For solving \(x\rightarrow\infty\), put all \(x\) terms in denominators\\
947 e.g. \[\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2\]
948
949 \subsubsection*{Limit theorems}
950
951 \begin{enumerate}
952 \item
953 For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\)
954 \item
955 \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\)
956 \item
957 \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\)
958 \item
959 \(\therefore \lim_{x \rightarrow a} c \times f(x)=cF\) where \(c=\) constant
960 \item
961 \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\)
962 \item
963 \(f(x)\) is continuous \(\iff L^-=L^+=f(x) \> \forall x\)
964 \end{enumerate}
965
966 \subsection*{Gradients}
967
968 \textbf{Secant (chord)} - line joining two points on curve\\
969 \textbf{Tangent} - line that intersects curve at one point
970
971 \subsubsection*{Points of Inflection}
972
973 \emph{Stationary point} - i.e.
974 \(f^\prime(x)=0\)\\
975 \emph{Point of inflection} - max \(|\)gradient\(|\) (i.e.
976 \(f^{\prime\prime} = 0\))
977
978 \subsubsection*{Strictly increasing/decreasing}
979
980 For \(x_2\) and \(x_1\) where \(x_2 > x_1\):
981
982 \textbf{strictly increasing}\\
983 \-\hspace{1em}where \(f(x_2) > f(x_1)\) or \(f^\prime(x)>0\)
984
985 \textbf{strictly decreasing}\\
986 \hspace{1em}where \(f(x_2) < f(x_1)\) or \(f^\prime(x)<0\)
987 \begin{warning}
988 Endpoints are included, even where \(\boldsymbol{\frac{dy}{dx}=0}\)
989 \end{warning}
990
991
992 \subsection*{Second derivative}
993 \begin{align*}f(x) \longrightarrow &f^\prime (x) \longrightarrow f^{\prime\prime}(x)\\
994 \implies y \longrightarrow &\frac{dy}{dx} \longrightarrow \frac{d^2 y}{dx^2}\end{align*}
995
996 \noindent Order of polynomial \(n\)th derivative decrements each time the derivative is taken
997
998
999 \subsection*{Slope fields}
1000
1001 \begin{tikzpicture}[declare function={diff(\x,\y) = \x+\y;}]
1002 \begin{axis}[axis equal, ymin=-4, ymax=4, xmin=-4, xmax=4, ticks=none, enlargelimits=true, ]
1003 \addplot[thick, orange, domain=-4:2] {e^(x)-x-1};
1004 \pgfplotsinvokeforeach{-4,...,4}{%
1005 \draw[gray] ( {#1 -0.1}, {4 - diff(#1, 4) *0.1}) -- ( {#1 +0.1}, {4 + diff(#1, 4) *0.1});
1006 \draw[gray] ( {#1 -0.1}, {3 - diff(#1, 3) *0.1}) -- ( {#1 +0.1}, {3 + diff(#1, 3) *0.1});
1007 \draw[gray] ( {#1 -0.1}, {2 - diff(#1, 2) *0.1}) -- ( {#1 +0.1}, {2 + diff(#1, 2) *0.1});
1008 \draw[gray] ( {#1 -0.1}, {1 - diff(#1, 1) *0.1}) -- ( {#1 +0.1}, {1 + diff(#1, 1) *0.1});
1009 \draw[gray] ( {#1 -0.1}, {0 - diff(#1, 0) *0.1}) -- ( {#1 +0.1}, {0 + diff(#1, 0) *0.1});
1010 \draw[gray] ( {#1 -0.1}, {-1 - diff(#1, -1) *0.1}) -- ( {#1 +0.1}, {-1 + diff(#1, -1) *0.1});
1011 \draw[gray] ( {#1 -0.1}, {-2 - diff(#1, -2) *0.1}) -- ( {#1 +0.1}, {-2 + diff(#1, -2) *0.1});
1012 \draw[gray] ( {#1 -0.1}, {-3 - diff(#1, -3) *0.1}) -- ( {#1 +0.1}, {-3 + diff(#1, -3) *0.1});
1013 \draw[gray] ( {#1 -0.1}, {-4 - diff(#1, -4) *0.1}) -- ( {#1 +0.1}, {-4 + diff(#1, -4) *0.1});
1014 }
1015 \end{axis}
1016 \end{tikzpicture}
1017
1018 \begin{table*}[ht]
1019 \centering
1020 \begin{tabularx}{\textwidth}{|r|Y|Y|Y|}
1021 \hline
1022 \rowcolor{lblue}
1023 & \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)} \\
1024 \hline
1025 \(\dfrac{dy}{dx}>0\) &
1026 \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)}&
1027 \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)}&
1028 \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}\\
1029 \hline
1030 \(\dfrac{dy}{dx}<0\) &
1031 \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)}&
1032 \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)}&
1033 \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}\\
1034 \hline
1035 \(\dfrac{dy}{dx}=0\)&
1036 \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}&
1037 \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}\\
1038 \hline
1039 \end{tabularx}
1040 \end{table*}
1041 \begin{itemize}
1042 \item
1043 \(f^\prime (a) = 0, \> f^{\prime\prime}(a) > 0\) \\
1044 \textbf{local min} at \((a, f(a))\) (concave up)
1045 \item
1046 \(f^\prime (a) = 0, \> f^{\prime\prime} (a) < 0\) \\
1047 \textbf{local max} at \((a, f(a))\) (concave down)
1048 \item
1049 \(f^{\prime\prime}(a) = 0\) \\
1050 \textbf{point of inflection} at \((a, f(a))\)
1051 \item
1052 \(f^{\prime\prime}(a) = 0, \> f^\prime(a)=0\) \\
1053 stationary point of inflection at \((a, f(a)\)
1054 \end{itemize}
1055
1056 \subsection*{Implicit Differentiation}
1057
1058 \noindent Used for differentiating circles etc.
1059
1060 If \(p\) and \(q\) are expressions in \(x\) and \(y\) such that \(p=q\),
1061 for all \(x\) and \(y\), then:
1062
1063 \[{\frac{dp}{dx}} = {\frac{dq}{dx}} \quad \text{and} \quad {\frac{dp}{dy}} = {\frac{dq}{dy}}\]
1064
1065 \begin{cas}
1066 Action \(\rightarrow\) Calculation \\
1067 \-\hspace{1em}\texttt{impDiff(y\^{}2+ax=5,\ x,\ y)}
1068 \end{cas}
1069
1070 \subsection*{Function of the dependent
1071 variable}
1072
1073 If \({\frac{dy}{dx}}=g(y)\), then
1074 \(\frac{dx}{dy} = 1 \div \frac{dy}{dx} = \frac{1}{g(y)}\). Integrate both sides to solve equation. Only add \(c\) on one side. Express
1075 \(e^c\) as \(A\).
1076
1077 \subsection*{Reciprocal derivatives}
1078
1079 \[\frac{1}{\frac{dy}{dx}} = \frac{dx}{dy}\]
1080
1081 \subsection*{Differentiating \(x=f(y)\)}
1082 Find \(\dfrac{dx}{dy}\), then \(\dfrac{dy}{dx} = \dfrac{1}{\left(\dfrac{dx}{dy}\right)}\)
1083
1084 \subsection*{Parametric equations}
1085
1086
1087 \begin{align*}
1088 \dfrac{dy}{dt} &= \dfrac{dy}{dx} \cdot \dfrac{dx}{dt} \\
1089 \therefore \dfrac{dy}{dx} &= \dfrac{\left(\dfrac{dy}{dt}\right)}{\left(\dfrac{dx}{dt}\right)} \text{ provided } \dfrac{dx}{dt} \ne 0 \\
1090 \dfrac{d^2y}{dx^2} &= \dfrac{\left(\dfrac{dy^\prime}{dt}\right)}{\left(\dfrac{dx}{dt}\right)} \text{ where } y^\prime = \dfrac{dy}{dx}
1091 \end{align*}
1092
1093 \subsection*{Integration}
1094
1095 \[\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)\]
1096
1097 \subsubsection*{Properties}
1098
1099 \begin{align*}
1100 \int^b_a f(x) \> dx &= \int^c_a f(x) \> dx + \int^b_c f(x) \> dx \\
1101 \int^a_a f(x) \> dx &= 0 \\
1102 \int^b_a k \cdot f(x) \> dx &= k \int^b_a f(x) \> dx \\
1103 \int^b_a f(x) \pm g(x) \> dx &= \int^b_a f(x) \> dx \pm \int^b_a g(x) \> dx \\
1104 \int^b_a f(x) \> dx &= - \int^a_b f(x) \> dx \\
1105 \end{align*}
1106
1107 \subsection*{Integration by substitution}
1108
1109 \[\int f(u) {\frac{du}{dx}} \cdot dx = \int f(u) \cdot du\]
1110
1111 \begin{warning}
1112 \(\boldsymbol{f(u)}\) must be 1:1 \(\boldsymbol{\implies}\) one \(\boldsymbol{x}\) for each \(\boldsymbol{y}\)
1113 \end{warning}
1114 \begin{align*}\text{e.g. for } y&=\int(2x+1)\sqrt{x+4} \cdot dx\\
1115 \text{let } u&=x+4\\
1116 \implies& {\frac{du}{dx}} = 1\\
1117 \implies& x = u - 4\\
1118 \text{then } &y=\int (2(u-4)+1)u^{\frac{1}{2}} \cdot du\\
1119 &\text{(solve as normal integral)}
1120 \end{align*}
1121
1122 \subsubsection*{Definite integrals by substitution}
1123
1124 For \(\int^b_a f(x) {\frac{du}{dx}} \cdot dx\), evaluate new \(a\) and
1125 \(b\) for \(f(u) \cdot du\).
1126
1127 \subsubsection*{Trigonometric integration}
1128
1129 \[\sin^m x \cos^n x \cdot dx\]
1130
1131 \paragraph{\textbf{\(m\) is odd:}}
1132 \(m=2k+1\) where \(k \in \mathbb{Z}\)\\
1133 \(\implies \sin^{2k+1} x = (\sin^2 z)^k \sin x = (1 - \cos^2 x)^k \sin x\)\\
1134 Substitute \(u=\cos x\)
1135
1136 \paragraph{\textbf{\(n\) is odd:}}
1137 \(n=2k+1\) where \(k \in \mathbb{Z}\)\\
1138 \(\implies \cos^{2k+1} x = (\cos^2 x)^k \cos x = (1-\sin^2 x)^k \cos x\)\\
1139 Substitute \(u=\sin x\)
1140
1141 \paragraph{\textbf{\(m\) and \(n\) are even:}}
1142 use identities...
1143
1144 \begin{itemize}
1145
1146 \item
1147 \(\sin^2x={1 \over 2}(1-\cos 2x)\)
1148 \item
1149 \(\cos^2x={1 \over 2}(1+\cos 2x)\)
1150 \item
1151 \(\sin 2x = 2 \sin x \cos x\)
1152 \end{itemize}
1153
1154 \subsection*{Separation of variables}
1155
1156 If \({\frac{dy}{dx}}=f(x)g(y)\), then:
1157
1158 \[\int f(x) \> dx = \int \frac{1}{g(y)} \> dy\]
1159
1160 \subsection*{Partial fractions}
1161
1162 To factorise \(f(x) = \frac{\delta}{\alpha \cdot \beta}\):
1163 \begin{align*}
1164 \dfrac{\delta}{\alpha \cdot \beta \cdot \gamma} &= \dfrac{A}{\alpha} + \dfrac{B}{\beta} + \dfrac{C}{\gamma} \tag{1} \\
1165 \text{Multiply by } & (\alpha \cdot \beta \cdot \gamma) \text{:} \\
1166 \delta &= \beta\gamma A + \alpha\gamma B +\alpha\beta C \tag{2} \\
1167 \text{Substitute } x &= \{\alpha, \beta, \gamma\} \text{ into (2) to find denominators}
1168 \end{align*}
1169
1170 \subsubsection*{Repeated linear factors}
1171
1172 \[ \dfrac{p(x)}{(x-a)^n} = \dfrac{A_1}{(x-a)} + \dfrac{A_2}{(x-a)^2} + \dots + \dfrac{A_n}{(x-a)^n} \]
1173
1174 \subsubsection*{Irreducible quadratic factors}
1175
1176 \[ \text{e.g. } \dfrac{3x-4}{(2x-3)(x^2+5)} = \dfrac{A}{2x-3} + \dfrac{Bx+C}{x^2+5} \]
1177
1178 \begin{cas}
1179 Action \(\rightarrow\) Transformation:\\
1180 \-\hspace{1em} \texttt{expand(..., x)}
1181
1182 To reverse, use \texttt{combine(...)}
1183 \end{cas}
1184
1185 \subsection*{Integrating \(\boldsymbol{\dfrac{dy}{dx} = g(y)}\)}
1186
1187 \[ \text{if } \dfrac{dy}{dx} = g(y), \text{ then } x = \int{\dfrac{1}{g(y)}} \> dy \]
1188
1189 \subsection*{Graphing integrals on CAS}
1190
1191 \begin{cas}
1192 \textbf{In main:} Interactive \(\rightarrow\) Calculation \(\rightarrow\) \(\int\)\\
1193 For restrictions, \texttt{Define\ f(x)=...} then \texttt{f(x)\textbar{}x\textgreater{}...}
1194 \end{cas}
1195
1196 \subsection*{Solids of revolution}
1197
1198 Approximate as sum of infinitesimally-thick cylinders
1199
1200 \subsubsection*{Rotation about \(\boldsymbol{x}\)-axis}
1201
1202 \[ V = \pi\int^{x=b}_{x=a} f(x)^2 \> dx \]
1203
1204 \subsubsection*{Rotation about \(\boldsymbol{y}\)-axis}
1205
1206 \begin{align*}
1207 V &= \pi \int^{y=b}_{y=a} x^2 \> dy \\
1208 &= \pi \int^{y=b}_{y=a} (f(y))^2 \> dy
1209 \end{align*}
1210
1211 \subsubsection*{Regions not bound by \(\boldsymbol{y=0}\)}
1212
1213 \[V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx\]
1214 \hfill where \(f(x) > g(x)\)
1215
1216 \subsection*{Length of a curve}
1217
1218 For length of \(f(x)\) from \(x=a \rightarrow x=b\):
1219 \begin{align*}
1220 &\text{Cartesian} \> & L &= \int^b_a \sqrt{1 + \left(\dfrac{dy}{dx}\right)^2} \> dx \\
1221 &\text{Parametric} \> & L & = \int^b_a \sqrt{\left(\dfrac{dx}{dt}\right)^2 + \left(\dfrac{dy}{dt}\right)^2} \> dt
1222 \end{align*}
1223
1224 \begin{cas}
1225 \begin{enumerate}[label=\alph*), leftmargin=5mm]
1226 \item Evaluate formula
1227 \item Interactive \(\rightarrow\) Calculation \(\rightarrow\) Line \(\rightarrow\) \texttt{arcLen}
1228 \end{enumerate}
1229 \end{cas}
1230
1231 \subsection*{Applications of antidifferentiation}
1232
1233 \begin{itemize}
1234
1235 \item
1236 \(x\)-intercepts of \(y=f(x)\) identify \(x\)-coordinates of
1237 stationary points on \(y=F(x)\)
1238 \item
1239 nature of stationary points is determined by sign of \(y=f(x)\) on
1240 either side of its \(x\)-intercepts
1241 \item
1242 if \(f(x)\) is a polynomial of degree \(n\), then \(F(x)\) has degree
1243 \(n+1\)
1244 \end{itemize}
1245
1246 To find stationary points of a function, substitute \(x\) value of given
1247 point into derivative. Solve for \({\frac{dy}{dx}}=0\). Integrate to find
1248 original function.
1249
1250 \subsection*{Rates}
1251
1252 \subsubsection*{Gradient at a point on parametric curve}
1253
1254 \[{\frac{dy}{dx}} = {{\frac{dy}{dt}} \div {\frac{dx}{dt}}} \> \vert \> {\frac{dx}{dt}} \ne 0 \text{ (chain rule)}\]
1255
1256 \[\frac{d^2}{dx^2} = \frac{d(y^\prime)}{dx} = {\frac{dy^\prime}{dt} \div {\frac{dx}{dt}}} \> \vert \> y^\prime = {\frac{dy}{dx}}\]
1257
1258 \subsection*{Rational functions}
1259
1260 \[f(x) = \frac{P(x)}{Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}\]
1261
1262 \subsection*{Euler's method}
1263
1264 \[\dfrac{f(x+h) - f(x)}{h} \approx f^\prime (x) \quad \text{for small } h\]
1265
1266 \[\implies f(x+h) \approx f(x) + hf^\prime(x)\]
1267
1268 \begin{theorembox}{}
1269 If \(\dfrac{dy}{dx} = g(x)\) with \(x_0 = a\) and \(y_0 = b\), then:
1270 \[\begin{cases}
1271 x_{n+1} = x_n + h \\
1272 y_{n+1} = y_n + hg(x_n)
1273 \end{cases}\]
1274 \end{theorembox}
1275
1276 \[
1277 \dfrac{d^2y}{dx^2}
1278 \begin{cases}
1279 > 0 \implies \text{ underestimate (concave up)} \\
1280 < 0 \implies \text{ overestimate (concave down)}
1281 \end{cases}
1282 \]
1283
1284 \begin{center}\begin{tikzpicture}
1285 \begin{axis}[xmin=0, xmax=1.6, ticks=none, enlargelimits=true, samples=100]
1286 \addplot[blue, domain=-0.25:1.5, postaction={decorate,decoration={text along path, text align={align=center, left indent=3cm}, text={|\sffamily|solution curve}}}] {e^(x-3/2)+1/4};
1287 \addplot[red] {(x+1/2)*e^(-1)+1/4} (1.7,1.0593) node [above, black] {\(\ell\)};
1288 \addplot[mark=*, black] coordinates {(0.5,0.6179)} node[above left]{\((x_0, y_0)\)};
1289 \addplot[mark=*, orange] coordinates {(1.4,1.1548)} node[left]{\color{black} \sffamily correct solution};
1290 \addplot[mark=*, black] coordinates {(1.4,0.94897)} node[above right] {\((x_1,y_1)\)};
1291 \draw [gray, dashed] (0.5,0) -- (0.5,0.6179) -- (1.6,0.6179);
1292 \draw [gray, dashed] (1.4,0) -- (1.4, 1.1548);
1293 \draw [<->] (0.5,0.48) -- (1.4,0.48) node[midway, fill=white] {\(h\)};
1294 \draw [gray, dashed] (1.4,0.94897) -- (1.6,0.94897);
1295 \draw [<->] (1.5,0.94897) -- (1.5,0.6179) node[midway, rotate=90, below] {\(hg(x_0)\)};
1296 \end{axis}
1297 \end{tikzpicture}\end{center}
1298
1299 \begin{cas}
1300 Menu \(\rightarrow\) Sequence \(\rightarrow\) Recursive
1301
1302 \textbf{To generate \(\boldsymbol{x}\)-values:}
1303 \begin{itemize}
1304 \item Enter \(a_{n+1}=a_n + h\) where \(h\) is the step size \\
1305 (input \(a_n\) from menu bar)
1306 \item In \(a_0\), set the initial value \(x_0\) as a constant
1307 \end{itemize}
1308
1309 \textbf{To generate \(\boldsymbol{y}\)-values:}
1310 \begin{itemize}
1311 \item In \(b_{n+1}\), enter \(\dfrac{dy}{dx}\), replacing \(x\) with \(a_n\)
1312 \item Set \(b_0 = y(x_0)\) as a constant
1313 \end{itemize}
1314
1315 To view table of values, tap table icon (top left) \\
1316 To compare approximations with actual values, enter in \(c_{n+1} = a_{n+1} - f(a_{n+1})\) where \(f(x) = \int \dfrac{dy}{dx} \> dx\)
1317
1318 \end{cas}
1319
1320 \subsection*{Fundamental theorem of calculus}
1321
1322 If \(f\) is continuous on \([a, b]\), then
1323
1324 \[\int^b_a f(x) \> dx = F(b) - F(a)\]
1325 \hfill where \(F = \int f \> dx\)
1326
1327 \subsection*{Differential equations}
1328
1329 \noindent\textbf{Order} - highest power inside derivative\\
1330 \textbf{Degree} - highest power of highest derivative\\
1331 e.g. \({\left(\dfrac{dy^2}{d^2} x\right)}^3\) \qquad order 2, degree 3
1332
1333 \begin{warning}
1334 To verify solutions, find \(\frac{dy}{dx}\) from \(y\) and substitute into original
1335 \end{warning}
1336
1337 \vspace*{1cm}
1338 \hspace*{-1cm}
1339
1340 { \tabulinesep=1.2mm
1341 \begin{tabu}{|c|c|}
1342
1343 \hline
1344 \taburowcolors 2{gray..white}
1345 \textbf{DE} & \textbf{Method} \\
1346 \hline
1347
1348 \tabureset
1349 \(\dfrac{dy}{dx} = f(x)\)
1350 &
1351 {\(\begin{aligned}
1352 y &= \int f(x) \> dx \\
1353 &= F(x) + c \quad \text{where } F^\prime(x) = f(x)
1354 \end{aligned}\)} \\
1355
1356 \hline
1357
1358 \(\dfrac{d^2y}{dx^2} = f(x)\)
1359 &
1360 {\(\begin{aligned}
1361 \dfrac{dy}{dx} &= \int f(x) \> dx \\
1362 &= F(x) + c \quad \text{where } F^\prime(x) = f(x) \\
1363 \therefore y &= \iint f(x) \> dx = \int \left( F(x) + c \right) \> dx \\
1364 &= G(x) + cx + d \\
1365 & \text{where } G^\prime(x) = F(x)
1366 \end{aligned}\)} \\
1367
1368 \hline
1369
1370 \(\dfrac{dy}{dx} = g(y)\)
1371 &
1372 {\(\begin{aligned}
1373 \dfrac{dx}{dy} &= \dfrac{1}{g(y)} \\
1374 \therefore x &= \int \dfrac{1}{g(y)} \> dy \\
1375 &= F(y) + c \\
1376 & \text{where } F^\prime(y) = \dfrac{1}{g(y)}
1377 \end{aligned}\)} \\
1378
1379 \hline
1380
1381 \(\dfrac{dy}{dx} = f(x) g(y)\)
1382 &
1383 {\(\begin{aligned}
1384 f(x) &= \dfrac{1}{g(y)} \cdot \dfrac{dy}{dx} \\
1385 \int f(x) \> dx &= \int \dfrac{1}{g(y)} \> dy
1386 \end{aligned}\)} \\
1387
1388 \hline
1389 \end{tabu}}
1390
1391 \subsubsection*{Mixing problems}
1392
1393 \[\left(\frac{dm}{dt}\right)_\Sigma = \left(\frac{dm}{dt}\right)_{\text{in}} - \left(\frac{dm}{dt}_{\text{out}}\right)\]
1394
1395 \include{calculus-rules}
1396
1397 \section{Kinematics \& Mechanics}
1398
1399 \subsection*{Constant acceleration}
1400
1401 \begin{itemize}
1402 \item \textbf{Position} - relative to origin
1403 \item \textbf{Displacement} - relative to starting point
1404 \end{itemize}
1405
1406 \subsubsection*{Velocity-time graphs}
1407
1408 \begin{description}[nosep, labelindent=0.5cm, leftmargin=0.5\columnwidth]
1409 \item[Displacement:] \textit{signed} area
1410 \item[Distance travelled:] \textit{total} area
1411 \end{description}
1412
1413 \[ \text{acceleration} = \frac{d^2x}{dt^2} = \frac{dv}{dt} = v\frac{dv}{dx} = \frac{d}{dx}\left(\frac{1}{2}v^2\right) \]
1414
1415 \begin{center}
1416 \renewcommand{\arraystretch}{1}
1417 \begin{tabular}{ l r }
1418 \hline & no \\ \hline
1419 \(v=u+at\) & \(x\) \\
1420 \(v^2 = u^2+2as\) & \(t\) \\
1421 \(s = \frac{1}{2} (v+u)t\) & \(a\) \\
1422 \(s = ut + \frac{1}{2} at^2\) & \(v\) \\
1423 \(s = vt- \frac{1}{2} at^2\) & \(u\) \\ \hline
1424 \end{tabular}
1425 \end{center}
1426
1427 \[ v_{\text{avg}} = \frac{\Delta\text{position}}{\Delta t} \]
1428 \begin{align*}
1429 \text{speed} &= |{\text{velocity}}| \\
1430 &= \sqrt{v_x^2 + v_y^2 + v_z^2}
1431 \end{align*}
1432
1433 \noindent \textbf{Distance travelled between \(t=a \rightarrow t=b\):}
1434 \begin{align*}
1435 &= \int^{b}_{a}{\sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}} \> dt \tag{2D} \\
1436 &= \int^{t=b}_{t=a}{\dfrac{dx}{dt}} \> dt \tag{linear}
1437 \end{align*}
1438
1439 \noindent \textbf{Shortest distance between \(\boldsymbol{r}(t_0)\) and \(\boldsymbol{r}(t_1)\):}
1440 \[ = |\boldsymbol{r}(t_1) - \boldsymbol{r}(t_2)| \]
1441
1442 \subsection*{Vector functions}
1443
1444 \[ \boldsymbol{r}(t) = x \boldsymbol{i} + y \boldsymbol{j} + z \boldsymbol{k} \]
1445
1446 \begin{itemize}
1447 \item If \(\boldsymbol{r}(t) \equiv\) position with time, then the graph of endpoints of \(\boldsymbol{r}(t) \equiv\) Cartesian path
1448 \item Domain of \(\boldsymbol{r}(t)\) is the range of \(x(t)\)
1449 \item Range of \(\boldsymbol{r}(t)\) is the range of \(y(t)\)
1450 \end{itemize}
1451
1452 \subsection*{Vector calculus}
1453
1454 \subsubsection*{Derivative}
1455
1456 Let \(\boldsymbol{r}(t)=x(t)\boldsymbol{i} + y(t)\boldsymbol(j)\). If both \(x(t)\) and \(y(t)\) are differentiable, then:
1457 \[ \boldsymbol{r}(t)=x(t)\boldsymbol{i}+y(t)\boldsymbol{j} \]
1458
1459 \subfile{dynamics}
1460 \subfile{statistics}
1461 \end{multicols}
1462\end{document}