e2ccd0eb7d21af85877eb08eae93d311be6d0727
1\documentclass[a4paper]{article}
2\usepackage[a4paper,margin=2cm]{geometry}
3\usepackage{multicol}
4\usepackage{multirow}
5\usepackage{amsmath}
6\usepackage{amssymb}
7\usepackage{harpoon}
8\usepackage{tabularx}
9\usepackage[dvipsnames, table]{xcolor}
10\usepackage{blindtext}
11\usepackage{graphicx}
12\usepackage{wrapfig}
13\usepackage{tikz}
14\usepackage{tikz-3dplot}
15\usepackage{pgfplots}
16\usetikzlibrary{calc}
17\usetikzlibrary{angles}
18\usetikzlibrary{datavisualization.formats.functions}
19\usetikzlibrary{decorations.markings}
20\usepgflibrary{arrows.meta}
21\usepackage{fancyhdr}
22\pagestyle{fancy}
23\fancyhead[LO,LE]{Year 12 Specialist}
24\fancyhead[CO,CE]{Andrew Lorimer}
25
26\usepackage{mathtools}
27\usepackage{xcolor} % used only to show the phantomed stuff
28\renewcommand\hphantom[1]{{\color[gray]{.6}#1}} % comment out!
29\setlength\fboxsep{0pt} \setlength\fboxrule{.2pt} % for the \fboxes
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32\newcolumntype{L}[1]{>{\hsize=#1\hsize\raggedright\arraybackslash}X}%
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34\definecolor{cas}{HTML}{e6f0fe}
35\linespread{1.5}
36\newcommand{\midarrow}{\tikz \draw[-triangle 90] (0,0) -- +(.1,0);}
37
38\begin{document}
39
40\begin{multicols}{2}
41
42 \section{Complex numbers}
43
44 \[\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}\]
45
46 \begin{align*}
47 \text{Cartesian form: } & a+bi\\
48 \text{Polar form: } & r\operatorname{cis}\theta
49 \end{align*}
50
51 \subsection*{Operations}
52
53 \definecolor{shade1}{HTML}{ffffff}
54 \definecolor{shade2}{HTML}{e6f2ff}
55 \definecolor{shade3}{HTML}{cce2ff}
56 \begin{tabularx}{\columnwidth}{r|X|X}
57 & \textbf{Cartesian} & \textbf{Polar} \\
58 \hline
59 \(z_1 \pm z_2\) & \((a \pm c)(b \pm d)i\) & convert to \(a+bi\)\\
60 \hline
61 \(+k \times z\) & \multirow{2}{*}{\(ka \pm kbi\)} & \(kr\operatorname{cis} \theta\)\\
62 \cline{1-1}\cline{3-3}
63 \(-k \times z\) & & \(kr \operatorname{cis}(\theta\pm \pi)\)\\
64 \hline
65 \(z_1 \cdot z_2\) & \(ac-bd+(ad+bc)i\) & \(r_1r_2 \operatorname{cis}(\theta_1 + \theta_2)\)\\
66 \hline
67 \(z_1 \div z_2\) & \((z_1 \overline{z_2}) \div |z_2|^2\) & \(\left(\frac{r_1}{r_2}\right) \operatorname{cis}(\theta_1 - \theta_2)\)
68 \end{tabularx}
69
70 \subsubsection*{Scalar multiplication in polar form}
71
72 For \(k \in \mathbb{R}^+\):
73 \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\theta\]
74
75 \noindent For \(k \in \mathbb{R}^-\):
76 \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\left(\begin{cases}\theta - \pi & |0<\operatorname{Arg}(z)\le \pi \\ \theta + \pi & |-\pi<\operatorname{Arg}(z)\le 0\end{cases}\right)\]
77
78 \subsection*{Conjugate}
79
80 \begin{align*}
81 \overline{z} &= a \mp bi\\
82 &= r \operatorname{cis}(-\theta)
83 \end{align*}
84
85 \noindent \colorbox{cas}{On CAS: \texttt{conjg(a+bi)}}
86
87 \subsubsection*{Properties}
88
89 \begin{align*}
90 \overline{z_1 \pm z_2} &= \overline{z_1}\pm\overline{z_2}\\
91 \overline{z_1 \cdot z_2} &= \overline{z_1}\cdot\overline{z_2}\\
92 \overline{kz} &= k\overline{z} \quad | \quad k \in \mathbb{R}\\
93 z\overline{z} &= (a+bi)(a-bi)\\
94 &= a^2 + b^2\\
95 &= |z|^2
96 \end{align*}
97
98 \subsection*{Modulus}
99
100 \[|z|=|\vec{Oz}|=\sqrt{a^2 + b^2}\]
101
102 \subsubsection*{Properties}
103
104 \begin{align*}
105 |z_1z_2|&=|z_1||z_2|\\
106 \left|\frac{z_1}{z_2}\right|&=\frac{|z_1|}{|z_2|}\\
107 |z_1+z_2|&\le|z_1|+|z_2|
108 \end{align*}
109
110 \subsection*{Multiplicative inverse}
111
112 \begin{align*}
113 z^{-1}&=\frac{a-bi}{a^2+b^2}\\
114 &=\frac{\overline{z}}{|z|^2}a\\
115 &=r \operatorname{cis}(-\theta)
116 \end{align*}
117
118 \subsection*{Dividing over \(\mathbb{C}\)}
119
120 \begin{align*}
121 \frac{z_1}{z_2}&=z_1z_2^{-1}\\
122 &=\frac{z_1\overline{z_2}}{|z_2|^2}\\
123 &=\frac{(a+bi)(c-di)}{c^2+d^2}\\
124 & \qquad \text{(rationalise denominator)}
125 \end{align*}
126
127 \subsection*{Polar form}
128
129 \begin{align*}
130 z&=r\operatorname{cis}\theta\\
131 &=r(\cos \theta + i \sin \theta)
132 \end{align*}
133
134 \begin{itemize}
135 \item{\(r=|z|=\sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}\)}
136 \item{\(\theta = \operatorname{arg}(z)\) \quad \colorbox{cas}{On CAS: \texttt{arg(a+bi)}}}
137 \item{\(\operatorname{Arg}(z) \in (-\pi,\pi)\) \quad \bf{(principal argument)}}
138 \item{\colorbox{cas}{Convert on CAS:}\\ \verb|compToTrig(a+bi)| \(\iff\) \verb|cExpand{r·cisX}|}
139 \item{Multiple representations:\\\(r\operatorname{cis}\theta=r\operatorname{cis}(\theta+2n\pi)\) with \(n \in \mathbb{Z}\) revolutions}
140 \item{\(\operatorname{cis}\pi=-1,\qquad \operatorname{cis}0=1\)}
141 \end{itemize}
142
143 \subsection*{de Moivres' theorem}
144
145 \[(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}\]
146
147 \subsection*{Complex polynomials}
148
149 Include \(\pm\) for all solutions, incl. imaginary
150
151 \begin{tabularx}{\columnwidth}{ R{0.55} X }
152 \hline
153 Sum of squares & \(\begin{aligned}
154 z^2 + a^2 &= z^2-(ai)^2\\
155 &= (z+ai)(z-ai) \end{aligned}\) \\
156 \hline
157 Sum of cubes & \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)\\
158 \hline
159 Division & \(P(z)=D(z)Q(z)+R(z)\) \\
160 \hline
161 Remainder theorem & Let \(\alpha \in \mathbb{C}\). Remainder of \(P(z) \div (z-\alpha)\) is \(P(\alpha)\)\\
162 \hline
163 Factor theorem & \(z-\alpha\) is a factor of \(P(z) \iff P(\alpha)=0\) for \(\alpha \in \mathbb{C}\)\\
164 \hline
165 Conjugate root theorem & \(P(z)=0 \text{ at } z=a\pm bi\) (\(\implies\) both \(z_1\) and \(\overline{z_1}\) are solutions)\\
166 \hline
167 \end{tabularx}
168
169 \subsection*{\(n\)th roots}
170
171 \(n\)th roots of \(z=r\operatorname{cis}\theta\) are:
172
173 \[z = r^{\frac{1}{n}} \operatorname{cis}\left(\frac{\theta+2k\pi}{n}\right)\]
174
175 \begin{itemize}
176
177 \item{Same modulus for all solutions}
178 \item{Arguments separated by \(\frac{2\pi}{n} \therefore\) there are \(n\) roots}
179 \item{If one square root is \(a+bi\), the other is \(-a-bi\)}
180 \item{Give one implicit \(n\)th root \(z_1\), function is \(z=z_1^n\)}
181 \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 \(\frac{2\pi}{n}\))}
182 \end{itemize}
183
184 \noindent For \(0=az^2+bz+c\), use quadratic formula:
185
186 \[z=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]
187
188 \subsection*{Fundamental theorem of algebra}
189
190 A polynomial of degree \(n\) can be factorised into \(n\) linear factors in \(\mathbb{C}\):
191
192 \[\implies P(z)=a_n(z-\alpha_1)(z-\alpha_2)(z-\alpha_3)\dots(z-\alpha_n)\]
193 \[\text{ where } \alpha_1,\alpha_2,\alpha_3,\dots,\alpha_n \in \mathbb{C}\]
194
195 \subsection*{Argand planes}
196
197 \begin{center}\begin{tikzpicture}[scale=2]
198 \draw [->] (-0.2,0) -- (1.5,0) node [right] {$\operatorname{Re}(z)$};
199 \draw [->] (0,-0.2) -- (0,1.5) node [above] {$\operatorname{Im}(z)$};
200 \coordinate (P) at (1,1);
201 \coordinate (a) at (1,0);
202 \coordinate (b) at (0,1);
203 \coordinate (O) at (0,0);
204 \draw (0,0) -- (P) node[pos=0.5, above left]{\(r\)} node[pos=1, right]{\(\begin{aligned}z&=a+bi\\&=r\operatorname{cis}\theta\end{aligned}\)};
205 \draw [gray, dashed] (1,1) -- (1,0) node[black, pos=1, below]{\(a\)};
206 \draw [gray, dashed] (1,1) -- (0,1) node[black, pos=1, left]{\(b\)};
207 \begin{scope}
208 \path[clip] (O) -- (P) -- (a);
209 \fill[red, opacity=0.5, draw=black] (O) circle (2mm);
210 \node at ($(O)+(20:3mm)$) {$\theta$};
211 \end{scope}
212 \filldraw (P) circle (0.5pt);
213 \end{tikzpicture}\end{center}
214
215 \begin{itemize}
216 \item{Multiplication by \(i \implies\) CCW rotation of \(\frac{\pi}{2}\)}
217 \item{Addition: \(z_1 + z_2 \equiv\) \overrightharp{\(Oz_1\)} + \overrightharp{\(Oz_2\)}}
218 \end{itemize}
219
220 \subsection*{Sketching complex graphs}
221
222 \subsubsection*{Linear}
223
224 \begin{itemize}
225 \item{\(\operatorname{Re}(z)=c\) or \(\operatorname{Im}(z)=c\) (perpendicular bisector)}
226 \item{\(\operatorname{Im}(z)=m\operatorname{Re}(z)\)}
227 \item{\(|z+a|=|z+b| \implies 2(a-b)x=b^2-a^2\)\\Geometric: equidistant from \(a,b\)}
228 \end{itemize}
229
230 \subsubsection*{Circles}
231
232 \begin{itemize}
233 \item \(|z-z_1|^2=c^2|z_2+2|^2\)
234 \item \(|z-(a+bi)|=c \implies (x-a)^2+_(y-b)^2=c^2\)
235 \end{itemize}
236
237 \noindent \textbf{Loci} \qquad \(\operatorname{Arg}(z)<\theta\)
238
239 \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
240 \draw [->] (0,0) -- (1,0) node [right] {$\operatorname{Re}(z)$};
241 \draw [->] (0,-0.5) -- (0,1) node [above] {$\operatorname{Im}(z)$};
242 \draw [<-, dashed, thick, blue] (-1,0) -- (0,0);
243 \draw [->, thick, blue] (0,0) -- (1,1);
244 \fill [gray, opacity=0.2, domain=-1:1, variable=\x] (-1,-0.5) -- (-1,0) -- (0, 0) -- (1,1) -- (1,-0.5) -- cycle;
245 \begin{scope}
246 \path[clip] (0,0) -- (1,1) -- (1,0);
247 \fill[red, opacity=0.5, draw=black] (0,0) circle (2mm);
248 \node at ($(0,0)+(20:3mm)$) {$\frac{\pi}{4}$};
249 \end{scope}
250 \node [font=\footnotesize] at (0.5,-0.25) {\(\operatorname{Arg}(z)\le\frac{\pi}{4}\)};
251 \node [blue, mydot] {};
252 \end{tikzpicture}\end{center}
253
254 \noindent \textbf{Rays} \qquad \(\operatorname{Arg}(z-b)=\theta\)
255
256 \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
257 \draw [->] (-0.75,0) -- (1.5,0) node [right] {$\operatorname{Re}(z)$};
258 \draw [->] (0,-1) -- (0,1) node [above] {$\operatorname{Im}(z)$};
259 \draw [->, thick, brown] (-0.25,0) -- (-0.75,-1);
260 \node [above, font=\footnotesize] at (-0.25,0) {\(\frac{1}{4}\)};
261 \begin{scope}
262 \path[clip] (-0.25,0) -- (-0.75,-1) -- (0,0);
263 \fill[orange, opacity=0.5, draw=black] (-0.25,0) circle (2mm);
264 \end{scope}
265 \node at (-0.08,-0.3) {\(\frac{\pi}{8}\)};
266 \node [font=\footnotesize, left] at (-0.75,-1) {\(\operatorname{Arg}(z+\frac{1}{4})=\frac{\pi}{8}\)};
267 \node [brown, mydot] at (-0.25,0) {};
268 \draw [<->, thick, green] (0,-1) -- (1.5,0.5) node [pos=0.25, black, font=\footnotesize, right] {\(|z-2|=|z-(1+i)|\)};
269 \node [left, font=\footnotesize] at (0,-1) {\(-1\)};
270 \node [below, font=\footnotesize] at (1,0) {\(1\)};
271 \end{tikzpicture}\end{center}
272
273 \section{Vectors}
274 \begin{center}\begin{tikzpicture}
275 \draw [->] (-0.5,0) -- (3,0) node [right] {\(x\)};
276 \draw [->] (0,-0.5) -- (0,3) node [above] {\(y\)};
277 \draw [orange, ->, thick] (0.5,0.5) -- (2.5,2.5) node [pos=0.5, above] {\(\vec{u}\)};
278 \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
279 \draw [gray, dashed, thick] (0.5,0.5) -- (2.5,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, below]{\(x\vec{i}\)};
280 \draw [gray, dashed, thick] (2.5,0.5) -- (2.5,2.5) node [pos=0.5] {\midarrow};
281 \end{scope}
282 \node[black, right] at (2.5,1.5) {\(y\vec{j}\)};
283 \end{tikzpicture}\end{center}
284 \subsection*{Column notation}
285
286 \[\begin{bmatrix}x\\ y \end{bmatrix} \iff x\boldsymbol{i} + y\boldsymbol{j}\]
287 \(\begin{bmatrix}x_2-x_1\\ y_2-y_1 \end{bmatrix}\) \quad between \(A(x_1,y_1), \> B(x_2,y_2)\)
288
289 \subsection*{Scalar multiplication}
290
291 \[k\cdot (x\boldsymbol{i}+y\boldsymbol{j})=kx\boldsymbol{i}+ky\boldsymbol{j}\]
292
293 \noindent For \(k \in \mathbb{R}^-\), direction is reversed
294
295 \subsection*{Vector addition}
296 \begin{center}\begin{tikzpicture}[scale=1]
297 \coordinate (A) at (0,0);
298 \coordinate (B) at (2,2);
299 \draw [->, thick, red] (0,0) -- (2,2) node [pos=0.5, below right] {\(\vec{u}=2\vec{i}+2\vec{j}\)};
300 \draw [->, thick, blue] (2,2) -- (1,4) node [pos=0.5, above right] {\(\vec{v}=-\vec{i}+2\vec{j}\)};
301 \draw [->, thick, orange] (0,0) -- (1,4) node [pos=0.5, left] {\(\vec{u}+\vec{v}=\vec{i}+4\vec{j}\)};
302 \end{tikzpicture}\end{center}
303
304 \[(x\boldsymbol{i}+y\boldsymbol{j}) \pm (a\boldsymbol{i}+b\boldsymbol{j})=(x \pm a)\boldsymbol{i}+(y \pm b)\boldsymbol{j}\]
305
306 \begin{itemize}
307 \item Draw each vector head to tail then join lines
308 \item Addition is commutative (parallelogram)
309 \item \(\boldsymbol{u}-\boldsymbol{v}=\boldsymbol{u}+(-\boldsymbol{v}) \implies \overrightharp{AB}=\boldsymbol{b}-\boldsymbol{a}\)
310 \end{itemize}
311
312 \subsection*{Magnitude}
313
314 \[|(x\boldsymbol{i} + y\boldsymbol{j})|=\sqrt{x^2+y^2}\]
315
316 \subsection*{Parallel vectors}
317
318 \[\boldsymbol{u} || \boldsymbol{v} \iff \boldsymbol{u} = k \boldsymbol{v} \text{ where } k \in \mathbb{R} \setminus \{0\}\]
319
320 For parallel vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\):\\
321 \[\boldsymbol{a \cdot b}=\begin{cases}
322 |\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction}\\
323 -|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions}
324 \end{cases}\]
325 %\includegraphics[width=0.2,height=\textheight]{graphics/parallelogram-vectors.jpg}
326 %\includegraphics[width=1]{graphics/vector-subtraction.jpg}
327
328 \subsection*{Perpendicular vectors}
329
330 \[\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b} = 0\ \quad \text{(since \(\cos 90 = 0\))}\]
331
332 \subsection*{Unit vector \(|\hat{\boldsymbol{a}}|=1\)}
333 \[\begin{split}\hat{\boldsymbol{a}} & = {\frac{1}{|\boldsymbol{a}|}}\boldsymbol{a} \\ & = \boldsymbol{a} \cdot {|\boldsymbol{a}|}\end{split}\]
334
335 \subsection*{Scalar product \(\boldsymbol{a} \cdot \boldsymbol{b}\)}
336
337
338 \begin{center}\begin{tikzpicture}[scale=2]
339 \draw [->] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{b}\)};
340 \draw [->] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{a}\)};
341 \begin{scope}
342 \path[clip] (1,0.5) -- (1,0) -- (0,0);
343 \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
344 \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
345 \end{scope}
346 \end{tikzpicture}\end{center}
347 \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*}
348 \noindent\colorbox{cas}{On CAS: \texttt{dotP({[}a\ b\ c{]},\ {[}d\ e\ f{]})}}
349
350 \subsubsection*{Properties}
351
352 \begin{enumerate}
353 \item
354 \(k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k\boldsymbol{b})\)
355 \item
356 \(\boldsymbol{a \cdot 0}=0\)
357 \item
358 \(\boldsymbol{a} \cdot (\boldsymbol{b} + \boldsymbol{c})=\boldsymbol{a} \cdot \boldsymbol{b} + \boldsymbol{a} \cdot \boldsymbol{c}\)
359 \item
360 \(\boldsymbol{i \cdot i} = \boldsymbol{j \cdot j} = \boldsymbol{k \cdot k}= 1\)
361 \item
362 \(\boldsymbol{a} \cdot \boldsymbol{b} = 0 \quad \implies \quad \boldsymbol{a} \perp \boldsymbol{b}\)
363 \item
364 \(\boldsymbol{a \cdot a} = |\boldsymbol{a}|^2 = a^2\)
365 \end{enumerate}
366
367 \subsection*{Angle between vectors}
368
369 \[\cos \theta = \frac{\boldsymbol{a} \cdot \boldsymbol{b}}{|\boldsymbol{a}| |\boldsymbol{b}|} = \frac{a_1 b_1 + a_2 b_2}{|\boldsymbol{a}| |\boldsymbol{b}|}\]
370
371 \noindent \colorbox{cas}{On CAS:} \texttt{angle([a b c], [a b c])}
372
373 (Action \(\rightarrow\) Vector \(\rightarrow\)Angle)
374
375 \subsection*{Angle between vector and axis}
376
377 \noindent For\(\boldsymbol{a} = a_1 \boldsymbol{i} + a_2 \boldsymbol{j} + a_3 \boldsymbol{k}\)
378 which makes angles \(\alpha, \beta, \gamma\) with positive side of
379 \(x, y, z\) axes:
380 \[\cos \alpha = \frac{a_1}{|\boldsymbol{a}|}, \quad \cos \beta = \frac{a_2}{|\boldsymbol{a}|}, \quad \cos \gamma = \frac{a_3}{|\boldsymbol{a}|}\]
381
382 \noindent \colorbox{cas}{On CAS:} \texttt{angle({[}a\ b\ c{]},\ {[}1\ 0\ 0{]})}\\for angle
383 between \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) and
384 \(x\)-axis
385
386 \subsection*{Projections \& resolutes}
387
388 \begin{tikzpicture}[scale=3]
389 \draw [->, purple] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{a}\)};
390 \draw [->, orange] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{u}\)};
391 \draw [->, blue] (1,0) -- (2,0) node [pos=0.5, below] {\(\boldsymbol{b}\)};
392 \begin{scope}
393 \path[clip] (1,0.5) -- (1,0) -- (0,0);
394 \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
395 \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
396 \end{scope}
397 \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
398 \draw [gray, dashed, thick] (1,0) -- (1,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, right, rotate=-90]{\(\boldsymbol{w}\)};
399 \end{scope}
400 \draw (0,0) coordinate (O)
401 (1,0) coordinate (A)
402 (1,0.5) coordinate (B)
403 pic [draw,red,angle radius=2mm] {right angle = O--A--B};
404 \end{tikzpicture}
405
406 \subsubsection*{\(\parallel\boldsymbol{b}\) (vector projection/resolute)}
407
408 \begin{align*}
409 \boldsymbol{u} & = \frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|^2}\boldsymbol{b} \\
410 & = \left(\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\right)\left(\frac{\boldsymbol{b}}{|\boldsymbol{b}|}\right) \\
411 & = (\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}
412 \end{align*}
413
414 \subsubsection*{\(\perp\boldsymbol{b}\) (perpendicular projection)}
415 \[\boldsymbol{w} = \boldsymbol{a} - \boldsymbol{u}\]
416
417 \subsubsection*{\(|\boldsymbol{u}|\) (scalar projection/resolute)}
418 \begin{align*}
419 s &= |\boldsymbol{u}|\\
420 &= \boldsymbol{a} \cdot \hat{\boldsymbol{b}}\\
421 &=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\\
422 &= |\boldsymbol{a}| \cos \theta
423 \end{align*}
424
425 \subsubsection*{Rectangular (\(\parallel,\perp\)) components}
426
427 \[\boldsymbol{a}=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}+\left(\boldsymbol{a}-\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}\right)\]
428
429
430 \subsection*{Vector proofs}
431
432 \textbf{Concurrent:} intersection of \(\ge\) 3 lines
433
434 \begin{tikzpicture}
435 \draw [blue] (0,0) -- (1,1);
436 \draw [red] (1,0) -- (0,1);
437 \draw [brown] (0.4,0) -- (0.6,1);
438 \filldraw (0.5,0.5) circle (2pt);
439 \end{tikzpicture}
440
441 \subsubsection*{Collinear points}
442
443 \(\ge\) 3 points lie on the same line
444
445 \begin{tikzpicture}
446 \draw [purple] (0,0) -- (4,1);
447 \filldraw (2,0.5) circle (2pt) node [above] {\(C\)};
448 \filldraw (1,0.25) circle (2pt) node [above] {\(A\)};
449 \filldraw (3,0.75) circle (2pt) node [above] {\(B\)};
450 \coordinate (O) at (2.8,-0.2);
451 \node at (O) [below] {\(O\)};
452 \begin{scope}[->, orange, thick]
453 \draw (O) -- (2,0.5) node [pos=0.5, above, font=\footnotesize, black] {\(\boldsymbol{c}\)};
454 \draw (O) -- (1,0.25) node [pos=0.5, below, font=\footnotesize, black] {\(\boldsymbol{a}\)};
455 \draw (O) -- (3,0.75) node [pos=0.5, right, font=\footnotesize, black] {\(\boldsymbol{b}\)};
456 \end{scope}
457 \end{tikzpicture}
458
459 \begin{align*}
460 \text{e.g. Prove that}\\
461 \overrightharp{AC}=m\overrightharp{AB} \iff \boldsymbol{c}&=(1-m)\boldsymbol{a}+m\boldsymbol{b}\\
462 \implies \boldsymbol{c} &= \overrightharp{OA} + \overrightharp{AC}\\
463 &= \overrightharp{OA} + m\overrightharp{AB}\\
464 &=\boldsymbol{a}+m(\boldsymbol{b}-\boldsymbol{a})\\
465 &=\boldsymbol{a}+m\boldsymbol{b}-m\boldsymbol{a}\\
466 &=(1-m)\boldsymbol{a}+m{b}
467 \end{align*}
468 \begin{align*}
469 \text{Also, } \implies \overrightharp{OC} &= \lambda \vec{OA} + \mu \overrightharp{OB} \\
470 \text{where } \lambda + \mu &= 1\\
471 \text{If } C \text{ lies along } \overrightharp{AB}, & \implies 0 < \mu < 1
472 \end{align*}
473
474
475 \subsubsection*{Parallelograms}
476
477 \begin{center}\begin{tikzpicture}
478 \coordinate (O) at (0,0) node [below left] {\(O\)};
479 \coordinate (A) at (4,0);
480 \coordinate (B) at (6,2);
481 \coordinate (C) at (2,2);
482 \coordinate (D) at (6,0);
483
484 \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (O)--(A) node [below left] {\(A\)};
485 \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (A)--(B) node [above right] {\(B\)};
486 \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (B)--(C) node [above left] {\(C\)};
487 \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (C)--(O);
488
489 \draw [gray, dashed] (O) -- (B) node [pos=0.75] {\(\diagdown\diagdown\)} node [pos=0.25] {\(\diagdown\diagdown\)};
490 \draw [gray, dashed] (A) -- (C) node [pos=0.75] {\(\diagup\)} node [pos=0.25] {\(\diagup\)};
491 \begin{scope}
492 \path[clip] (C) -- (A) -- (O);
493 \fill[orange, opacity=0.5, draw=black] (0,0) circle (4mm);
494 \node at ($(0,0)+(20:8mm)$) {\(\theta\)};
495 \end{scope}
496 \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] {\(|\boldsymbol{c}|\cos\theta\)};
497 \draw pic [draw,thick,red,angle radius=2mm] {right angle=O--D--B};
498 \end{tikzpicture}\end{center}
499
500 \begin{itemize}
501 \item
502 Diagonals \(\overrightharp{OB}, \overrightharp{AC}\) bisect each other
503 \item
504 If diagonals are equal length, it is a rectangle
505 \item
506 \(|\overrightharp{OB}|^2 + |\overrightharp{CA}|^2 = |\overrightharp{OA}|^2 + |\overrightharp{AB}|^2 + |\overrightharp{CB}|^2 + |\overrightharp{OC}|^2\)
507 \item
508 Area \(=\boldsymbol{c} \cdot \boldsymbol{a}\)
509 \end{itemize}
510
511 \subsubsection*{Useful vector properties}
512
513 \begin{itemize}
514 \item
515 \(\boldsymbol{a} \parallel \boldsymbol{b} \implies \boldsymbol{b}=k\boldsymbol{a}\) for some
516 \(k \in \mathbb{R} \setminus \{0\}\)
517 \item
518 If \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are parallel with at
519 least one point in common, then they lie on the same straight line
520 \item
521 \(\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b}=0\)
522 \item
523 \(\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2\)
524 \end{itemize}
525
526 \subsection*{Linear dependence}
527
528 \(\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\) are linearly dependent if they are \(\nparallel\) and:
529 \begin{align*}
530 0&=k\boldsymbol{a}+l\boldsymbol{b}+m\boldsymbol{c}\\
531 \therefore \boldsymbol{c} &= m\boldsymbol{a} + n\boldsymbol{b} \quad \text{(simultaneous)}
532 \end{align*}
533
534 \noindent \(\boldsymbol{a}, \boldsymbol{b},\) and \(\boldsymbol{c}\) are linearly
535 independent if no vector in the set is expressible as a linear
536 combination of other vectors in set, or if they are parallel.
537
538 \subsection*{Three-dimensional vectors}
539
540 Right-hand rule for axes: \(z\) is up or out of page.
541
542 \tdplotsetmaincoords{60}{120}
543 \begin{center}\begin{tikzpicture} [scale=3, tdplot_main_coords, axis/.style={->,thick},
544 vector/.style={-stealth,red,very thick},
545 vector guide/.style={dashed,gray,thick}]
546
547 %standard tikz coordinate definition using x, y, z coords
548 \coordinate (O) at (0,0,0);
549
550 %tikz-3dplot coordinate definition using x, y, z coords
551
552 \pgfmathsetmacro{\ax}{1}
553 \pgfmathsetmacro{\ay}{1}
554 \pgfmathsetmacro{\az}{1}
555
556 \coordinate (P) at (\ax,\ay,\az);
557
558 %draw axes
559 \draw[axis] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$};
560 \draw[axis] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$};
561 \draw[axis] (0,0,0) -- (0,0,1) node[anchor=south]{$z$};
562
563 %draw a vector from O to P
564 \draw[vector] (O) -- (P);
565
566 %draw guide lines to components
567 \draw[vector guide] (O) -- (\ax,\ay,0);
568 \draw[vector guide] (\ax,\ay,0) -- (P);
569 \draw[vector guide] (P) -- (0,0,\az);
570 \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
571 \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
572 \draw[vector guide] (\ax,\ay,0) -- (\ax,0,0);
573 \node[tdplot_main_coords,above right]
574 at (\ax,\ay,\az){(\ax, \ay, \az)};
575 \end{tikzpicture}\end{center}
576
577 \subsection*{Parametric vectors}
578
579 Parametric equation of line through point \((x_0, y_0, z_0)\) and
580 parallel to \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) is:
581
582 \[\begin{cases}x = x_o + a \cdot t \\ y = y_0 + b \cdot t \\ z = z_0 + c \cdot t\end{cases}\]
583
584 \section{Circular functions}
585
586 \(\sin(bx)\) or \(\cos(bx)\): period \(=\frac{2\pi}{b}\)
587
588 \noindent \(\tan(nx)\): period \(=\frac{\pi}{n}\)\\
589 \indent\indent\indent asymptotes at \(x=\frac{(2k+1)\pi}{2n} \> \vert \> k \in \mathbb{Z}\)
590
591 \subsection*{Reciprocal functions}
592
593 \subsubsection*{Cosecant}
594
595 \[\operatorname{cosec} \theta = \frac{1}{\sin \theta} \> \vert \> \sin \theta \ne 0\]
596
597 \begin{itemize}
598 \item
599 \textbf{Domain} \(= \mathbb{R} \setminus {n\pi : n \in \mathbb{Z}}\)
600 \item
601 \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
602 \item
603 \textbf{Turning points} at
604 \(\theta = \frac{(2n + 1)\pi}{2} \> \vert \> n \in \mathbb{Z}\)
605 \item
606 \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
607 \end{itemize}
608
609 \subsubsection*{Secant}
610 \begin{center}\includegraphics[width=0.7\columnwidth]{graphics/sec.png}\end{center}
611
612 \[\operatorname{sec} \theta = \frac{1}{\cos \theta} \> \vert \> \cos \theta \ne 0\]
613
614 \begin{itemize}
615
616 \item
617 \textbf{Domain}
618 \(= \mathbb{R} \setminus \frac{(2n + 1) \pi}{2} : n \in \mathbb{Z}\}\)
619 \item
620 \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
621 \item
622 \textbf{Turning points} at
623 \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
624 \item
625 \textbf{Asymptotes} at
626 \(\theta = \frac{(2n + 1) \pi}{2} \> \vert \> n \in \mathbb{Z}\)
627 \end{itemize}
628
629 \subsubsection*{Cotangent}
630
631 \begin{center}\includegraphics[width=0.7\columnwidth]{graphics/cot.png}\end{center}
632
633 \[\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0\]
634
635 \begin{itemize}
636
637 \item
638 \textbf{Domain} \(= \mathbb{R} \setminus \{n \pi: n \in \mathbb{Z}\}\)
639 \item
640 \textbf{Range} \(= \mathbb{R}\)
641 \item
642 \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
643 \end{itemize}
644
645 \subsubsection*{Symmetry properties}
646
647 \[\begin{split}
648 \operatorname{sec} (\pi \pm x) & = -\operatorname{sec} x \\
649 \operatorname{sec} (-x) & = \operatorname{sec} x \\
650 \operatorname{cosec} (\pi \pm x) & = \mp \operatorname{cosec} x \\
651 \operatorname{cosec} (-x) & = - \operatorname{cosec} x \\
652 \operatorname{cot} (\pi \pm x) & = \pm \operatorname{cot} x \\
653 \operatorname{cot} (-x) & = - \operatorname{cot} x
654 \end{split}\]
655
656 \subsubsection*{Complementary properties}
657
658 \[\begin{split}
659 \operatorname{sec} \left({\pi \over 2} - x\right) & = \operatorname{cosec} x \\
660 \operatorname{cosec} \left({\pi \over 2} - x\right) & = \operatorname{sec} x \\
661 \operatorname{cot} \left({\pi \over 2} - x\right) & = \tan x \\
662 \tan \left({\pi \over 2} - x\right) & = \operatorname{cot} x
663 \end{split}\]
664
665 \subsubsection*{Pythagorean identities}
666
667 \[\begin{split}
668 1 + \operatorname{cot}^2 x & = \operatorname{cosec}^2 x, \quad \text{where } \sin x \ne 0 \\
669 1 + \tan^2 x & = \operatorname{sec}^2 x, \quad \text{where } \cos x \ne 0
670 \end{split}\]
671
672 \subsection*{Compound angle formulas}
673
674 \[\cos(x \pm y) = \cos x + \cos y \mp \sin x \sin y\]
675 \[\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y\]
676 \[\tan(x \pm y) = {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}\]
677
678 \subsection*{Double angle formulas}
679
680 \[\begin{split}
681 \cos 2x &= \cos^2 x - \sin^2 x \\
682 & = 1 - 2\sin^2 x \\
683 & = 2 \cos^2 x -1
684 \end{split}\]
685
686 \[\sin 2x = 2 \sin x \cos x\]
687
688 \[\tan 2x = {{2 \tan x} \over {1 - \tan^2 x}}\]
689
690 \subsection*{Inverse circular functions}
691
692 \pgfplotsset{every axis/.append style={
693 axis x line=middle, % put the x axis in the middle
694 axis y line=middle, % put the y axis in the middle
695 axis line style={<->}, % arrows on the axis
696 xlabel={$x$}, % default put x on x-axis
697 ylabel={$y$}, % default put y on y-axis
698 }}
699
700% arrows as stealth fighters
701\tikzset{>=stealth}
702
703\begin{tikzpicture}
704 \begin{axis}[domain = -1:1, samples = 500]
705 \addplot[color = red] {rad(asin(x))} node [pos=0.25, below right] {\(\sin^{-1}x\)};
706 \addplot[color = blue] {rad(acos(x))} node [pos=0.25, below left] {\(\cos^{-1}x\)};
707 \end{axis}
708\end{tikzpicture}
709
710 Inverse functions: \(f(f^{-1}(x)) = x\) (restrict domain)
711
712 \[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y\]
713 \hfill where \(\sin y = x, \> y \in [{-\pi \over 2}, {\pi \over 2}]\)
714
715 \[\cos^{-1}: [-1,1] \rightarrow \mathbb{R}, \quad \cos^{-1} x = y\]
716 \hfill where \(\cos y = x, \> y \in [0, \pi]\)
717
718 \[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\]
719 \hfill where \(\tan y = x, \> y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\)
720
721
722 \section{Differential calculus}
723
724 \subsection*{Limits}
725
726 \[\lim_{x \rightarrow a}f(x)\]
727 \(L^-,\quad L^+\) \qquad limit from below/above\\
728 \(\lim_{x \to a} f(x)\) \quad limit of a point\\
729
730 \noindent For solving \(x\rightarrow\infty\), put all \(x\) terms in denominators\\
731 e.g. \[\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2\]
732
733 \subsubsection*{Limit theorems}
734
735 \begin{enumerate}
736 \item
737 For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\)
738 \item
739 \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\)
740 \item
741 \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\)
742 \item
743 \(\therefore \lim_{x \rightarrow a} c \times f(x)=cF\) where \(c=\) constant
744 \item
745 \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\)
746 \item
747 \(f(x)\) is continuous \(\iff L^-=L^+=f(x) \> \forall x\)
748 \end{enumerate}
749
750 \subsection*{Gradients of secants and tangents}
751
752 \textbf{Secant (chord)} - line joining two points on curve\\
753 \textbf{Tangent} - line that intersects curve at one point
754
755 \subsection*{First principles derivative}
756
757 \[f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={\frac{dy}{dx}}\]
758
759 \subsubsection*{Logarithmic identities}
760
761 \(\log_b (xy)=\log_b x + \log_b y\)\\
762 \(\log_b x^n = n \log_b x\)\\
763 \(\log_b y^{x^n} = x^n \log_b y\)
764
765 \subsubsection*{Index identities}
766
767 \(b^{m+n}=b^m \cdot b^n\)\\
768 \((b^m)^n=b^{m \cdot n}\)\\
769 \((b \cdot c)^n = b^n \cdot c^n\)\\
770 \({a^m \div a^n} = {a^{m-n}}\)
771
772 \subsection*{Derivative rules}
773
774 \renewcommand{\arraystretch}{1.4}
775 \begin{tabularx}{\columnwidth}{rX}
776 \hline
777 \(f(x)\) & \(f^\prime(x)\)\\
778 \hline
779 \(\sin x\) & \(\cos x\)\\
780 \(\sin ax\) & \(a\cos ax\)\\
781 \(\cos x\) & \(-\sin x\)\\
782 \(\cos ax\) & \(-a \sin ax\)\\
783 \(\tan f(x)\) & \(f^2(x) \sec^2f(x)\)\\
784 \(e^x\) & \(e^x\)\\
785 \(e^{ax}\) & \(ae^{ax}\)\\
786 \(ax^{nx}\) & \(an \cdot e^{nx}\)\\
787 \(\log_e x\) & \(\dfrac{1}{x}\)\\
788 \(\log_e {ax}\) & \(\dfrac{1}{x}\)\\
789 \(\log_e f(x)\) & \(\dfrac{f^\prime (x)}{f(x)}\)\\
790 \(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\\
791 \(\sin^{-1} x\) & \(\dfrac{1}{\sqrt{1-x^2}}\)\\
792 \(\cos^{-1} x\) & \(\dfrac{-1}{sqrt{1-x^2}}\)\\
793 \(\tan^{-1} x\) & \(\dfrac{1}{1 + x^2}\)\\
794 \(\frac{d}{dy}f(y)\) & \(\dfrac{1}{\frac{dx}{dy}}\) (reciprocal)\\
795 \(uv\) & \(u \frac{dv}{dx}+v\frac{du}{dx} (product rule)\)\\
796 \(\dfrac{u}{v}\) & \(\dfrac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\) (quotient rule)\\
797 \(f(g(x))\) & \(f^\prime(g(x))\cdot g^\prime(x)\)\\
798 \hline
799 \end{tabularx}
800
801 \subsection*{Reciprocal derivatives}
802
803 \[\frac{1}{\frac{dy}{dx}} = \frac{dx}{dy}\]
804
805 \subsection*{Differentiating \(x=f(y)\)}
806 \begin{align*}
807 \text{Find }& \frac{dx}{dy}\\
808 \text{Then, }\frac{dx}{dy} &= \frac{1}{\frac{dy}{dx}} \\
809 \implies {\frac{dy}{dx}} &= \frac{1}{\frac{dx}{dy}}\\
810 \therefore {\frac{dy}{dx}} &= \frac{1}{\frac{dx}{dy}}
811 \end{align*}
812
813 \subsubsection*{Second derivative}
814 \begin{align*}f(x) \longrightarrow &f^\prime (x) \longrightarrow f^{\prime\prime}(x)\\
815 \implies y \longrightarrow &\frac{dy}{dx} \longrightarrow \frac{d^2 y}{dx^2}\end{align*}
816
817 \noindent Order of polynomial \(n\)th derivative decrements each time the derivative is taken
818
819 \subsubsection*{Points of Inflection}
820
821 \emph{Stationary point} - i.e.
822 \(f^\prime(x)=0\)\\
823 \emph{Point of inflection} - max \(|\)gradient\(|\) (i.e.
824 \(f^{\prime\prime} = 0\))
825 %\begin{table*}[ht]
826 %\centering
827 % \begin{tabularx}{\textwidth}{XXXX}
828 %\hline
829 % \rowcolor{shade2}
830 % & \(\dfrac{d^2 y}{dx^2} > 0\) & \(\dfrac{d^2y}{dx^2}<0\) & \(\dfrac{d^2y}{dx^2}=0\) (inflection) \\
831 %\hline
832 % \(\frac{dy}{dx}>0\) & \begin{tikzpicture} \draw[domain=1:2,smooth,variable=\x,blue] plot ({\x},{(1/10)*\x*\x*\x}) plot ({\x},{0.675*\x-0.677}); \end{tikzpicture} & cell 3\\
833 %cell 1 & cell 2 & cell 3\\
834 %\hline
835 %\end{tabularx}
836 %\end{table*}
837
838
839\begin{itemize}
840 \item
841 if \(f^\prime (a) = 0\) and \(f^{\prime\prime}(a) > 0\), then point
842 \((a, f(a))\) is a local min (curve is concave up)
843 \item
844 if \(f^\prime (a) = 0\) and \(f^{\prime\prime} (a) < 0\), then point
845 \((a, f(a))\) is local max (curve is concave down)
846 \item
847 if \(f^{\prime\prime}(a) = 0\), then point \((a, f(a))\) is a point of
848 inflection
849 \item
850 if also \(f^\prime(a)=0\), then it is a stationary point of inflection
851 \end{itemize}
852
853 \subsection*{Implicit Differentiation}
854
855 \noindent Used for differentiating circles etc.
856
857 If \(p\) and \(q\) are expressions in \(x\) and \(y\) such that \(p=q\),
858 for all \(x\) and \(y\), then:
859
860 \[{\frac{dp}{dx}} = {\frac{dq}{dx}} \quad \text{and} \quad {\frac{dp}{dy}} = {\frac{dq}{dy}}\]
861
862 \noindent \colorbox{cas}{\textbf{On CAS:}}\\
863 Action \(\rightarrow\) Calculation \(\rightarrow\) \texttt{impDiff(y\^{}2+ax=5,\ x,\ y)}\\
864 Returns \(y^\prime= \dots\).
865
866 \subsection*{Integration}
867
868 \[\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)\]
869
870 \subsection*{Integral laws}
871
872 \renewcommand{\arraystretch}{1.4}
873 \begin{tabularx}{\columnwidth}{rX}
874 \hline
875 \(f(x)\) & \(\int f(x) \cdot dx\) \\
876 \hline
877 \(k\) (constant) & \(kx + c\)\\
878 \(x^n\) & \(\dfrac{1}{n+1} x^{n+1}\) \\
879 \(a x^{-n}\) &\(a \cdot \log_e |x| + c\)\\
880 \(\dfrac{1}{ax+b}\) &\(\dfrac{1}{a} \log_e (ax+b) + c\)\\
881 \((ax+b)^n\) & \(\dfrac{1}{a(n+1)}(ax+b)^{n-1} + c\>|\>n\ne 1\)\\
882 \((ax+b)^{-1}\) & \(\dfrac{1}{a}\log_e |ax+b|+c\)\\
883 \(e^{kx}\) & \(\dfrac{1}{k} e^{kx} + c\)\\
884 \(e^k\) & \(e^kx + c\)\\
885 \(\sin kx\) & \(\dfrac{-1}{k} \cos (kx) + c\)\\
886 \(\cos kx\) & \(\dfrac{1}{k} \sin (kx) + c\)\\
887 \(\sec^2 kx\) & \(\dfrac{1}{k} \tan(kx) + c\)\\
888 \(\dfrac{1}{\sqrt{a^2-x^2}}\) & \(\sin^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
889 \(\dfrac{-1}{\sqrt{a^2-x^2}}\) & \(\cos^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
890 \(\frac{a}{a^2-x^2}\) & \(\tan^{-1} \frac{x}{a} + c\)\\
891 \(\frac{f^\prime (x)}{f(x)}\) & \(\log_e f(x) + c\)\\
892 \(\int f(u) \cdot \frac{du}{dx} \cdot dx\) & \(\int f(u) \cdot du\) (substitution)\\
893 \(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\
894 \hline
895 \end{tabularx}
896
897 Note \(\sin^{-1} {x \over a} + \cos^{-1} {x \over a}\) is constant \(\forall x \in (-a, a)\)
898
899 \subsection*{Definite integrals}
900
901 \[\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)\]
902
903 \begin{itemize}
904
905 \item
906 Signed area enclosed by\\
907 \(\> y=f(x), \quad y=0, \quad x=a, \quad x=b\).
908 \item
909 \emph{Integrand} is \(f\).
910 \end{itemize}
911
912 \subsubsection*{Properties}
913
914 \[\int^b_a f(x) \> dx = \int^c_a f(x) \> dx + \int^b_c f(x) \> dx\]
915
916 \[\int^a_a f(x) \> dx = 0\]
917
918 \[\int^b_a k \cdot f(x) \> dx = k \int^b_a f(x) \> dx\]
919
920 \[\int^b_a f(x) \pm g(x) \> dx = \int^b_a f(x) \> dx \pm \int^b_a g(x) \> dx\]
921
922 \[\int^b_a f(x) \> dx = - \int^a_b f(x) \> dx\]
923
924 \subsection*{Integration by substitution}
925
926 \[\int f(u) {\frac{du}{dx}} \cdot dx = \int f(u) \cdot du\]
927
928 \noindent Note \(f(u)\) must be 1:1 \(\implies\) one \(x\) for each \(y\)
929 \begin{align*}\text{e.g. for } y&=\int(2x+1)\sqrt{x+4} \cdot dx\\
930 \text{let } u&=x+4\\
931 \implies& {\frac{du}{dx}} = 1\\
932 \implies& x = u - 4\\
933 \text{then } &y=\int (2(u-4)+1)u^{\frac{1}{2}} \cdot du\\
934 &\text{(solve as normal integral)}
935 \end{align*}
936
937 \subsubsection*{Definite integrals by substitution}
938
939 For \(\int^b_a f(x) {\frac{du}{dx}} \cdot dx\), evaluate new \(a\) and
940 \(b\) for \(f(u) \cdot du\).
941
942 \subsubsection*{Trigonometric integration}
943
944 \[\sin^m x \cos^n x \cdot dx\]
945
946 \paragraph{\textbf{\(m\) is odd:}}
947 \(m=2k+1\) where \(k \in \mathbb{Z}\)\\
948 \(\implies \sin^{2k+1} x = (\sin^2 z)^k \sin x = (1 - \cos^2 x)^k \sin x\)\\
949 Substitute \(u=\cos x\)
950
951 \paragraph{\textbf{\(n\) is odd:}}
952 \(n=2k+1\) where \(k \in \mathbb{Z}\)\\
953 \(\implies \cos^{2k+1} x = (\cos^2 x)^k \cos x = (1-\sin^2 x)^k \cos x\)\\
954 Substitute \(u=\sin x\)
955
956 \paragraph{\textbf{\(m\) and \(n\) are even:}}
957 use identities...
958
959 \begin{itemize}
960
961 \item
962 \(\sin^2x={1 \over 2}(1-\cos 2x)\)
963 \item
964 \(\cos^2x={1 \over 2}(1+\cos 2x)\)
965 \item
966 \(\sin 2x = 2 \sin x \cos x\)
967 \end{itemize}
968
969 \subsection*{Partial fractions}
970
971 \colorbox{cas}{On CAS:}\\
972 \indent Action \(\rightarrow\) Transformation \(\rightarrow\)
973 \texttt{expand/combine}\\
974 \indent Interactive \(\rightarrow\) Transformation \(\rightarrow\)
975 Expand \(\rightarrow\) Partial
976
977 \subsection*{Graphing integrals on CAS}
978
979 \colorbox{cas}{In main:} Interactive \(\rightarrow\) Calculation \(\rightarrow\)
980 \(\int\) (\(\rightarrow\) Definite)\\
981 Restrictions: \texttt{Define\ f(x)=..} then \texttt{f(x)\textbar{}x\textgreater{}..}
982
983 \subsection*{Applications of antidifferentiation}
984
985 \begin{itemize}
986
987 \item
988 \(x\)-intercepts of \(y=f(x)\) identify \(x\)-coordinates of
989 stationary points on \(y=F(x)\)
990 \item
991 nature of stationary points is determined by sign of \(y=f(x)\) on
992 either side of its \(x\)-intercepts
993 \item
994 if \(f(x)\) is a polynomial of degree \(n\), then \(F(x)\) has degree
995 \(n+1\)
996 \end{itemize}
997
998 To find stationary points of a function, substitute \(x\) value of given
999 point into derivative. Solve for \({\frac{dy}{dx}}=0\). Integrate to find
1000 original function.
1001
1002 \subsection*{Solids of revolution}
1003
1004 Approximate as sum of infinitesimally-thick cylinders
1005
1006 \subsubsection*{Rotation about \(x\)-axis}
1007
1008 \begin{align*}
1009 V &= \int^{x=b}_{x-a} \pi y^2 \> dx \\
1010 &= \pi \int^b_a (f(x))^2 \> dx
1011 \end{align*}
1012
1013 \subsubsection*{Rotation about \(y\)-axis}
1014
1015 \begin{align*}
1016 V &= \int^{y=b}_{y=a} \pi x^2 \> dy \\
1017 &= \pi \int^b_a (f(y))^2 \> dy
1018 \end{align*}
1019
1020 \subsubsection*{Regions not bound by \(y=0\)}
1021
1022 \[V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx\]
1023 \hfill where \(f(x) > g(x)\)
1024
1025 \subsection*{Length of a curve}
1026
1027 \[L = \int^b_a \sqrt{1 + ({\frac{dy}{dx}})^2} \> dx \quad \text{(Cartesian)}\]
1028
1029 \[L = \int^b_a \sqrt{{\frac{dx}{dt}} + ({\frac{dy}{dt}})^2} \> dt \quad \text{(parametric)}\]
1030
1031 \noindent \colorbox{cas}{On CAS:}\\
1032 \indent Evaluate formula,\\
1033 \indent or Interactive \(\rightarrow\) Calculation
1034 \(\rightarrow\) Line \(\rightarrow\) \texttt{arcLen}
1035
1036 \subsection*{Rates}
1037
1038 \subsubsection*{Gradient at a point on parametric curve}
1039
1040 \[{\frac{dy}{dx}} = {{\frac{dy}{dt}} \div {\frac{dx}{dt}}} \> \vert \> {\frac{dx}{dt}} \ne 0\]
1041
1042 \[\frac{d^2}{dx^2} = \frac{d(y^\prime)}{dx} = {\frac{dy^\prime}{dt} \div {\frac{dx}{dt}}} \> \vert \> y^\prime = {\frac{dy}{dx}}\]
1043
1044 \subsection*{Rational functions}
1045
1046 \[f(x) = \frac{P(x)}{Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}\]
1047
1048 \subsubsection*{Addition of ordinates}
1049
1050 \begin{itemize}
1051
1052 \item
1053 when two graphs have the same ordinate, \(y\)-coordinate is double the
1054 ordinate
1055 \item
1056 when two graphs have opposite ordinates, \(y\)-coordinate is 0 i.e.
1057 (\(x\)-intercept)
1058 \item
1059 when one of the ordinates is 0, the resulting ordinate is equal to the
1060 other ordinate
1061 \end{itemize}
1062
1063 \subsection*{Fundamental theorem of calculus}
1064
1065 If \(f\) is continuous on \([a, b]\), then
1066
1067 \[\int^b_a f(x) \> dx = F(b) - F(a)\]
1068 \hfill where \(F = \int f \> dx\)
1069
1070 \subsection*{Differential equations}
1071
1072 \noindent\textbf{Order} - highest power inside derivative\\
1073 \textbf{Degree} - highest power of highest derivative\\
1074 e.g. \({\left(\dfrac{dy^2}{d^2} x\right)}^3\) \qquad order 2, degree 3
1075
1076 \subsubsection*{Verifying solutions}
1077
1078 Start with \(y=\dots\), and differentiate. Substitute into original
1079 equation.
1080
1081 \subsubsection*{Function of the dependent
1082 variable}
1083
1084 If \({\frac{dy}{dx}}=g(y)\), then
1085 \(\frac{dx}{dy} = 1 \div \frac{dy}{dx} = \frac{1}{g(y)}\). Integrate both sides to solve equation. Only add \(c\) on one side. Express
1086 \(e^c\) as \(A\).
1087
1088 \begin{table*}[ht]
1089 \centering
1090 \includegraphics[width=0.7\textwidth]{graphics/second-derivatives.png}
1091 \end{table*}
1092
1093 \subsubsection*{Mixing problems}
1094
1095 \[\left(\frac{dm}{dt}\right)_\Sigma = \left(\frac{dm}{dt}\right)_{\text{in}} - \left(\frac{dm}{dt}_{\text{out}}\right)\]
1096
1097 \subsubsection*{Separation of variables}
1098
1099 If \({\frac{dy}{dx}}=f(x)g(y)\), then:
1100
1101 \[\int f(x) \> dx = \int \frac{1}{g(y)} \> dy\]
1102
1103 \subsubsection*{Euler's method for solving DEs}
1104
1105 \[\frac{f(x+h) - f(x)}{h} \approx f^\prime (x) \quad \text{for small } h\]
1106
1107 \[\implies f(x+h) \approx f(x) + hf^\prime(x)\]
1108
1109 \end{multicols}
1110 \end{document}