e7957131bcaf7d5e88a4803bfc9fa01cbbca4aaa
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{makecell}
10\usepackage[dvipsnames, table]{xcolor}
11\usepackage{blindtext}
12\usepackage{graphicx}
13\usepackage{wrapfig}
14\usepackage{tikz}
15\usepackage{tikz-3dplot}
16\usepackage{pgfplots}
17\usetikzlibrary{calc}
18\usetikzlibrary{angles}
19\usetikzlibrary{datavisualization.formats.functions}
20\usetikzlibrary{decorations.markings}
21\usepgflibrary{arrows.meta}
22\usepackage{fancyhdr}
23\pagestyle{fancy}
24\fancyhead[LO,LE]{Year 12 Specialist}
25\fancyhead[CO,CE]{Andrew Lorimer}
26
27\usepackage{mathtools}
28\usepackage{xcolor} % used only to show the phantomed stuff
29\renewcommand\hphantom[1]{{\color[gray]{.6}#1}} % comment out!
30\setlength\fboxsep{0pt} \setlength\fboxrule{.2pt} % for the \fboxes
31\newcommand*\leftlap[3][\,]{#1\hphantom{#2}\mathllap{#3}}
32\newcommand*\rightlap[2]{\mathrlap{#2}\hphantom{#1}}
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34\newcolumntype{R}[1]{>{\hsize=#1\hsize\raggedleft\arraybackslash}X}%
35\definecolor{cas}{HTML}{e6f0fe}
36\linespread{1.5}
37\newcommand{\midarrow}{\tikz \draw[-triangle 90] (0,0) -- +(.1,0);}
38\newcommand{\tg}{\mathop{\mathrm{tg}}}
39\newcommand{\cotg}{\mathop{\mathrm{cotg}}}
40\newcommand{\arctg}{\mathop{\mathrm{arctg}}}
41\newcommand{\arccotg}{\mathop{\mathrm{arccotg}}}
42
43
44 \pgfplotsset{every axis/.append style={
45 axis x line=middle, % put the x axis in the middle
46 axis y line=middle, % put the y axis in the middle
47 axis line style={->}, % arrows on the axis
48 xlabel={$x$}, % default put x on x-axis
49 ylabel={$y$}, % default put y on y-axis
50 }}
51\begin{document}
52
53\begin{multicols}{2}
54
55 \section{Complex numbers}
56
57 \[\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}\]
58
59 \begin{align*}
60 \text{Cartesian form: } & a+bi\\
61 \text{Polar form: } & r\operatorname{cis}\theta
62 \end{align*}
63
64 \subsection*{Operations}
65
66 \definecolor{shade1}{HTML}{ffffff}
67 \definecolor{shade2}{HTML}{e6f2ff}
68 \definecolor{shade3}{HTML}{cce2ff}
69 \begin{tabularx}{\columnwidth}{r|X|X}
70 & \textbf{Cartesian} & \textbf{Polar} \\
71 \hline
72 \(z_1 \pm z_2\) & \((a \pm c)(b \pm d)i\) & convert to \(a+bi\)\\
73 \hline
74 \(+k \times z\) & \multirow{2}{*}{\(ka \pm kbi\)} & \(kr\operatorname{cis} \theta\)\\
75 \cline{1-1}\cline{3-3}
76 \(-k \times z\) & & \(kr \operatorname{cis}(\theta\pm \pi)\)\\
77 \hline
78 \(z_1 \cdot z_2\) & \(ac-bd+(ad+bc)i\) & \(r_1r_2 \operatorname{cis}(\theta_1 + \theta_2)\)\\
79 \hline
80 \(z_1 \div z_2\) & \((z_1 \overline{z_2}) \div |z_2|^2\) & \(\left(\frac{r_1}{r_2}\right) \operatorname{cis}(\theta_1 - \theta_2)\)
81 \end{tabularx}
82
83 \subsubsection*{Scalar multiplication in polar form}
84
85 For \(k \in \mathbb{R}^+\):
86 \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\theta\]
87
88 \noindent For \(k \in \mathbb{R}^-\):
89 \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\left(\begin{cases}\theta - \pi & |0<\operatorname{Arg}(z)\le \pi \\ \theta + \pi & |-\pi<\operatorname{Arg}(z)\le 0\end{cases}\right)\]
90
91 \subsection*{Conjugate}
92
93 \begin{align*}
94 \overline{z} &= a \mp bi\\
95 &= r \operatorname{cis}(-\theta)
96 \end{align*}
97
98 \noindent \colorbox{cas}{On CAS: \texttt{conjg(a+bi)}}
99
100 \subsubsection*{Properties}
101
102 \begin{align*}
103 \overline{z_1 \pm z_2} &= \overline{z_1}\pm\overline{z_2}\\
104 \overline{z_1 \cdot z_2} &= \overline{z_1}\cdot\overline{z_2}\\
105 \overline{kz} &= k\overline{z} \quad | \quad k \in \mathbb{R}\\
106 z\overline{z} &= (a+bi)(a-bi)\\
107 &= a^2 + b^2\\
108 &= |z|^2
109 \end{align*}
110
111 \subsection*{Modulus}
112
113 \[|z|=|\vec{Oz}|=\sqrt{a^2 + b^2}\]
114
115 \subsubsection*{Properties}
116
117 \begin{align*}
118 |z_1z_2|&=|z_1||z_2|\\
119 \left|\frac{z_1}{z_2}\right|&=\frac{|z_1|}{|z_2|}\\
120 |z_1+z_2|&\le|z_1|+|z_2|
121 \end{align*}
122
123 \subsection*{Multiplicative inverse}
124
125 \begin{align*}
126 z^{-1}&=\frac{a-bi}{a^2+b^2}\\
127 &=\frac{\overline{z}}{|z|^2}a\\
128 &=r \operatorname{cis}(-\theta)
129 \end{align*}
130
131 \subsection*{Dividing over \(\mathbb{C}\)}
132
133 \begin{align*}
134 \frac{z_1}{z_2}&=z_1z_2^{-1}\\
135 &=\frac{z_1\overline{z_2}}{|z_2|^2}\\
136 &=\frac{(a+bi)(c-di)}{c^2+d^2}\\
137 & \qquad \text{(rationalise denominator)}
138 \end{align*}
139
140 \subsection*{Polar form}
141
142 \begin{align*}
143 z&=r\operatorname{cis}\theta\\
144 &=r(\cos \theta + i \sin \theta)
145 \end{align*}
146
147 \begin{itemize}
148 \item{\(r=|z|=\sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}\)}
149 \item{\(\theta = \operatorname{arg}(z)\) \quad \colorbox{cas}{On CAS: \texttt{arg(a+bi)}}}
150 \item{\(\operatorname{Arg}(z) \in (-\pi,\pi)\) \quad \bf{(principal argument)}}
151 \item{\colorbox{cas}{Convert on CAS:}\\ \verb|compToTrig(a+bi)| \(\iff\) \verb|cExpand{r·cisX}|}
152 \item{Multiple representations:\\\(r\operatorname{cis}\theta=r\operatorname{cis}(\theta+2n\pi)\) with \(n \in \mathbb{Z}\) revolutions}
153 \item{\(\operatorname{cis}\pi=-1,\qquad \operatorname{cis}0=1\)}
154 \end{itemize}
155
156 \subsection*{de Moivres' theorem}
157
158 \[(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}\]
159
160 \subsection*{Complex polynomials}
161
162 Include \(\pm\) for all solutions, incl. imaginary
163
164 \begin{tabularx}{\columnwidth}{ R{0.55} X }
165 \hline
166 Sum of squares & \(\begin{aligned}
167 z^2 + a^2 &= z^2-(ai)^2\\
168 &= (z+ai)(z-ai) \end{aligned}\) \\
169 \hline
170 Sum of cubes & \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)\\
171 \hline
172 Division & \(P(z)=D(z)Q(z)+R(z)\) \\
173 \hline
174 Remainder theorem & Let \(\alpha \in \mathbb{C}\). Remainder of \(P(z) \div (z-\alpha)\) is \(P(\alpha)\)\\
175 \hline
176 Factor theorem & \(z-\alpha\) is a factor of \(P(z) \iff P(\alpha)=0\) for \(\alpha \in \mathbb{C}\)\\
177 \hline
178 Conjugate root theorem & \(P(z)=0 \text{ at } z=a\pm bi\) (\(\implies\) both \(z_1\) and \(\overline{z_1}\) are solutions)\\
179 \hline
180 \end{tabularx}
181
182 \subsection*{\(n\)th roots}
183
184 \(n\)th roots of \(z=r\operatorname{cis}\theta\) are:
185
186 \[z = r^{\frac{1}{n}} \operatorname{cis}\left(\frac{\theta+2k\pi}{n}\right)\]
187
188 \begin{itemize}
189
190 \item{Same modulus for all solutions}
191 \item{Arguments separated by \(\frac{2\pi}{n} \therefore\) there are \(n\) roots}
192 \item{If one square root is \(a+bi\), the other is \(-a-bi\)}
193 \item{Give one implicit \(n\)th root \(z_1\), function is \(z=z_1^n\)}
194 \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 \(\frac{2\pi}{n}\))}
195 \end{itemize}
196
197 \noindent For \(0=az^2+bz+c\), use quadratic formula:
198
199 \[z=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]
200
201 \subsection*{Fundamental theorem of algebra}
202
203 A polynomial of degree \(n\) can be factorised into \(n\) linear factors in \(\mathbb{C}\):
204
205 \[\implies P(z)=a_n(z-\alpha_1)(z-\alpha_2)(z-\alpha_3)\dots(z-\alpha_n)\]
206 \[\text{ where } \alpha_1,\alpha_2,\alpha_3,\dots,\alpha_n \in \mathbb{C}\]
207
208 \subsection*{Argand planes}
209
210 \begin{center}\begin{tikzpicture}[scale=2]
211 \draw [->] (-0.2,0) -- (1.5,0) node [right] {$\operatorname{Re}(z)$};
212 \draw [->] (0,-0.2) -- (0,1.5) node [above] {$\operatorname{Im}(z)$};
213 \coordinate (P) at (1,1);
214 \coordinate (a) at (1,0);
215 \coordinate (b) at (0,1);
216 \coordinate (O) at (0,0);
217 \draw (0,0) -- (P) node[pos=0.5, above left]{\(r\)} node[pos=1, right]{\(\begin{aligned}z&=a+bi\\&=r\operatorname{cis}\theta\end{aligned}\)};
218 \draw [gray, dashed] (1,1) -- (1,0) node[black, pos=1, below]{\(a\)};
219 \draw [gray, dashed] (1,1) -- (0,1) node[black, pos=1, left]{\(b\)};
220 \begin{scope}
221 \path[clip] (O) -- (P) -- (a);
222 \fill[red, opacity=0.5, draw=black] (O) circle (2mm);
223 \node at ($(O)+(20:3mm)$) {$\theta$};
224 \end{scope}
225 \filldraw (P) circle (0.5pt);
226 \end{tikzpicture}\end{center}
227
228 \begin{itemize}
229 \item{Multiplication by \(i \implies\) CCW rotation of \(\frac{\pi}{2}\)}
230 \item{Addition: \(z_1 + z_2 \equiv\) \overrightharp{\(Oz_1\)} + \overrightharp{\(Oz_2\)}}
231 \end{itemize}
232
233 \subsection*{Sketching complex graphs}
234
235 \subsubsection*{Linear}
236
237 \begin{itemize}
238 \item{\(\operatorname{Re}(z)=c\) or \(\operatorname{Im}(z)=c\) (perpendicular bisector)}
239 \item{\(\operatorname{Im}(z)=m\operatorname{Re}(z)\)}
240 \item{\(|z+a|=|z+b| \implies 2(a-b)x=b^2-a^2\)\\Geometric: equidistant from \(a,b\)}
241 \end{itemize}
242
243 \subsubsection*{Circles}
244
245 \begin{itemize}
246 \item \(|z-z_1|^2=c^2|z_2+2|^2\)
247 \item \(|z-(a+bi)|=c \implies (x-a)^2+_(y-b)^2=c^2\)
248 \end{itemize}
249
250 \noindent \textbf{Loci} \qquad \(\operatorname{Arg}(z)<\theta\)
251
252 \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
253 \draw [->] (0,0) -- (1,0) node [right] {$\operatorname{Re}(z)$};
254 \draw [->] (0,-0.5) -- (0,1) node [above] {$\operatorname{Im}(z)$};
255 \draw [<-, dashed, thick, blue] (-1,0) -- (0,0);
256 \draw [->, thick, blue] (0,0) -- (1,1);
257 \fill [gray, opacity=0.2, domain=-1:1, variable=\x] (-1,-0.5) -- (-1,0) -- (0, 0) -- (1,1) -- (1,-0.5) -- cycle;
258 \begin{scope}
259 \path[clip] (0,0) -- (1,1) -- (1,0);
260 \fill[red, opacity=0.5, draw=black] (0,0) circle (2mm);
261 \node at ($(0,0)+(20:3mm)$) {$\frac{\pi}{4}$};
262 \end{scope}
263 \node [font=\footnotesize] at (0.5,-0.25) {\(\operatorname{Arg}(z)\le\frac{\pi}{4}\)};
264 \node [blue, mydot] {};
265 \end{tikzpicture}\end{center}
266
267 \noindent \textbf{Rays} \qquad \(\operatorname{Arg}(z-b)=\theta\)
268
269 \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
270 \draw [->] (-0.75,0) -- (1.5,0) node [right] {$\operatorname{Re}(z)$};
271 \draw [->] (0,-1) -- (0,1) node [above] {$\operatorname{Im}(z)$};
272 \draw [->, thick, brown] (-0.25,0) -- (-0.75,-1);
273 \node [above, font=\footnotesize] at (-0.25,0) {\(\frac{1}{4}\)};
274 \begin{scope}
275 \path[clip] (-0.25,0) -- (-0.75,-1) -- (0,0);
276 \fill[orange, opacity=0.5, draw=black] (-0.25,0) circle (2mm);
277 \end{scope}
278 \node at (-0.08,-0.3) {\(\frac{\pi}{8}\)};
279 \node [font=\footnotesize, left] at (-0.75,-1) {\(\operatorname{Arg}(z+\frac{1}{4})=\frac{\pi}{8}\)};
280 \node [brown, mydot] at (-0.25,0) {};
281 \draw [<->, thick, green] (0,-1) -- (1.5,0.5) node [pos=0.25, black, font=\footnotesize, right] {\(|z-2|=|z-(1+i)|\)};
282 \node [left, font=\footnotesize] at (0,-1) {\(-1\)};
283 \node [below, font=\footnotesize] at (1,0) {\(1\)};
284 \end{tikzpicture}\end{center}
285
286 \section{Vectors}
287 \begin{center}\begin{tikzpicture}
288 \draw [->] (-0.5,0) -- (3,0) node [right] {\(x\)};
289 \draw [->] (0,-0.5) -- (0,3) node [above] {\(y\)};
290 \draw [orange, ->, thick] (0.5,0.5) -- (2.5,2.5) node [pos=0.5, above] {\(\vec{u}\)};
291 \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
292 \draw [gray, dashed, thick] (0.5,0.5) -- (2.5,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, below]{\(x\vec{i}\)};
293 \draw [gray, dashed, thick] (2.5,0.5) -- (2.5,2.5) node [pos=0.5] {\midarrow};
294 \end{scope}
295 \node[black, right] at (2.5,1.5) {\(y\vec{j}\)};
296 \end{tikzpicture}\end{center}
297 \subsection*{Column notation}
298
299 \[\begin{bmatrix}x\\ y \end{bmatrix} \iff x\boldsymbol{i} + y\boldsymbol{j}\]
300 \(\begin{bmatrix}x_2-x_1\\ y_2-y_1 \end{bmatrix}\) \quad between \(A(x_1,y_1), \> B(x_2,y_2)\)
301
302 \subsection*{Scalar multiplication}
303
304 \[k\cdot (x\boldsymbol{i}+y\boldsymbol{j})=kx\boldsymbol{i}+ky\boldsymbol{j}\]
305
306 \noindent For \(k \in \mathbb{R}^-\), direction is reversed
307
308 \subsection*{Vector addition}
309 \begin{center}\begin{tikzpicture}[scale=1]
310 \coordinate (A) at (0,0);
311 \coordinate (B) at (2,2);
312 \draw [->, thick, red] (0,0) -- (2,2) node [pos=0.5, below right] {\(\vec{u}=2\vec{i}+2\vec{j}\)};
313 \draw [->, thick, blue] (2,2) -- (1,4) node [pos=0.5, above right] {\(\vec{v}=-\vec{i}+2\vec{j}\)};
314 \draw [->, thick, orange] (0,0) -- (1,4) node [pos=0.5, left] {\(\vec{u}+\vec{v}=\vec{i}+4\vec{j}\)};
315 \end{tikzpicture}\end{center}
316
317 \[(x\boldsymbol{i}+y\boldsymbol{j}) \pm (a\boldsymbol{i}+b\boldsymbol{j})=(x \pm a)\boldsymbol{i}+(y \pm b)\boldsymbol{j}\]
318
319 \begin{itemize}
320 \item Draw each vector head to tail then join lines
321 \item Addition is commutative (parallelogram)
322 \item \(\boldsymbol{u}-\boldsymbol{v}=\boldsymbol{u}+(-\boldsymbol{v}) \implies \overrightharp{AB}=\boldsymbol{b}-\boldsymbol{a}\)
323 \end{itemize}
324
325 \subsection*{Magnitude}
326
327 \[|(x\boldsymbol{i} + y\boldsymbol{j})|=\sqrt{x^2+y^2}\]
328
329 \subsection*{Parallel vectors}
330
331 \[\boldsymbol{u} || \boldsymbol{v} \iff \boldsymbol{u} = k \boldsymbol{v} \text{ where } k \in \mathbb{R} \setminus \{0\}\]
332
333 For parallel vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\):\\
334 \[\boldsymbol{a \cdot b}=\begin{cases}
335 |\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction}\\
336 -|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions}
337 \end{cases}\]
338 %\includegraphics[width=0.2,height=\textheight]{graphics/parallelogram-vectors.jpg}
339 %\includegraphics[width=1]{graphics/vector-subtraction.jpg}
340
341 \subsection*{Perpendicular vectors}
342
343 \[\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b} = 0\ \quad \text{(since \(\cos 90 = 0\))}\]
344
345 \subsection*{Unit vector \(|\hat{\boldsymbol{a}}|=1\)}
346 \[\begin{split}\hat{\boldsymbol{a}} & = {\frac{1}{|\boldsymbol{a}|}}\boldsymbol{a} \\ & = \boldsymbol{a} \cdot {|\boldsymbol{a}|}\end{split}\]
347
348 \subsection*{Scalar product \(\boldsymbol{a} \cdot \boldsymbol{b}\)}
349
350
351 \begin{center}\begin{tikzpicture}[scale=2]
352 \draw [->] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{b}\)};
353 \draw [->] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{a}\)};
354 \begin{scope}
355 \path[clip] (1,0.5) -- (1,0) -- (0,0);
356 \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
357 \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
358 \end{scope}
359 \end{tikzpicture}\end{center}
360 \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*}
361 \noindent\colorbox{cas}{On CAS: \texttt{dotP({[}a\ b\ c{]},\ {[}d\ e\ f{]})}}
362
363 \subsubsection*{Properties}
364
365 \begin{enumerate}
366 \item
367 \(k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k\boldsymbol{b})\)
368 \item
369 \(\boldsymbol{a \cdot 0}=0\)
370 \item
371 \(\boldsymbol{a} \cdot (\boldsymbol{b} + \boldsymbol{c})=\boldsymbol{a} \cdot \boldsymbol{b} + \boldsymbol{a} \cdot \boldsymbol{c}\)
372 \item
373 \(\boldsymbol{i \cdot i} = \boldsymbol{j \cdot j} = \boldsymbol{k \cdot k}= 1\)
374 \item
375 \(\boldsymbol{a} \cdot \boldsymbol{b} = 0 \quad \implies \quad \boldsymbol{a} \perp \boldsymbol{b}\)
376 \item
377 \(\boldsymbol{a \cdot a} = |\boldsymbol{a}|^2 = a^2\)
378 \end{enumerate}
379
380 \subsection*{Angle between vectors}
381
382 \[\cos \theta = \frac{\boldsymbol{a} \cdot \boldsymbol{b}}{|\boldsymbol{a}| |\boldsymbol{b}|} = \frac{a_1 b_1 + a_2 b_2}{|\boldsymbol{a}| |\boldsymbol{b}|}\]
383
384 \noindent \colorbox{cas}{On CAS:} \texttt{angle([a b c], [a b c])}
385
386 (Action \(\rightarrow\) Vector \(\rightarrow\)Angle)
387
388 \subsection*{Angle between vector and axis}
389
390 \noindent For\(\boldsymbol{a} = a_1 \boldsymbol{i} + a_2 \boldsymbol{j} + a_3 \boldsymbol{k}\)
391 which makes angles \(\alpha, \beta, \gamma\) with positive side of
392 \(x, y, z\) axes:
393 \[\cos \alpha = \frac{a_1}{|\boldsymbol{a}|}, \quad \cos \beta = \frac{a_2}{|\boldsymbol{a}|}, \quad \cos \gamma = \frac{a_3}{|\boldsymbol{a}|}\]
394
395 \noindent \colorbox{cas}{On CAS:} \texttt{angle({[}a\ b\ c{]},\ {[}1\ 0\ 0{]})}\\for angle
396 between \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) and
397 \(x\)-axis
398
399 \subsection*{Projections \& resolutes}
400
401 \begin{tikzpicture}[scale=3]
402 \draw [->, purple] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{a}\)};
403 \draw [->, orange] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{u}\)};
404 \draw [->, blue] (1,0) -- (2,0) node [pos=0.5, below] {\(\boldsymbol{b}\)};
405 \begin{scope}
406 \path[clip] (1,0.5) -- (1,0) -- (0,0);
407 \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
408 \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
409 \end{scope}
410 \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
411 \draw [gray, dashed, thick] (1,0) -- (1,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, right, rotate=-90]{\(\boldsymbol{w}\)};
412 \end{scope}
413 \draw (0,0) coordinate (O)
414 (1,0) coordinate (A)
415 (1,0.5) coordinate (B)
416 pic [draw,red,angle radius=2mm] {right angle = O--A--B};
417 \end{tikzpicture}
418
419 \subsubsection*{\(\parallel\boldsymbol{b}\) (vector projection/resolute)}
420
421 \begin{align*}
422 \boldsymbol{u} & = \frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|^2}\boldsymbol{b} \\
423 & = \left(\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\right)\left(\frac{\boldsymbol{b}}{|\boldsymbol{b}|}\right) \\
424 & = (\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}
425 \end{align*}
426
427 \subsubsection*{\(\perp\boldsymbol{b}\) (perpendicular projection)}
428 \[\boldsymbol{w} = \boldsymbol{a} - \boldsymbol{u}\]
429
430 \subsubsection*{\(|\boldsymbol{u}|\) (scalar projection/resolute)}
431 \begin{align*}
432 s &= |\boldsymbol{u}|\\
433 &= \boldsymbol{a} \cdot \hat{\boldsymbol{b}}\\
434 &=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\\
435 &= |\boldsymbol{a}| \cos \theta
436 \end{align*}
437
438 \subsubsection*{Rectangular (\(\parallel,\perp\)) components}
439
440 \[\boldsymbol{a}=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}+\left(\boldsymbol{a}-\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}\right)\]
441
442
443 \subsection*{Vector proofs}
444
445 \textbf{Concurrent:} intersection of \(\ge\) 3 lines
446
447 \begin{tikzpicture}
448 \draw [blue] (0,0) -- (1,1);
449 \draw [red] (1,0) -- (0,1);
450 \draw [brown] (0.4,0) -- (0.6,1);
451 \filldraw (0.5,0.5) circle (2pt);
452 \end{tikzpicture}
453
454 \subsubsection*{Collinear points}
455
456 \(\ge\) 3 points lie on the same line
457
458 \begin{tikzpicture}
459 \draw [purple] (0,0) -- (4,1);
460 \filldraw (2,0.5) circle (2pt) node [above] {\(C\)};
461 \filldraw (1,0.25) circle (2pt) node [above] {\(A\)};
462 \filldraw (3,0.75) circle (2pt) node [above] {\(B\)};
463 \coordinate (O) at (2.8,-0.2);
464 \node at (O) [below] {\(O\)};
465 \begin{scope}[->, orange, thick]
466 \draw (O) -- (2,0.5) node [pos=0.5, above, font=\footnotesize, black] {\(\boldsymbol{c}\)};
467 \draw (O) -- (1,0.25) node [pos=0.5, below, font=\footnotesize, black] {\(\boldsymbol{a}\)};
468 \draw (O) -- (3,0.75) node [pos=0.5, right, font=\footnotesize, black] {\(\boldsymbol{b}\)};
469 \end{scope}
470 \end{tikzpicture}
471
472 \begin{align*}
473 \text{e.g. Prove that}\\
474 \overrightharp{AC}=m\overrightharp{AB} \iff \boldsymbol{c}&=(1-m)\boldsymbol{a}+m\boldsymbol{b}\\
475 \implies \boldsymbol{c} &= \overrightharp{OA} + \overrightharp{AC}\\
476 &= \overrightharp{OA} + m\overrightharp{AB}\\
477 &=\boldsymbol{a}+m(\boldsymbol{b}-\boldsymbol{a})\\
478 &=\boldsymbol{a}+m\boldsymbol{b}-m\boldsymbol{a}\\
479 &=(1-m)\boldsymbol{a}+m{b}
480 \end{align*}
481 \begin{align*}
482 \text{Also, } \implies \overrightharp{OC} &= \lambda \vec{OA} + \mu \overrightharp{OB} \\
483 \text{where } \lambda + \mu &= 1\\
484 \text{If } C \text{ lies along } \overrightharp{AB}, & \implies 0 < \mu < 1
485 \end{align*}
486
487
488 \subsubsection*{Parallelograms}
489
490 \begin{center}\begin{tikzpicture}
491 \coordinate (O) at (0,0) node [below left] {\(O\)};
492 \coordinate (A) at (4,0);
493 \coordinate (B) at (6,2);
494 \coordinate (C) at (2,2);
495 \coordinate (D) at (6,0);
496
497 \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (O)--(A) node [below left] {\(A\)};
498 \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (A)--(B) node [above right] {\(B\)};
499 \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (B)--(C) node [above left] {\(C\)};
500 \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (C)--(O);
501
502 \draw [gray, dashed] (O) -- (B) node [pos=0.75] {\(\diagdown\diagdown\)} node [pos=0.25] {\(\diagdown\diagdown\)};
503 \draw [gray, dashed] (A) -- (C) node [pos=0.75] {\(\diagup\)} node [pos=0.25] {\(\diagup\)};
504 \begin{scope}
505 \path[clip] (C) -- (A) -- (O);
506 \fill[orange, opacity=0.5, draw=black] (0,0) circle (4mm);
507 \node at ($(0,0)+(20:8mm)$) {\(\theta\)};
508 \end{scope}
509 \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] {\(|\boldsymbol{c}|\cos\theta\)};
510 \draw pic [draw,thick,red,angle radius=2mm] {right angle=O--D--B};
511 \end{tikzpicture}\end{center}
512
513 \begin{itemize}
514 \item
515 Diagonals \(\overrightharp{OB}, \overrightharp{AC}\) bisect each other
516 \item
517 If diagonals are equal length, it is a rectangle
518 \item
519 \(|\overrightharp{OB}|^2 + |\overrightharp{CA}|^2 = |\overrightharp{OA}|^2 + |\overrightharp{AB}|^2 + |\overrightharp{CB}|^2 + |\overrightharp{OC}|^2\)
520 \item
521 Area \(=\boldsymbol{c} \cdot \boldsymbol{a}\)
522 \end{itemize}
523
524 \subsubsection*{Useful vector properties}
525
526 \begin{itemize}
527 \item
528 \(\boldsymbol{a} \parallel \boldsymbol{b} \implies \boldsymbol{b}=k\boldsymbol{a}\) for some
529 \(k \in \mathbb{R} \setminus \{0\}\)
530 \item
531 If \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are parallel with at
532 least one point in common, then they lie on the same straight line
533 \item
534 \(\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b}=0\)
535 \item
536 \(\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2\)
537 \end{itemize}
538
539 \subsection*{Linear dependence}
540
541 \(\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\) are linearly dependent if they are \(\nparallel\) and:
542 \begin{align*}
543 0&=k\boldsymbol{a}+l\boldsymbol{b}+m\boldsymbol{c}\\
544 \therefore \boldsymbol{c} &= m\boldsymbol{a} + n\boldsymbol{b} \quad \text{(simultaneous)}
545 \end{align*}
546
547 \noindent \(\boldsymbol{a}, \boldsymbol{b},\) and \(\boldsymbol{c}\) are linearly
548 independent if no vector in the set is expressible as a linear
549 combination of other vectors in set, or if they are parallel.
550
551 \subsection*{Three-dimensional vectors}
552
553 Right-hand rule for axes: \(z\) is up or out of page.
554
555 \tdplotsetmaincoords{60}{120}
556 \begin{center}\begin{tikzpicture} [scale=3, tdplot_main_coords, axis/.style={->,thick},
557 vector/.style={-stealth,red,very thick},
558 vector guide/.style={dashed,gray,thick}]
559
560 %standard tikz coordinate definition using x, y, z coords
561 \coordinate (O) at (0,0,0);
562
563 %tikz-3dplot coordinate definition using x, y, z coords
564
565 \pgfmathsetmacro{\ax}{1}
566 \pgfmathsetmacro{\ay}{1}
567 \pgfmathsetmacro{\az}{1}
568
569 \coordinate (P) at (\ax,\ay,\az);
570
571 %draw axes
572 \draw[axis] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$};
573 \draw[axis] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$};
574 \draw[axis] (0,0,0) -- (0,0,1) node[anchor=south]{$z$};
575
576 %draw a vector from O to P
577 \draw[vector] (O) -- (P);
578
579 %draw guide lines to components
580 \draw[vector guide] (O) -- (\ax,\ay,0);
581 \draw[vector guide] (\ax,\ay,0) -- (P);
582 \draw[vector guide] (P) -- (0,0,\az);
583 \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
584 \draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
585 \draw[vector guide] (\ax,\ay,0) -- (\ax,0,0);
586 \node[tdplot_main_coords,above right]
587 at (\ax,\ay,\az){(\ax, \ay, \az)};
588 \end{tikzpicture}\end{center}
589
590 \subsection*{Parametric vectors}
591
592 Parametric equation of line through point \((x_0, y_0, z_0)\) and
593 parallel to \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) is:
594
595 \[\begin{cases}x = x_o + a \cdot t \\ y = y_0 + b \cdot t \\ z = z_0 + c \cdot t\end{cases}\]
596
597 \section{Circular functions}
598
599 \(\sin(bx)\) or \(\cos(bx)\): period \(=\frac{2\pi}{b}\)
600
601 \noindent \(\tan(nx)\): period \(=\frac{\pi}{n}\)\\
602 \indent\indent\indent asymptotes at \(x=\frac{(2k+1)\pi}{2n} \> \vert \> k \in \mathbb{Z}\)
603
604 \subsection*{Reciprocal functions}
605
606 \subsubsection*{Cosecant}
607
608 \[\operatorname{cosec} \theta = \frac{1}{\sin \theta} \> \vert \> \sin \theta \ne 0\]
609
610 \begin{itemize}
611 \item
612 \textbf{Domain} \(= \mathbb{R} \setminus {n\pi : n \in \mathbb{Z}}\)
613 \item
614 \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
615 \item
616 \textbf{Turning points} at
617 \(\theta = \frac{(2n + 1)\pi}{2} \> \vert \> n \in \mathbb{Z}\)
618 \item
619 \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
620 \end{itemize}
621
622 \subsubsection*{Secant}
623
624\begin{tikzpicture}
625 \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={\(-2\pi\), \(-\pi\), \(\pi\), \(2\pi\)}]
626% \addplot[blue, domain=-6.2830:6.2830,unbounded coords=jump,samples=80] {sec(deg(x))};
627 \addplot[blue, restrict y to domain=-10:10, domain=-7:7,samples=100] {sec(deg(x))} node [pos=0.93, black, right] {\(\operatorname{sec} x\)};
628 \addplot[red, dashed, domain=-7:7,samples=100] {cos(deg(x))};
629 \draw [gray, dotted, thick] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{rel axis cs:0,1});
630 \draw [gray, dotted, thick] ({axis cs:4.71239,0}|-{rel axis cs:0,0}) -- ({axis cs:4.71239,0}|-{rel axis cs:0,1});
631 \draw [gray, dotted, thick] ({axis cs:-4.71239,0}|-{rel axis cs:0,0}) -- ({axis cs:-4.71239,0}|-{rel axis cs:0,1});
632 \draw [gray, dotted, thick] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1});
633\end{axis}
634 \node [black] at (7,3.5) {\(\cos x\)};
635\end{tikzpicture}
636
637 \[\operatorname{sec} \theta = \frac{1}{\cos \theta} \> \vert \> \cos \theta \ne 0\]
638
639 \begin{itemize}
640
641 \item
642 \textbf{Domain}
643 \(= \mathbb{R} \setminus \frac{(2n + 1) \pi}{2} : n \in \mathbb{Z}\}\)
644 \item
645 \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
646 \item
647 \textbf{Turning points} at
648 \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
649 \item
650 \textbf{Asymptotes} at
651 \(\theta = \frac{(2n + 1) \pi}{2} \> \vert \> n \in \mathbb{Z}\)
652 \end{itemize}
653
654 \subsubsection*{Cotangent}
655
656\begin{tikzpicture}
657 \begin{axis}[xmin=-3,xmax=3,ymin=-1.5,ymax=1.5,enlargelimits=true, xtick={-3.1415, -1.5708, 1.5708, 3.1415},xticklabels={\(-\pi\), \(-\frac{\pi}{2}\), \(\frac{\pi}{2}\), \(\pi\)}]
658 \addplot[blue, smooth, domain=-3:-0.1,unbounded coords=jump,samples=105] {cot(deg(x))} node [pos=0.3, left] {\(\operatorname{cot} x\)};
659\addplot[blue, smooth, domain=0.1:3,unbounded coords=jump,samples=105] {cot(deg(x))};
660\addplot[red, smooth, dashed] gnuplot [domain=-1.5:1.5,unbounded coords=jump,samples=105] {tan(x)};
661\addplot[red, smooth, dashed] gnuplot [domain=-3.5:-1.8,unbounded coords=jump,samples=105] {tan(x)} node [pos=0.5, right] {\(\tan x\)};
662\addplot[red, smooth, dashed] gnuplot [domain=1.8:3.5,unbounded coords=jump,samples=105] {tan(x)};
663 \draw [thick, red, dotted] ({axis cs:-1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:-1.5708,0}|-{rel axis cs:0,1});
664 \draw [thick, blue, dotted] ({axis cs:-3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:-3.1415,0}|-{rel axis cs:0,1});
665 \draw [thick, blue, dotted] ({axis cs:0,0}|-{rel axis cs:0,0}) -- ({axis cs:0,0}|-{rel axis cs:0,1});
666 \draw [thick, blue, dotted] ({axis cs:3.1415,0}|-{rel axis cs:0,0}) -- ({axis cs:3.1415,0}|-{rel axis cs:0,1});
667 \draw [thick, red, dotted] ({axis cs:1.5708,0}|-{rel axis cs:0,0}) -- ({axis cs:1.5708,0}|-{rel axis cs:0,1});
668\end{axis}
669\end{tikzpicture}
670
671 \[\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0\]
672
673 \begin{itemize}
674
675 \item
676 \textbf{Domain} \(= \mathbb{R} \setminus \{n \pi: n \in \mathbb{Z}\}\)
677 \item
678 \textbf{Range} \(= \mathbb{R}\)
679 \item
680 \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
681 \end{itemize}
682
683 \subsubsection*{Symmetry properties}
684
685 \[\begin{split}
686 \operatorname{sec} (\pi \pm x) & = -\operatorname{sec} x \\
687 \operatorname{sec} (-x) & = \operatorname{sec} x \\
688 \operatorname{cosec} (\pi \pm x) & = \mp \operatorname{cosec} x \\
689 \operatorname{cosec} (-x) & = - \operatorname{cosec} x \\
690 \operatorname{cot} (\pi \pm x) & = \pm \operatorname{cot} x \\
691 \operatorname{cot} (-x) & = - \operatorname{cot} x
692 \end{split}\]
693
694 \subsubsection*{Complementary properties}
695
696 \[\begin{split}
697 \operatorname{sec} \left({\pi \over 2} - x\right) & = \operatorname{cosec} x \\
698 \operatorname{cosec} \left({\pi \over 2} - x\right) & = \operatorname{sec} x \\
699 \operatorname{cot} \left({\pi \over 2} - x\right) & = \tan x \\
700 \tan \left({\pi \over 2} - x\right) & = \operatorname{cot} x
701 \end{split}\]
702
703 \subsubsection*{Pythagorean identities}
704
705 \[\begin{split}
706 1 + \operatorname{cot}^2 x & = \operatorname{cosec}^2 x, \quad \text{where } \sin x \ne 0 \\
707 1 + \tan^2 x & = \operatorname{sec}^2 x, \quad \text{where } \cos x \ne 0
708 \end{split}\]
709
710 \subsection*{Compound angle formulas}
711
712 \[\cos(x \pm y) = \cos x + \cos y \mp \sin x \sin y\]
713 \[\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y\]
714 \[\tan(x \pm y) = {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}\]
715
716 \subsection*{Double angle formulas}
717
718 \[\begin{split}
719 \cos 2x &= \cos^2 x - \sin^2 x \\
720 & = 1 - 2\sin^2 x \\
721 & = 2 \cos^2 x -1
722 \end{split}\]
723
724 \[\sin 2x = 2 \sin x \cos x\]
725
726 \[\tan 2x = {{2 \tan x} \over {1 - \tan^2 x}}\]
727
728 \subsection*{Inverse circular functions}
729
730 \begin{tikzpicture}
731 \begin{axis}[ymin=-2, ymax=4, xmin=-1.1, xmax=1.1, ytick={-1.5708, 1.5708, 3.14159},yticklabels={$-\frac{\pi}{2}$, $\frac{\pi}{2}$, $\pi$}]
732 \addplot[color=red, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {asin(x)} node [pos=0.25, below right] {\(\sin^{-1}x\)};
733 \addplot[color=blue, smooth] gnuplot [domain=-2:2,unbounded coords=jump,samples=500] {acos(x)} node [pos=0.25, below left] {\(\cos^{-1}x\)};
734 \addplot[mark=*, red] coordinates {(-1,-1.5708)} node[right, font=\footnotesize]{\((-1,-\frac{\pi}{2})\)} ;
735 \addplot[mark=*, red] coordinates {(1,1.5708)} node[left, font=\footnotesize]{\((1,\frac{\pi}{2})\)} ;
736 \addplot[mark=*, blue] coordinates {(1,0)};
737 \addplot[mark=*, blue] coordinates {(-1,3.1415)} node[right, font=\footnotesize]{\((-1,\pi)\)} ;
738 \end{axis}
739 \end{tikzpicture}\\
740
741 Inverse functions: \(f(f^{-1}(x)) = x\) (restrict domain)
742
743 \[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y\]
744 \hfill where \(\sin y = x, \> y \in [{-\pi \over 2}, {\pi \over 2}]\)
745
746 \[\cos^{-1}: [-1,1] \rightarrow \mathbb{R}, \quad \cos^{-1} x = y\]
747 \hfill where \(\cos y = x, \> y \in [0, \pi]\)
748
749 \[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\]
750 \hfill where \(\tan y = x, \> y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\)
751
752 \begin{tikzpicture}
753 \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={\(-\frac{\pi}{2}\),\(\frac{\pi}{2}\)}]
754 \addplot[color=orange, smooth] gnuplot [domain=-35:35, unbounded coords=jump,samples=350] {atan(x)} node [pos=0.5, above left] {\(\tan^{-1}x\)};
755 \addplot[->, gray, dotted, thick, domain=-35:35] {1.5708};
756 \addplot[->, gray, dotted, thick, domain=-35:35] {-1.5708};
757 \end{axis}
758 \end{tikzpicture}
759\columnbreak
760 \section{Differential calculus}
761
762 \subsection*{Limits}
763
764 \[\lim_{x \rightarrow a}f(x)\]
765 \(L^-,\quad L^+\) \qquad limit from below/above\\
766 \(\lim_{x \to a} f(x)\) \quad limit of a point\\
767
768 \noindent For solving \(x\rightarrow\infty\), put all \(x\) terms in denominators\\
769 e.g. \[\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2\]
770
771 \subsubsection*{Limit theorems}
772
773 \begin{enumerate}
774 \item
775 For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\)
776 \item
777 \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\)
778 \item
779 \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\)
780 \item
781 \(\therefore \lim_{x \rightarrow a} c \times f(x)=cF\) where \(c=\) constant
782 \item
783 \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\)
784 \item
785 \(f(x)\) is continuous \(\iff L^-=L^+=f(x) \> \forall x\)
786 \end{enumerate}
787
788 \subsection*{Gradients of secants and tangents}
789
790 \textbf{Secant (chord)} - line joining two points on curve\\
791 \textbf{Tangent} - line that intersects curve at one point
792
793 \subsection*{First principles derivative}
794
795 \[f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={\frac{dy}{dx}}\]
796
797 \subsubsection*{Logarithmic identities}
798
799 \(\log_b (xy)=\log_b x + \log_b y\)\\
800 \(\log_b x^n = n \log_b x\)\\
801 \(\log_b y^{x^n} = x^n \log_b y\)
802
803 \subsubsection*{Index identities}
804
805 \(b^{m+n}=b^m \cdot b^n\)\\
806 \((b^m)^n=b^{m \cdot n}\)\\
807 \((b \cdot c)^n = b^n \cdot c^n\)\\
808 \({a^m \div a^n} = {a^{m-n}}\)
809
810 \subsection*{Derivative rules}
811
812 \renewcommand{\arraystretch}{1.4}
813 \begin{tabularx}{\columnwidth}{rX}
814 \hline
815 \(f(x)\) & \(f^\prime(x)\)\\
816 \hline
817 \(\sin x\) & \(\cos x\)\\
818 \(\sin ax\) & \(a\cos ax\)\\
819 \(\cos x\) & \(-\sin x\)\\
820 \(\cos ax\) & \(-a \sin ax\)\\
821 \(\tan f(x)\) & \(f^2(x) \sec^2f(x)\)\\
822 \(e^x\) & \(e^x\)\\
823 \(e^{ax}\) & \(ae^{ax}\)\\
824 \(ax^{nx}\) & \(an \cdot e^{nx}\)\\
825 \(\log_e x\) & \(\dfrac{1}{x}\)\\
826 \(\log_e {ax}\) & \(\dfrac{1}{x}\)\\
827 \(\log_e f(x)\) & \(\dfrac{f^\prime (x)}{f(x)}\)\\
828 \(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\\
829 \(\sin^{-1} x\) & \(\dfrac{1}{\sqrt{1-x^2}}\)\\
830 \(\cos^{-1} x\) & \(\dfrac{-1}{sqrt{1-x^2}}\)\\
831 \(\tan^{-1} x\) & \(\dfrac{1}{1 + x^2}\)\\
832 \(\frac{d}{dy}f(y)\) & \(\dfrac{1}{\frac{dx}{dy}}\) (reciprocal)\\
833 \(uv\) & \(u \frac{dv}{dx}+v\frac{du}{dx} (product rule)\)\\
834 \(\dfrac{u}{v}\) & \(\dfrac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\) (quotient rule)\\
835 \(f(g(x))\) & \(f^\prime(g(x))\cdot g^\prime(x)\)\\
836 \hline
837 \end{tabularx}
838
839 \subsection*{Reciprocal derivatives}
840
841 \[\frac{1}{\frac{dy}{dx}} = \frac{dx}{dy}\]
842
843 \subsection*{Differentiating \(x=f(y)\)}
844 \begin{align*}
845 \text{Find }& \frac{dx}{dy}\\
846 \text{Then, }\frac{dx}{dy} &= \frac{1}{\frac{dy}{dx}} \\
847 \implies {\frac{dy}{dx}} &= \frac{1}{\frac{dx}{dy}}\\
848 \therefore {\frac{dy}{dx}} &= \frac{1}{\frac{dx}{dy}}
849 \end{align*}
850
851 \subsection*{Second derivative}
852 \begin{align*}f(x) \longrightarrow &f^\prime (x) \longrightarrow f^{\prime\prime}(x)\\
853 \implies y \longrightarrow &\frac{dy}{dx} \longrightarrow \frac{d^2 y}{dx^2}\end{align*}
854
855 \noindent Order of polynomial \(n\)th derivative decrements each time the derivative is taken
856
857 \subsubsection*{Points of Inflection}
858
859 \emph{Stationary point} - i.e.
860 \(f^\prime(x)=0\)\\
861 \emph{Point of inflection} - max \(|\)gradient\(|\) (i.e.
862 \(f^{\prime\prime} = 0\))
863
864
865 \pgfplotsset{every axis/.append style={
866 axis x line=none, % put the x axis in the middle
867 axis y line=none, % put the y axis in the middle
868 }}
869 \begin{table*}[ht]
870 \centering
871 \begin{tabularx}{\textwidth}{rXXX}
872 \hline
873 \rowcolor{shade2}
874 & \centering\(\dfrac{d^2 y}{dx^2} > 0\) & \centering \(\dfrac{d^2y}{dx^2}<0\) & \(\dfrac{d^2y}{dx^2}=0\) (inflection) \\
875 \hline
876 \(\dfrac{dy}{dx}>0\) &
877 \makecell{\\\begin{tikzpicture}\begin{axis}[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)}&
878 \makecell{\\\begin{tikzpicture}\begin{axis}[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)}&
879 \makecell{\\\begin{tikzpicture}\begin{axis}[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}\\
880 \hline
881 \(\dfrac{dy}{dx}<0\) &
882 \makecell{\\\begin{tikzpicture}\begin{axis}[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)}&
883 \makecell{\\\begin{tikzpicture}\begin{axis}[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)}&
884 \makecell{\\\begin{tikzpicture}\begin{axis}[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}\\
885 \hline
886 \(\dfrac{dy}{dx}=0\)&
887 \makecell{\\\begin{tikzpicture}\begin{axis}[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}[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}&
888 \makecell{\\\begin{tikzpicture}\begin{axis}[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}[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}\\
889 \hline
890 \end{tabularx}
891 \end{table*}
892 \begin{itemize}
893 \item
894 if \(f^\prime (a) = 0\) and \(f^{\prime\prime}(a) > 0\), then point
895 \((a, f(a))\) is a local min (curve is concave up)
896 \item
897 if \(f^\prime (a) = 0\) and \(f^{\prime\prime} (a) < 0\), then point
898 \((a, f(a))\) is local max (curve is concave down)
899 \item
900 if \(f^{\prime\prime}(a) = 0\), then point \((a, f(a))\) is a point of
901 inflection
902 \item
903 if also \(f^\prime(a)=0\), then it is a stationary point of inflection
904 \end{itemize}
905
906 \subsection*{Implicit Differentiation}
907
908 \noindent Used for differentiating circles etc.
909
910 If \(p\) and \(q\) are expressions in \(x\) and \(y\) such that \(p=q\),
911 for all \(x\) and \(y\), then:
912
913 \[{\frac{dp}{dx}} = {\frac{dq}{dx}} \quad \text{and} \quad {\frac{dp}{dy}} = {\frac{dq}{dy}}\]
914
915 \noindent \colorbox{cas}{\textbf{On CAS:}}\\
916 Action \(\rightarrow\) Calculation \(\rightarrow\) \texttt{impDiff(y\^{}2+ax=5,\ x,\ y)}\\
917 Returns \(y^\prime= \dots\).
918
919 \subsection*{Integration}
920
921 \[\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)\]
922
923 \subsection*{Integral laws}
924
925 \renewcommand{\arraystretch}{1.4}
926 \begin{tabularx}{\columnwidth}{rX}
927 \hline
928 \(f(x)\) & \(\int f(x) \cdot dx\) \\
929 \hline
930 \(k\) (constant) & \(kx + c\)\\
931 \(x^n\) & \(\dfrac{1}{n+1} x^{n+1}\) \\
932 \(a x^{-n}\) &\(a \cdot \log_e |x| + c\)\\
933 \(\dfrac{1}{ax+b}\) &\(\dfrac{1}{a} \log_e (ax+b) + c\)\\
934 \((ax+b)^n\) & \(\dfrac{1}{a(n+1)}(ax+b)^{n-1} + c\>|\>n\ne 1\)\\
935 \((ax+b)^{-1}\) & \(\dfrac{1}{a}\log_e |ax+b|+c\)\\
936 \(e^{kx}\) & \(\dfrac{1}{k} e^{kx} + c\)\\
937 \(e^k\) & \(e^kx + c\)\\
938 \(\sin kx\) & \(\dfrac{-1}{k} \cos (kx) + c\)\\
939 \(\cos kx\) & \(\dfrac{1}{k} \sin (kx) + c\)\\
940 \(\sec^2 kx\) & \(\dfrac{1}{k} \tan(kx) + c\)\\
941 \(\dfrac{1}{\sqrt{a^2-x^2}}\) & \(\sin^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
942 \(\dfrac{-1}{\sqrt{a^2-x^2}}\) & \(\cos^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
943 \(\frac{a}{a^2-x^2}\) & \(\tan^{-1} \frac{x}{a} + c\)\\
944 \(\frac{f^\prime (x)}{f(x)}\) & \(\log_e f(x) + c\)\\
945 \(\int f(u) \cdot \frac{du}{dx} \cdot dx\) & \(\int f(u) \cdot du\) (substitution)\\
946 \(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\
947 \hline
948 \end{tabularx}
949
950 Note \(\sin^{-1} {x \over a} + \cos^{-1} {x \over a}\) is constant \(\forall x \in (-a, a)\)
951
952 \subsection*{Definite integrals}
953
954 \[\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)\]
955
956 \begin{itemize}
957
958 \item
959 Signed area enclosed by\\
960 \(\> y=f(x), \quad y=0, \quad x=a, \quad x=b\).
961 \item
962 \emph{Integrand} is \(f\).
963 \end{itemize}
964
965 \subsubsection*{Properties}
966
967 \[\int^b_a f(x) \> dx = \int^c_a f(x) \> dx + \int^b_c f(x) \> dx\]
968
969 \[\int^a_a f(x) \> dx = 0\]
970
971 \[\int^b_a k \cdot f(x) \> dx = k \int^b_a f(x) \> dx\]
972
973 \[\int^b_a f(x) \pm g(x) \> dx = \int^b_a f(x) \> dx \pm \int^b_a g(x) \> dx\]
974
975 \[\int^b_a f(x) \> dx = - \int^a_b f(x) \> dx\]
976
977 \subsection*{Integration by substitution}
978
979 \[\int f(u) {\frac{du}{dx}} \cdot dx = \int f(u) \cdot du\]
980
981 \noindent Note \(f(u)\) must be 1:1 \(\implies\) one \(x\) for each \(y\)
982 \begin{align*}\text{e.g. for } y&=\int(2x+1)\sqrt{x+4} \cdot dx\\
983 \text{let } u&=x+4\\
984 \implies& {\frac{du}{dx}} = 1\\
985 \implies& x = u - 4\\
986 \text{then } &y=\int (2(u-4)+1)u^{\frac{1}{2}} \cdot du\\
987 &\text{(solve as normal integral)}
988 \end{align*}
989
990 \subsubsection*{Definite integrals by substitution}
991
992 For \(\int^b_a f(x) {\frac{du}{dx}} \cdot dx\), evaluate new \(a\) and
993 \(b\) for \(f(u) \cdot du\).
994
995 \subsubsection*{Trigonometric integration}
996
997 \[\sin^m x \cos^n x \cdot dx\]
998
999 \paragraph{\textbf{\(m\) is odd:}}
1000 \(m=2k+1\) where \(k \in \mathbb{Z}\)\\
1001 \(\implies \sin^{2k+1} x = (\sin^2 z)^k \sin x = (1 - \cos^2 x)^k \sin x\)\\
1002 Substitute \(u=\cos x\)
1003
1004 \paragraph{\textbf{\(n\) is odd:}}
1005 \(n=2k+1\) where \(k \in \mathbb{Z}\)\\
1006 \(\implies \cos^{2k+1} x = (\cos^2 x)^k \cos x = (1-\sin^2 x)^k \cos x\)\\
1007 Substitute \(u=\sin x\)
1008
1009 \paragraph{\textbf{\(m\) and \(n\) are even:}}
1010 use identities...
1011
1012 \begin{itemize}
1013
1014 \item
1015 \(\sin^2x={1 \over 2}(1-\cos 2x)\)
1016 \item
1017 \(\cos^2x={1 \over 2}(1+\cos 2x)\)
1018 \item
1019 \(\sin 2x = 2 \sin x \cos x\)
1020 \end{itemize}
1021
1022 \subsection*{Partial fractions}
1023
1024 \colorbox{cas}{On CAS:}\\
1025 \indent Action \(\rightarrow\) Transformation \(\rightarrow\)
1026 \texttt{expand/combine}\\
1027 \indent Interactive \(\rightarrow\) Transformation \(\rightarrow\)
1028 Expand \(\rightarrow\) Partial
1029
1030 \subsection*{Graphing integrals on CAS}
1031
1032 \colorbox{cas}{In main:} Interactive \(\rightarrow\) Calculation \(\rightarrow\)
1033 \(\int\) (\(\rightarrow\) Definite)\\
1034 Restrictions: \texttt{Define\ f(x)=..} then \texttt{f(x)\textbar{}x\textgreater{}..}
1035
1036 \subsection*{Applications of antidifferentiation}
1037
1038 \begin{itemize}
1039
1040 \item
1041 \(x\)-intercepts of \(y=f(x)\) identify \(x\)-coordinates of
1042 stationary points on \(y=F(x)\)
1043 \item
1044 nature of stationary points is determined by sign of \(y=f(x)\) on
1045 either side of its \(x\)-intercepts
1046 \item
1047 if \(f(x)\) is a polynomial of degree \(n\), then \(F(x)\) has degree
1048 \(n+1\)
1049 \end{itemize}
1050
1051 To find stationary points of a function, substitute \(x\) value of given
1052 point into derivative. Solve for \({\frac{dy}{dx}}=0\). Integrate to find
1053 original function.
1054
1055 \subsection*{Solids of revolution}
1056
1057 Approximate as sum of infinitesimally-thick cylinders
1058
1059 \subsubsection*{Rotation about \(x\)-axis}
1060
1061 \begin{align*}
1062 V &= \int^{x=b}_{x-a} \pi y^2 \> dx \\
1063 &= \pi \int^b_a (f(x))^2 \> dx
1064 \end{align*}
1065
1066 \subsubsection*{Rotation about \(y\)-axis}
1067
1068 \begin{align*}
1069 V &= \int^{y=b}_{y=a} \pi x^2 \> dy \\
1070 &= \pi \int^b_a (f(y))^2 \> dy
1071 \end{align*}
1072
1073 \subsubsection*{Regions not bound by \(y=0\)}
1074
1075 \[V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx\]
1076 \hfill where \(f(x) > g(x)\)
1077
1078 \subsection*{Length of a curve}
1079
1080 \[L = \int^b_a \sqrt{1 + ({\frac{dy}{dx}})^2} \> dx \quad \text{(Cartesian)}\]
1081
1082 \[L = \int^b_a \sqrt{{\frac{dx}{dt}} + ({\frac{dy}{dt}})^2} \> dt \quad \text{(parametric)}\]
1083
1084 \noindent \colorbox{cas}{On CAS:}\\
1085 \indent Evaluate formula,\\
1086 \indent or Interactive \(\rightarrow\) Calculation
1087 \(\rightarrow\) Line \(\rightarrow\) \texttt{arcLen}
1088
1089 \subsection*{Rates}
1090
1091 \subsubsection*{Gradient at a point on parametric curve}
1092
1093 \[{\frac{dy}{dx}} = {{\frac{dy}{dt}} \div {\frac{dx}{dt}}} \> \vert \> {\frac{dx}{dt}} \ne 0 \text{ (chain rule)}\]
1094
1095 \[\frac{d^2}{dx^2} = \frac{d(y^\prime)}{dx} = {\frac{dy^\prime}{dt} \div {\frac{dx}{dt}}} \> \vert \> y^\prime = {\frac{dy}{dx}}\]
1096
1097 \subsection*{Rational functions}
1098
1099 \[f(x) = \frac{P(x)}{Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}\]
1100
1101 \subsubsection*{Addition of ordinates}
1102
1103 \begin{itemize}
1104
1105 \item
1106 when two graphs have the same ordinate, \(y\)-coordinate is double the
1107 ordinate
1108 \item
1109 when two graphs have opposite ordinates, \(y\)-coordinate is 0 i.e.
1110 (\(x\)-intercept)
1111 \item
1112 when one of the ordinates is 0, the resulting ordinate is equal to the
1113 other ordinate
1114 \end{itemize}
1115
1116 \subsection*{Fundamental theorem of calculus}
1117
1118 If \(f\) is continuous on \([a, b]\), then
1119
1120 \[\int^b_a f(x) \> dx = F(b) - F(a)\]
1121 \hfill where \(F = \int f \> dx\)
1122
1123 \subsection*{Differential equations}
1124
1125 \noindent\textbf{Order} - highest power inside derivative\\
1126 \textbf{Degree} - highest power of highest derivative\\
1127 e.g. \({\left(\dfrac{dy^2}{d^2} x\right)}^3\) \qquad order 2, degree 3
1128
1129 \subsubsection*{Verifying solutions}
1130
1131 Start with \(y=\dots\), and differentiate. Substitute into original
1132 equation.
1133
1134 \subsubsection*{Function of the dependent
1135 variable}
1136
1137 If \({\frac{dy}{dx}}=g(y)\), then
1138 \(\frac{dx}{dy} = 1 \div \frac{dy}{dx} = \frac{1}{g(y)}\). Integrate both sides to solve equation. Only add \(c\) on one side. Express
1139 \(e^c\) as \(A\).
1140
1141
1142
1143 \subsubsection*{Mixing problems}
1144
1145 \[\left(\frac{dm}{dt}\right)_\Sigma = \left(\frac{dm}{dt}\right)_{\text{in}} - \left(\frac{dm}{dt}_{\text{out}}\right)\]
1146
1147 \subsubsection*{Separation of variables}
1148
1149 If \({\frac{dy}{dx}}=f(x)g(y)\), then:
1150
1151 \[\int f(x) \> dx = \int \frac{1}{g(y)} \> dy\]
1152
1153 \subsubsection*{Euler's method for solving DEs}
1154
1155 \[\frac{f(x+h) - f(x)}{h} \approx f^\prime (x) \quad \text{for small } h\]
1156
1157 \[\implies f(x+h) \approx f(x) + hf^\prime(x)\]
1158
1159
1160 \section{Kinematics \& Mechanics}
1161
1162 \subsection*{Constant acceleration}
1163 {\centering \begin{tabular}{ l r } % TODO need to fix centering here
1164 \hline & no \\ \hline
1165 $v=u+at$ & $x$ \\
1166 $s = {1 \over 2}(v+u)t$ & $a$ \\
1167 $s=ut+{1 \over 2}at^2$ & $v$ \\
1168 $s=vt-{1 \over 2}at^2$ & $u$ \\
1169 $v^2=u^2+2as$ & $t$ \\ \hline
1170 \end{tabular}}
1171
1172 \[ v_{\text{avg}} = \frac{\Delta\text{position}}{\Delta t} \]
1173 \begin{align*}
1174 \text{speed} &= |{\text{velocity}}| \\
1175 &= \sqrt{v_x^2 + v_y^2 + v_z^2}
1176 \end{align*}
1177 \textbf{Distance travelled between \(t=a \rightarrow t=b\):}
1178 \[= \int^b_a \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \cdot dt \]
1179
1180 \subsection*{Vector functions}
1181
1182 \[ \boldsymbol{r}(t) = x \boldsymbol{i} + y \boldsymbol{j} + z \boldsymbol{k} \]
1183
1184 \begin{itemize}
1185 \item If \(\boldsymbol{r}(t) \equiv\) position with time, then the graph of endpoints of \(\boldsymbol{r}(t) \equiv\) Cartesian path
1186 \item Domain of \(\boldsymbol{r}(t)\) is the range of \(x(t)\)
1187 \item Range of \(\boldsymbol{r}(t)\) is the range of \(y(t)\)
1188 \end{itemize}
1189
1190 \subsection*{Vector calculus}
1191
1192 \subsubsection*{Derivative}
1193
1194 Let \(\boldsymbol{r}(t)=x(t)\boldsymbol{i} + y(t)\boldsymbol(j)\). If both \(x(t)\) and \(y(t)\) are differentiable, then:
1195 \[ \boldsymbol{r}(t)=x(t)\boldsymbol{i}+y(t)\boldsymbol{j} \]
1196
1197 \end{multicols}
1198\end{document}