eca12dcc48d1438480a906486cf0a5013aa0bc21
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{graphicx}
11\usepackage{wrapfig}
12\usepackage{tikz}
13\usepackage{tikz-3dplot}
14\usetikzlibrary{calc}
15\usetikzlibrary{angles}
16\usepgflibrary{arrows.meta}
17\usepackage{fancyhdr}
18\pagestyle{fancy}
19\fancyhead[LO,LE]{Year 12 Specialist}
20\fancyhead[CO,CE]{Andrew Lorimer}
21
22\usepackage{mathtools}
23\usepackage{xcolor} % used only to show the phantomed stuff
24\renewcommand\hphantom[1]{{\color[gray]{.6}#1}} % comment out!
25\setlength\fboxsep{0pt} \setlength\fboxrule{.2pt} % for the \fboxes
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28\newcolumntype{L}[1]{>{\hsize=#1\hsize\raggedright\arraybackslash}X}%
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30\definecolor{cas}{HTML}{e6f0fe}
31\linespread{1.5}
32\newcommand{\midarrow}{\tikz \draw[-triangle 90] (0,0) -- +(.1,0);}
33
34\begin{document}
35
36\begin{multicols}{2}
37
38 \section{Complex numbers}
39
40 \[\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}\]
41
42 \begin{align*}
43 \text{Cartesian form: } & a+bi\\
44 \text{Polar form: } & r\operatorname{cis}\theta
45 \end{align*}
46
47 \subsection*{Operations}
48
49\definecolor{shade1}{HTML}{ffffff}
50\definecolor{shade2}{HTML}{e6f2ff}
51 \definecolor{shade3}{HTML}{cce2ff}
52 \begin{tabularx}{\columnwidth}{r|X|X}
53 & \textbf{Cartesian} & \textbf{Polar} \\
54 \hline
55 \(z_1 \pm z_2\) & \((a \pm c)(b \pm d)i\) & convert to \(a+bi\)\\
56 \hline
57 \(+k \times z\) & \multirow{2}{*}{\(ka \pm kbi\)} & \(kr\operatorname{cis} \theta\)\\
58 \cline{1-1}\cline{3-3}
59 \(-k \times z\) & & \(kr \operatorname{cis}(\theta\pm \pi)\)\\
60 \hline
61 \(z_1 \cdot z_2\) & \(ac-bd+(ad+bc)i\) & \(r_1r_2 \operatorname{cis}(\theta_1 + \theta_2)\)\\
62 \hline
63 \(z_1 \div z_2\) & \((z_1 \overline{z_2}) \div |z_2|^2\) & \(\left(\frac{r_1}{r_2}\right) \operatorname{cis}(\theta_1 - \theta_2)\)
64 \end{tabularx}
65
66 \subsubsection*{Scalar multiplication in polar form}
67
68 For \(k \in \mathbb{R}^+\):
69 \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\theta\]
70
71 \noindent For \(k \in \mathbb{R}^-\):
72 \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\left(\begin{cases}\theta - \pi & |0<\operatorname{Arg}(z)\le \pi \\ \theta + \pi & |-\pi<\operatorname{Arg}(z)\le 0\end{cases}\right)\]
73
74 \subsection*{Conjugate}
75
76 \begin{align*}
77 \overline{z} &= a \mp bi\\
78 &= r \operatorname{cis}(-\theta)
79 \end{align*}
80
81 \noindent \colorbox{cas}{On CAS: \texttt{conjg(a+bi)}}
82
83 \subsubsection*{Properties}
84
85 \begin{align*}
86 \overline{z_1 \pm z_2} &= \overline{z_1}\pm\overline{z_2}\\
87 \overline{z_1 \cdot z_2} &= \overline{z_1}\cdot\overline{z_2}\\
88 \overline{kz} &= k\overline{z} \quad | \quad k \in \mathbb{R}\\
89 z\overline{z} &= (a+bi)(a-bi)\\
90 &= a^2 + b^2\\
91 &= |z|^2
92 \end{align*}
93
94 \subsection*{Modulus}
95
96 \[|z|=|\vec{Oz}|=\sqrt{a^2 + b^2}\]
97
98 \subsubsection*{Properties}
99
100 \begin{align*}
101 |z_1z_2|&=|z_1||z_2|\\
102 \left|\frac{z_1}{z_2}\right|&=\frac{|z_1|}{|z_2|}\\
103 |z_1+z_2|&\le|z_1|+|z_2|
104 \end{align*}
105
106 \subsection*{Multiplicative inverse}
107
108 \begin{align*}
109 z^{-1}&=\frac{a-bi}{a^2+b^2}\\
110 &=\frac{\overline{z}}{|z|^2}a\\
111 &=r \operatorname{cis}(-\theta)
112 \end{align*}
113
114 \subsection*{Dividing over \(\mathbb{C}\)}
115
116 \begin{align*}
117 \frac{z_1}{z_2}&=z_1z_2^{-1}\\
118 &=\frac{z_1\overline{z_2}}{|z_2|^2}\\
119 &=\frac{(a+bi)(c-di)}{c^2+d^2}\\
120 & \qquad \text{(rationalise denominator)}
121 \end{align*}
122
123 \subsection*{Polar form}
124
125 \begin{align*}
126 z&=r\operatorname{cis}\theta\\
127 &=r(\cos \theta + i \sin \theta)
128 \end{align*}
129
130 \begin{itemize}
131 \item{\(r=|z|=\sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}\)}
132 \item{\(\theta = \operatorname{arg}(z)\) \quad \colorbox{cas}{On CAS: \texttt{arg(a+bi)}}}
133 \item{\(\operatorname{Arg}(z) \in (-\pi,\pi)\) \quad \bf{(principal argument)}}
134 \item{\colorbox{cas}{Convert on CAS:}\\ \verb|compToTrig(a+bi)| \(\iff\) \verb|cExpand{r·cisX}|}
135 \item{Multiple representations:\\\(r\operatorname{cis}\theta=r\operatorname{cis}(\theta+2n\pi)\) with \(n \in \mathbb{Z}\) revolutions}
136 \item{\(\operatorname{cis}\pi=-1,\qquad \operatorname{cis}0=1\)}
137 \end{itemize}
138
139 \subsection*{de Moivres' theorem}
140
141 \[(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}\]
142
143 \subsection*{Complex polynomials}
144
145 Include \(\pm\) for all solutions, incl. imaginary
146
147 \begin{tabularx}{\columnwidth}{ R{0.55} X }
148 \hline
149 Sum of squares & \(\begin{aligned}
150 z^2 + a^2 &= z^2-(ai)^2\\
151 &= (z+ai)(z-ai) \end{aligned}\) \\
152 \hline
153 Sum of cubes & \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)\\
154 \hline
155 Division & \(P(z)=D(z)Q(z)+R(z)\) \\
156 \hline
157 Remainder theorem & Let \(\alpha \in \mathbb{C}\). Remainder of \(P(z) \div (z-\alpha)\) is \(P(\alpha)\)\\
158 \hline
159 Factor theorem & \(z-\alpha\) is a factor of \(P(z) \iff P(\alpha)=0\) for \(\alpha \in \mathbb{C}\)\\
160 \hline
161 Conjugate root theorem & \(P(z)=0 \text{ at } z=a\pm bi\) (\(\implies\) both \(z_1\) and \(\overline{z_1}\) are solutions)
162 \end{tabularx}
163
164 \subsection*{Roots}
165
166 \(n\)th roots of \(z=r\operatorname{cis}\theta\) are:
167
168 \[z = r^{\frac{1}{n}} \operatorname{cis}\left(\frac{\theta+2k\pi}{n}\right)\]
169
170 \begin{itemize}
171
172 \item{Same modulus for all solutions}
173 \item{Arguments are separated by \(\frac{2\pi}{n}\)}
174 \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 \(\frac{2\pi}{n}\))}
175 \end{itemize}
176
177 \noindent For \(0=az^2+bz+c\), use quadratic formula:
178
179 \[z=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]
180
181 \subsection*{Fundamental theorem of algebra}
182
183 A polynomial of degree \(n\) can be factorised into \(n\) linear factors in \(\mathbb{C}\):
184
185 \[\implies P(z)=a_n(z-\alpha_1)(z-\alpha_2)(z-\alpha_3)\dots(z-\alpha_n)\]
186 \[\text{ where } \alpha_1,\alpha_2,\alpha_3,\dots,\alpha_n \in \mathbb{C}\]
187
188 \subsection*{Argand planes}
189
190 \begin{center}\begin{tikzpicture}[scale=2]
191 \draw [->] (-0.2,0) -- (1.5,0) node [right] {$\operatorname{Re}(z)$};
192 \draw [->] (0,-0.2) -- (0,1.5) node [above] {$\operatorname{Im}(z)$};
193 \coordinate (P) at (1,1);
194 \coordinate (a) at (1,0);
195 \coordinate (b) at (0,1);
196 \coordinate (O) at (0,0);
197 \draw (0,0) -- (P) node[pos=0.5, above left]{\(r\)} node[pos=1, right]{\(\begin{aligned}z&=a+bi\\&=r\operatorname{cis}\theta\end{aligned}\)};
198 \draw [gray, dashed] (1,1) -- (1,0) node[black, pos=1, below]{\(a\)};
199 \draw [gray, dashed] (1,1) -- (0,1) node[black, pos=1, left]{\(b\)};
200 \begin{scope}
201 \path[clip] (O) -- (P) -- (a);
202 \fill[red, opacity=0.5, draw=black] (O) circle (2mm);
203 \node at ($(O)+(20:3mm)$) {$\theta$};
204 \end{scope}
205 \filldraw (P) circle (0.5pt);
206 \end{tikzpicture}\end{center}
207
208 \begin{itemize}
209 \item{Multiplication by \(i \implies\) CCW rotation of \(\frac{\pi}{2}\)}
210 \item{Addition: \(z_1 + z_2 \equiv\) \overrightharp{\(Oz_1\)} + \overrightharp{\(Oz_2\)}}
211 \end{itemize}
212
213 \subsection*{Sketching complex graphs}
214
215 \subsubsection*{Linear}
216
217 \begin{itemize}
218 \item{\(\operatorname{Re}(z)=c\) or \(\operatorname{Im}(z)=c\) (perpendicular bisector)}
219 \item{\(\operatorname{Im}(z)=m\operatorname{Re}(z)\)}
220 \item{\(|z+a|=|z+b| \implies 2(a-b)x=b^2-a^2\)}
221 \end{itemize}
222
223 \subsubsection*{Circles}
224
225 \begin{itemize}
226 \item \(|z-z_1|^2=c^2|z_2+2|^2\)
227 \item \(|z-(a+bi)|=c\)
228 \end{itemize}
229
230 \noindent \textbf{Loci} \qquad \(\operatorname{Arg}(z)<\theta\)
231
232 \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
233 \draw [->] (0,0) -- (1,0) node [right] {$\operatorname{Re}(z)$};
234 \draw [->] (0,-0.5) -- (0,1) node [above] {$\operatorname{Im}(z)$};
235 \draw [<-, dashed, thick, blue] (-1,0) -- (0,0);
236 \draw [->, thick, blue] (0,0) -- (1,1);
237 \fill [gray, opacity=0.2, domain=-1:1, variable=\x] (-1,-0.5) -- (-1,0) -- (0, 0) -- (1,1) -- (1,-0.5) -- cycle;
238 \begin{scope}
239 \path[clip] (0,0) -- (1,1) -- (1,0);
240 \fill[red, opacity=0.5, draw=black] (0,0) circle (2mm);
241 \node at ($(0,0)+(20:3mm)$) {$\frac{\pi}{4}$};
242 \end{scope}
243 \node [font=\footnotesize] at (0.5,-0.25) {\(\operatorname{Arg}(z)\le\frac{\pi}{4}\)};
244 \node [blue, mydot] {};
245 \end{tikzpicture}\end{center}
246
247 \noindent \textbf{Rays} \qquad \(\operatorname{Arg}(z-b)=\theta\)
248
249 \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
250 \draw [->] (-0.75,0) -- (1.5,0) node [right] {$\operatorname{Re}(z)$};
251 \draw [->] (0,-1) -- (0,1) node [above] {$\operatorname{Im}(z)$};
252 \draw [->, thick, brown] (-0.25,0) -- (-0.75,-1);
253 \node [above, font=\footnotesize] at (-0.25,0) {\(\frac{1}{4}\)};
254 \begin{scope}
255 \path[clip] (-0.25,0) -- (-0.75,-1) -- (0,0);
256 \fill[orange, opacity=0.5, draw=black] (-0.25,0) circle (2mm);
257 \end{scope}
258 \node at (-0.08,-0.3) {\(\frac{\pi}{8}\)};
259 \node [font=\footnotesize, left] at (-0.75,-1) {\(\operatorname{Arg}(z+\frac{1}{4})=\frac{\pi}{8}\)};
260 \node [brown, mydot] at (-0.25,0) {};
261 \draw [<->, thick, green] (0,-1) -- (1.5,0.5) node [pos=0.25, black, font=\footnotesize, right] {\(|z-2|=|z-(1+i)|\)};
262 \node [left, font=\footnotesize] at (0,-1) {\(-1\)};
263 \node [below, font=\footnotesize] at (1,0) {\(1\)};
264 \end{tikzpicture}\end{center}
265
266 \section{Vectors}
267\begin{center}\begin{tikzpicture}
268 \draw [->] (-0.5,0) -- (3,0) node [right] {\(x\)};
269 \draw [->] (0,-0.5) -- (0,3) node [above] {\(y\)};
270 \draw [orange, ->, thick] (0.5,0.5) -- (2.5,2.5) node [pos=0.5, above] {\(\vec{u}\)};
271 \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
272 \draw [gray, dashed, thick] (0.5,0.5) -- (2.5,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, below]{\(x\vec{i}\)};
273 \draw [gray, dashed, thick] (2.5,0.5) -- (2.5,2.5) node [pos=0.5] {\midarrow};
274 \end{scope}
275 \node[black, right] at (2.5,1.5) {\(y\vec{j}\)};
276
277\end{tikzpicture}\end{center}
278
279\subsection*{Column notation}
280
281\[\begin{bmatrix}x\\ y \end{bmatrix} \iff x\boldsymbol{i} + y\boldsymbol{j}\]
282\(\begin{bmatrix}x_2-x_1\\ y_2-y_1 \end{bmatrix}\) \quad between \(A(x_1,y_1), \> B(x_2,y_2)\)
283
284\subsection*{Scalar multiplication}
285
286\[k\cdot (x\boldsymbol{i}+y\boldsymbol{j})=kx\boldsymbol{i}+ky\boldsymbol{j}\]
287
288\noindent For \(k \in \mathbb{R}^-\), direction is reversed
289
290\subsection*{Vector addition}
291\begin{center}\begin{tikzpicture}[scale=1]
292 \coordinate (A) at (0,0);
293 \coordinate (B) at (2,2);
294 \draw [->, thick, red] (0,0) -- (2,2) node [pos=0.5, below right] {\(\vec{u}=2\vec{i}+2\vec{j}\)};
295 \draw [->, thick, blue] (2,2) -- (1,4) node [pos=0.5, above right] {\(\vec{v}=-\vec{i}+2\vec{j}\)};
296 \draw [->, thick, orange] (0,0) -- (1,4) node [pos=0.5, left] {\(\vec{u}+\vec{v}=\vec{i}+4\vec{j}\)};
297\end{tikzpicture}\end{center}
298
299\[(x\boldsymbol{i}+y\boldsymbol{j}) \pm (a\boldsymbol{i}+b\boldsymbol{j})=(x \pm a)\boldsymbol{i}+(y \pm b)\boldsymbol{j}\]
300
301\begin{itemize}
302 \item Draw each vector head to tail then join lines
303 \item Addition is commutative (parallelogram)
304 \item \(\boldsymbol{u}-\boldsymbol{v}=\boldsymbol{u}+(-\boldsymbol{v})\)
305\end{itemize}
306
307\subsection*{Magnitude}
308
309\[|(x\boldsymbol{i} + y\boldsymbol{j})|=\sqrt{x^2+y^2}\]
310
311\subsection*{Parallel vectors}
312
313\[\boldsymbol{u} || \boldsymbol{v} \iff \boldsymbol{u} = k \boldsymbol{v} \text{ where } k \in \mathbb{R} \setminus \{0\}\]
314
315For parallel vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\):\\
316\[\boldsymbol{a \cdot b}=\begin{cases}
317|\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction}\\
318-|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions}
319\end{cases}\]
320%\includegraphics[width=0.2,height=\textheight]{graphics/parallelogram-vectors.jpg}
321%\includegraphics[width=1]{graphics/vector-subtraction.jpg}
322
323\subsection*{Perpendicular vectors}
324
325\[\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b} = 0\ \quad \text{(since \(\cos 90 = 0\))}\]
326
327\subsection*{Unit vector \(|\hat{\boldsymbol{a}}|=1\)}
328\[\begin{split}\hat{\boldsymbol{a}} & = {\frac{1}{|\boldsymbol{a}|}}\boldsymbol{a} \\ & = \boldsymbol{a} \cdot {|\boldsymbol{a}|}\end{split}\]
329
330 \subsection*{Scalar product \(\boldsymbol{a} \cdot \boldsymbol{b}\)}
331
332
333\begin{center}\begin{tikzpicture}[scale=2]
334 \draw [->] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{b}\)};
335 \draw [->] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{a}\)};
336 \begin{scope}
337 \path[clip] (1,0.5) -- (1,0) -- (0,0);
338 \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
339 \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
340 \end{scope}
341\end{tikzpicture}\end{center}
342\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*}
343\noindent\colorbox{cas}{On CAS: \texttt{dotP({[}a\ b\ c{]},\ {[}d\ e\ f{]})}}
344
345\subsubsection*{Properties}
346
347\begin{enumerate}
348\item
349 \(k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k\boldsymbol{b})\)
350\item
351 \(\boldsymbol{a \cdot 0}=0\)
352\item
353 \(\boldsymbol{a} \cdot (\boldsymbol{b} + \boldsymbol{c})=\boldsymbol{a} \cdot \boldsymbol{b} + \boldsymbol{a} \cdot \boldsymbol{c}\)
354\item
355 \(\boldsymbol{i \cdot i} = \boldsymbol{j \cdot j} = \boldsymbol{k \cdot k}= 1\)
356\item
357 \(\boldsymbol{a} \cdot \boldsymbol{b} = 0 \quad \implies \quad \boldsymbol{a} \perp \boldsymbol{b}\)
358\item
359 \(\boldsymbol{a \cdot a} = |\boldsymbol{a}|^2 = a^2\)
360\end{enumerate}
361
362\subsection*{Angle between vectors}
363
364\[\cos \theta = {{\boldsymbol{a} \cdot \frac{\boldsymbol{b}}{|\boldsymbol{a}| |\boldsymbol{b}|}} = {\frac{a_1 b_1 + a_2 b_2}{|\boldsymbol{a}| |\boldsymbol{b}|}}\]
365
366\noindent \colorbox{cas}{On CAS:} \texttt{angle([a b c], [a b c])}
367
368(Action \(\rightarrow\) Vector \(\rightarrow\)Angle)
369
370\subsection*{Angle between vector and axis}
371
372\noindent For\(\boldsymbol{a} = a_1 \boldsymbol{i} + a_2 \boldsymbol{j} + a_3 \boldsymbol{k}\)
373which makes angles \(\alpha, \beta, \gamma\) with positive side of
374\(x, y, z\) axes:
375\[\cos \alpha = \frac{a_1}{|\boldsymbol{a}|}, \quad \cos \beta = \frac{a_2}{|\boldsymbol{a}|}, \quad \cos \gamma = \frac{a_3}{|\boldsymbol{a}|}\]
376
377\noindent \colorbox{cas}{On CAS:} \texttt{angle({[}a\ b\ c{]},\ {[}1\ 0\ 0{]})}\\for angle
378between \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) and
379\(x\)-axis
380
381\subsection*{Projections \& resolutes}
382
383\begin{tikzpicture}[scale=3]
384 \draw [->, purple] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{a}\)};
385 \draw [->, orange] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{u}\)};
386 \draw [->, blue] (1,0) -- (2,0) node [pos=0.5, below] {\(\boldsymbol{b}\)};
387 \begin{scope}
388 \path[clip] (1,0.5) -- (1,0) -- (0,0);
389 \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
390 \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
391 \end{scope}
392 \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
393 \draw [gray, dashed, thick] (1,0) -- (1,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, right, rotate=-90]{\(\boldsymbol{w}\)};
394 \end{scope}
395\draw (0,0) coordinate (O)
396 (1,0) coordinate (A)
397 (1,0.5) coordinate (B)
398 pic [draw,red,angle radius=2mm] {right angle = O--A--B};
399\end{tikzpicture}
400
401\subsubsection*{\(\parallel\boldsymbol{b}\) (vector projection/resolute)}
402\begin{align*}
403 \boldsymbol{u}&={{\boldsymbol{a}\cdot\boldsymbol{b}}\over |\boldsymbol{b}|^2}\boldsymbol{b}\\
404 &=\left({\boldsymbol{a}\cdot{\boldsymbol{b} \over |\boldsymbol{b}|}}\right)\left({\boldsymbol{b} \over |\boldsymbol{b}|}\right)\\
405 &=(\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}
406\end{align*}
407
408\subsubsection*{\(\perp\boldsymbol{b}\) (perpendicular projection)}
409\[\boldsymbol{w} = \boldsymbol{a} - \boldsymbol{u}\]
410
411\subsubsection*{\(|\boldsymbol{u}|\) (scalar resolute)}
412\begin{align*}
413 r_s &= |\boldsymbol{u}|\\
414 &= \boldsymbol{a} \cdot \hat{\boldsymbol{b}}\\
415 &=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}
416\end{align*}
417
418\subsubsection*{Rectangular (\(\parallel,\perp\)) components}
419
420\[\boldsymbol{a}=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}+\left(\boldsymbol{a}-\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}\right)\]
421
422
423\subsection*{Vector proofs}
424
425\textbf{Concurrent:} intersection of \(\ge\) 3 lines
426
427\begin{tikzpicture}
428 \draw [blue] (0,0) -- (1,1);
429 \draw [red] (1,0) -- (0,1);
430 \draw [brown] (0.4,0) -- (0.6,1);
431 \filldraw (0.5,0.5) circle (2pt);
432\end{tikzpicture}
433
434\subsubsection*{Collinear points}
435
436\(\ge\) 3 points lie on the same line
437
438\begin{tikzpicture}
439 \draw [purple] (0,0) -- (4,1);
440 \filldraw (2,0.5) circle (2pt) node [above] {\(C\)};
441 \filldraw (1,0.25) circle (2pt) node [above] {\(A\)};
442 \filldraw (3,0.75) circle (2pt) node [above] {\(B\)};
443 \coordinate (O) at (2.8,-0.2);
444 \node at (O) [below] {\(O\)};
445 \begin{scope}[->, orange, thick]
446 \draw (O) -- (2,0.5) node [pos=0.5, above, font=\footnotesize, black] {\(\boldsymbol{c}\)};
447 \draw (O) -- (1,0.25) node [pos=0.5, below, font=\footnotesize, black] {\(\boldsymbol{a}\)};
448 \draw (O) -- (3,0.75) node [pos=0.5, right, font=\footnotesize, black] {\(\boldsymbol{b}\)};
449 \end{scope}
450\end{tikzpicture}
451
452\begin{align*}
453 \text{e.g. Prove that}\\
454 \overrightharp{AC}=m\overrightharp{AB} \iff \boldsymbol{c}&=(1-m)\boldsymbol{a}+m\boldsymbol{b}\\
455 \implies \boldsymbol{c} &= \overrightharp{OA} + \overrightharp{AC}\\
456 &= \overrightharp{OA} + m\overrightharp{AB}\\
457 &=\boldsymbol{a}+m(\boldsymbol{b}-\boldsymbol{a})\\
458 &=\boldsymbol{a}+m\boldsymbol{b}-m\boldsymbol{a}\\
459 &=(1-m)\boldsymbol{a}+m{b}
460\end{align*}
461
462\begin{align*}
463 \text{Also, } \implies \overrightharp{OC} &= \lambda \vec{OA} + \mu \overrightharp{OB} \\
464 \text{where } \lambda + \mu &= 1\\
465 \text{If } C \text{ lies along } \overrightharp{AB}, & \implies 0 < \mu < 1
466\end{align*}
467
468
469 \subsubsection*{Useful vector properties}
470
471\begin{itemize}
472\item
473 \(\boldsymbol{a} \parallel \boldsymbol{b} \implies \boldsymbol{b}=k\boldsymbol{a}\) for some
474 \(k \in \mathbb{R} \setminus \{0\}\)
475\item
476 If \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are parallel with at
477 least one point in common, then they lie on the same straight line
478\item
479 Two vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are
480 perpendicular if \(\boldsymbol{a} \cdot \boldsymbol{b}=0\)
481\item
482 \(\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2\)
483\end{itemize}
484
485\subsection*{Linear dependence}
486
487Vectors \(\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\) are linearly
488dependent if they are non-parallel and:
489
490\[k\boldsymbol{a}+l\boldsymbol{b}+m\boldsymbol{c} = 0\]
491\[\therefore \boldsymbol{c} = m\boldsymbol{a} + n\boldsymbol{b} \quad \text{(simultaneous)}\]
492
493\(\boldsymbol{a}, \boldsymbol{b},\) and \(\boldsymbol{c}\) are linearly
494independent if no vector in the set is expressible as a linear
495combination of other vectors in set, or if they are parallel.
496
497Vector \(\boldsymbol{w}\) is a linear combination of vectors
498\(\boldsymbol{v_1}, \boldsymbol{v_2}, \boldsymbol{v_3}\)
499
500\subsection*{Three-dimensional vectors}
501
502Right-hand rule for axes: \(z\) is up or out of page.
503
504\tdplotsetmaincoords{60}{120}
505\begin{center}\begin{tikzpicture} [scale=3, tdplot_main_coords, axis/.style={->,thick},
506vector/.style={-stealth,red,very thick},
507vector guide/.style={dashed,gray,thick}]
508
509%standard tikz coordinate definition using x, y, z coords
510\coordinate (O) at (0,0,0);
511
512%tikz-3dplot coordinate definition using x, y, z coords
513
514\pgfmathsetmacro{\ax}{1}
515\pgfmathsetmacro{\ay}{1}
516\pgfmathsetmacro{\az}{1}
517
518\coordinate (P) at (\ax,\ay,\az);
519
520%draw axes
521\draw[axis] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$};
522\draw[axis] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$};
523\draw[axis] (0,0,0) -- (0,0,1) node[anchor=south]{$z$};
524
525%draw a vector from O to P
526\draw[vector] (O) -- (P);
527
528%draw guide lines to components
529\draw[vector guide] (O) -- (\ax,\ay,0);
530\draw[vector guide] (\ax,\ay,0) -- (P);
531\draw[vector guide] (P) -- (0,0,\az);
532\draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
533\draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
534\draw[vector guide] (\ax,\ay,0) -- (\ax,0,0);
535\node[tdplot_main_coords,above right]
536at (\ax,\ay,\az){(\ax, \ay, \az)};
537\end{tikzpicture}\end{center}
538
539\subsection*{Parametric vectors}
540
541Parametric equation of line through point \((x_0, y_0, z_0)\) and
542parallel to \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) is:
543
544\begin{equation}\begin{cases}x = x_o + a \cdot t \\ y = y_0 + b \cdot t \\ z = z_0 + c \cdot t\end{cases}\end{equation}
545
546\section{Circular functions}
547
548Period of \(a\sin(bx)\) is \(\frac{{2\pi}{b}\)
549
550Period of \(a\tan(nx)\) is \(\frac{\pi}{n}\)\\
551Asymptotes at \(x=\frac{2k+1)\pi}{2n} \> \vert \> k \in \mathbb{Z}\)
552
553\subsection*{Reciprocal functions}
554
555\subsubsection*{Cosecant}
556
557\begin{figure}
558\centering
559\includegraphics{graphics/csc.png}
560\caption{}
561\end{figure}
562
563\[\operatorname{cosec} \theta = \frac{1}{\sin \theta} \> \vert \> \sin \theta \ne 0\]
564
565\begin{itemize}
566\tightlist
567\item
568 \textbf{Domain} \(= \mathbb{R} \setminus {n\pi : n \in \mathbb{Z}}\)
569\item
570 \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
571\item
572 \textbf{Turning points} at
573 \(\theta = {\frac{(2n + 1)\pi}{2} \> \vert \> n \in \mathbb{Z}\)
574\item
575 \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
576\end{itemize}
577
578\subsubsection*{Secant}
579
580\begin{figure}
581\centering
582\includegraphics{graphics/sec.png}
583\caption{}
584\end{figure}
585
586\[\operatorname{sec} \theta = \frac{1}{\cos \theta} \> \vert \> \cos \theta \ne 0\]
587
588\begin{itemize}
589\tightlist
590\item
591 \textbf{Domain}
592 \(= \mathbb{R} \setminus \{{{(2n + 1) \pi} \over 2 } : n \in \mathbb{Z}\}\)
593\item
594 \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
595\item
596 \textbf{Turning points} at
597 \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
598\item
599 \textbf{Asymptotes} at
600 \(\theta = {{(2n + 1) \pi} \over 2} \> \vert \> n \in \mathbb{Z}\)
601\end{itemize}
602
603\subsubsection*{Cotangent}
604
605\begin{figure}
606\centering
607\includegraphics{graphics/cot.png}
608\caption{}
609\end{figure}
610
611\[\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0\]
612
613\begin{itemize}
614\tightlist
615\item
616 \textbf{Domain} \(= \mathbb{R} \setminus \{n \pi: n \in \mathbb{Z}\}\)
617\item
618 \textbf{Range} \(= \mathbb{R}\)
619\item
620 \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
621\end{itemize}
622
623\subsubsection*{Symmetry properties}
624
625\begin{equation}\begin{split}
626 \operatorname{sec} (\pi \pm x) & = -\operatorname{sec} x \\
627 \operatorname{sec} (-x) & = \operatorname{sec} x \\
628 \operatorname{cosec} (\pi \pm x) & = \mp \operatorname{cosec} x \\
629 \operatorname{cosec} (-x) & = - \operatorname{cosec} x \\
630 \operatorname{cot} (\pi \pm x) & = \pm \operatorname{cot} x \\
631 \operatorname{cot} (-x) & = - \operatorname{cot} x
632\end{split}\end{equation}
633
634\subsubsection*{Complementary properties}
635
636\begin{equation}\begin{split}
637 \operatorname{sec} \left({\pi \over 2} - x\right) & = \operatorname{cosec} x \\
638 \operatorname{cosec} \left({\pi \over 2} - x\right) & = \operatorname{sec} x \\
639 \operatorname{cot} \left({\pi \over 2} - x\right) & = \tan x \\
640 \tan \left({\pi \over 2} - x\right) & = \operatorname{cot} x
641\end{split}\end{equation}
642
643\subsubsection*{Pythagorean identities}
644
645\begin{equation}\begin{split}
646 1 + \operatorname{cot}^2 x & = \operatorname{cosec}^2 x, \quad \text{where } \sin x \ne 0 \\
647 1 + \tan^2 x & = \operatorname{sec}^2 x, \quad \text{where } \cos x \ne 0
648\end{split}\end{equation}
649
650\subsection*{Compound angle formulas}
651
652\[\cos(x \pm y) = \cos x + \cos y \mp \sin x \sin y\]
653\[\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y\]
654\[\tan(x \pm y) = {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}\]
655
656\subsection*{Double angle formulas}
657
658\begin{equation}\begin{split}
659 \cos 2x &= \cos^2 x - \sin^2 x \\
660 & = 1 - 2\sin^2 x \\
661 & = 2 \cos^2 x -1
662\end{split}\end{equation}
663
664\[\sin 2x = 2 \sin x \cos x\]
665
666\[\tan 2x = {{2 \tan x} \over {1 - \tan^2 x}}\]
667
668\subsection*{Inverse circular functions}
669
670Inverse functions: \(f(f^{-1}(x)) = x, \quad f(f^{-1}(x)) = x\)\\
671Must be 1:1 to find inverse (reflection in \(y=x\)
672
673Domain is restricted to make functions 1:1.
674
675\subsubsection*{\(\arcsin\)}
676
677\[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y, \quad \text{where } \sin y = x \text{ and } y \in [{-\pi \over 2}, {\pi \over 2}]\]
678
679\subsubsection*{\(\arccos\)}
680
681\[\cos^{-1} \rightarrow \mathbb{R}, \quad \cos^{-1} x = y, \quad \text{where } \cos y = x \text{ and } y \in [0, \pi]\]
682
683\subsubsection*{\(\arctan\)}
684
685\[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y, \quad \text{where } \tan y = x \text{ and } y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\]
686
687
688\section{Differential calculus}
689
690\subsection*{Limits}
691
692\[\lim_{x \rightarrow a}f(x)\]
693
694\(L^-\) - limit from below
695
696\(L^+\) - limit from above
697
698\(\lim_{x \to a} f(x)\) - limit of a point
699
700\begin{itemize}
701\item
702 Limit exists if \(L^-=L^+\)
703\item
704 If limit exists, point does not.
705\item
706 For solving \(x\rightarrow\infty\), factorise so that all \(x\) terms are in denominators\\
707 e.g. \[\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2\]
708 \item
709Limits can be solved using normal techniques (if div 0, factorise)
710\end{itemize}
711
712
713\begin{enumerate}
714\item
715 For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\)
716\item
717 \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\)
718\item
719 \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\)
720 \item
721\(\therefore \lim_{x \rightarrow a} c \times f(x)=cF\) where \(c=\) constant
722\ite
723 \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\)
724\item
725A function is continuous if \(L^-=L^+=f(x)\) for all values of \(x\).
726\end{enumerate}
727
728\subsection{Gradients of secants and tangents}
729
730\textbf{Secant (chord)} - line joining two points on curve\\
731\textbf{Tangent} - line that intersects curve at one point
732
733\(m\left(\overrightharp{PQ}\right){m_{PQ}}={\operatorname{rise} \over \operatorname{run}} = {\delta y \over \delta x} \text{ for } P(x,y),\quad Q(x+\delta x, y+ \delta y)\)
734
735As \(Q \rightarrow P, \delta x \rightarrow 0\). Chord becomes tangent
736(two infinitesimal points are equal).
737
738\subsection{First principles derivative}
739
740\[f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={\frac{dy}{dx}}\]
741
742\subsubsection*{Logarithmic identities}
743
744\(\log_b (xy)=\log_b x + \log_b y\)\\
745\(\log_b x^n = n \log_b x\)\\
746\(\log_b y^{x^n} = x^n \log_b y\)
747
748\subsubsection*{Index identities}}
749
750\(b^{m+n}=b^m \cdot b^n\)\\
751\((b^m)^n=b^{m \cdot n}\)\\
752\((b \cdot c)^n = b^n \cdot c^n\)\\
753\({a^m \div a^n} = {a^{m-n}}\)
754
755\subsubsection{\texorpdfstring{\(e\) as a
756logarithm}{e as a logarithm}}\label{e-as-a-logarithm}
757
758\[\operatorname{if} y=e^x, \quad \operatorname{then} x=\log_e y\]
759\[\ln x = \log_e x\]
760
761\subsection*{Derivative rules}
762
763\begin{longtable}[]{@{}ll@{}}
764\toprule
765\(f(x)\) & \(f^\prime(x)\)\tabularnewline
766\midrule
767\endhead
768\(\sin x\) & \(\cos x\)\tabularnewline
769\(\sin ax\) & \(a\cos ax\)\tabularnewline
770\(\cos x\) & \(-\sin x\)\tabularnewline
771\(\cos ax\) & \(-a \sin ax\)\tabularnewline
772\(\tan f(x)\) & \(f^2(x) \sec^2f(x)\)\tabularnewline
773\(e^x\) & \(e^x\)\tabularnewline
774\(e^{ax}\) & \(ae^{ax}\)\tabularnewline
775\(ax^{nx}\) & \(an \cdot e^{nx}\)\tabularnewline
776\(\log_e x\) & \(1 \over x\)\tabularnewline
777\(\log_e {ax}\) & \(1 \over x\)\tabularnewline
778\(\log_e f(x)\) & \(f^\prime (x) \over f(x)\)\tabularnewline
779\(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\tabularnewline
780\(\sin^{-1} x\) & \(1 \over {\sqrt{1-x^2}}\)\tabularnewline
781\(\cos^{-1} x\) & \(-1 \over {sqrt{1-x^2}}\)\tabularnewline
782\(\tan^{-1} x\) & \(1 \over {1 + x^2}\)\tabularnewline
783\bottomrule
784\end{longtable}
785
786\subsection*{Reciprocal derivatives}
787
788\[\frac{1}{\frac{dy}{dx}} = \frac{dx}{dy}\]
789
790\subsection*{Differentiating \(x=f(y)\)}
791
792Find \(\frac{dx}{dy}\). Then:
793
794\begin{align*}
795 {\frac{dx}{dy}} =& {1 \over {\frac{dy}{dx}}} \\
796 \implies {\frac{dy}{dx}} &= {1 \over {\frac{dx}{dy}}}\).
797
798\[{\frac{dy}{dx}} = {1 \over {\frac{dx}{dy}}}\]
799
800\subsection*{Second derivative}}
801
802\[f(x) \longrightarrow f^\prime (x) \longrightarrow f^{\prime\prime}(x)\]
803
804\[\therefore y \longrightarrow {\frac{dy}{dx}} \longrightarrow {d({\frac{dy}{dx}}) \over dx} \longrightarrow {d^2 y \over dx^2}\]
805
806Order of polynomial \(n\)th derivative decrements each time the
807derivative is taken
808
809\subsubsection*{Points of Inflection}
810
811\emph{Stationary point} - point of zero gradient (i.e.
812\(f^\prime(x)=0\))\\
813\emph{Point of inflection} - point of maximum \(|\)gradient\(|\) (i.e.
814\(f^{\prime\prime} = 0\))
815
816\begin{itemize}
817\tightlist
818\item
819 if \(f^\prime (a) = 0\) and \(f^{\prime\prime}(a) > 0\), then point
820 \((a, f(a))\) is a local min (curve is concave up)
821\item
822 if \(f^\prime (a) = 0\) and \(f^{\prime\prime} (a) < 0\), then point
823 \((a, f(a))\) is local max (curve is concave down)
824\item
825 if \(f^{\prime\prime}(a) = 0\), then point \((a, f(a))\) is a point of
826 inflection
827\item
828 if also \(f^\prime(a)=0\), then it is a stationary point of inflection
829\end{itemize}
830
831\begin{figure}
832\centering
833\includegraphics{graphics/second-derivatives.png}
834\caption{}
835\end{figure}
836
837\subsection*{Implicit Differentiation}
838
839\textbf{On CAS:} Action \(\rightarrow\) Calculation \(\rightarrow\)
840\texttt{impDiff(y\^{}2+ax=5,\ x,\ y)}. Returns \(y^\prime= \dots\).
841
842Used for differentiating circles etc.
843
844If \(p\) and \(q\) are expressions in \(x\) and \(y\) such that \(p=q\),
845for all \(x\) nd \(y\), then:
846
847\[{\frac{dp}{dx}} = {\frac{dq}{dx}} \quad \text{and} \quad {\frac{dp}{dy}} = {\frac{dq}{dy}}\]
848
849\subsection*{Integration}
850
851\[\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)\]
852
853\[\int x^n \cdot dx = {x^{n+1} \over n+1} + c\]
854
855\begin{itemize}
856\tightlist
857\item
858 area enclosed by curves
859\item
860 \(+c\) should be shown on each step without \(\int\)
861\end{itemize}
862
863\subsubsection*{Integral laws}
864
865\(\int f(x) + g(x) dx = \int f(x) dx + \int g(x) dx\)\\
866\(\int k f(x) dx = k \int f(x) dx\)
867
868\begin{longtable}[]{@{}ll@{}}
869\toprule
870\begin{minipage}[b]{0.42\columnwidth}\raggedright\strut
871\(f(x)\)\strut
872\end{minipage} & \begin{minipage}[b]{0.38\columnwidth}\raggedright\strut
873\(\int f(x) \cdot dx\)\strut
874\end{minipage}\tabularnewline
875\midrule
876\endhead
877\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut
878\(k\) (constant)\strut
879\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut
880\(kx + c\)\strut
881\end{minipage}\tabularnewline
882\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut
883\(x^n\)\strut
884\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut
885\({x^{n+1} \over {n+1}} + c\)\strut
886\end{minipage}\tabularnewline
887\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut
888\(a x^{-n}\)\strut
889\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut
890\(a \cdot \log_e x + c\)\strut
891\end{minipage}\tabularnewline
892\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut
893\({1 \over {ax+b}}\)\strut
894\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut
895\({1 \over a} \log_e (ax+b) + c\)\strut
896\end{minipage}\tabularnewline
897\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut
898\((ax+b)^n\)\strut
899\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut
900\({1 \over {a(n+1)}}(ax+b)^{n-1} + c\)\strut
901\end{minipage}\tabularnewline
902\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut
903\(e^{kx}\)\strut
904\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut
905\({1 \over k} e^{kx} + c\)\strut
906\end{minipage}\tabularnewline
907\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut
908\(e^k\)\strut
909\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut
910\(e^kx + c\)\strut
911\end{minipage}\tabularnewline
912\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut
913\(\sin kx\)\strut
914\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut
915\(-{1 \over k} \cos (kx) + c\)\strut
916\end{minipage}\tabularnewline
917\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut
918\(\cos kx\)\strut
919\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut
920\({1 \over k} \sin (kx) + c\)\strut
921\end{minipage}\tabularnewline
922\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut
923\(\sec^2 kx\)\strut
924\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut
925\({1 \over k} \tan(kx) + c\)\strut
926\end{minipage}\tabularnewline
927\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut
928\(1 \over \sqrt{a^2-x^2}\)\strut
929\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut
930\(\sin^{-1} {x \over a} + c \>\vert\> a>0\)\strut
931\end{minipage}\tabularnewline
932\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut
933\(-1 \over \sqrt{a^2-x^2}\)\strut
934\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut
935\(\cos^{-1} {x \over a} + c \>\vert\> a>0\)\strut
936\end{minipage}\tabularnewline
937\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut
938\(a \over {a^2-x^2}\)\strut
939\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut
940\(\tan^{-1} {x \over a} + c\)\strut
941\end{minipage}\tabularnewline
942\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut
943\({f^\prime (x)} \over {f(x)}\)\strut
944\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut
945\(\log_e f(x) + c\)\strut
946\end{minipage}\tabularnewline
947\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut
948\(g^\prime(x)\cdot f^\prime(g(x)\)\strut
949\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut
950\(f(g(x))\) (chain rule)\strut
951\end{minipage}\tabularnewline
952\begin{minipage}[t]{0.42\columnwidth}\raggedright\strut
953\(f(x) \cdot g(x)\)\strut
954\end{minipage} & \begin{minipage}[t]{0.38\columnwidth}\raggedright\strut
955\(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\strut
956\end{minipage}\tabularnewline
957\bottomrule
958\end{longtable}
959
960Note \(\sin^{-1} {x \over a} + \cos^{-1} {x \over a}\) is constant for
961all \(x \in (-a, a)\).
962
963\subsubsection*{Definite integrals}}
964
965\[\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)\]
966
967\begin{itemize}
968\tightlist
969\item
970 Signed area enclosed by:
971 \(\> y=f(x), \quad y=0, \quad x=a, \quad x=b\).
972\item
973 \emph{Integrand} is \(f\).
974\item
975 \(F(x)\) may be any integral, i.e. \(c\) is inconsequential
976\end{itemize}
977
978\paragraph{Properties}\label{properties}
979
980\[\int^b_a f(x) \> dx = \int^c_a f(x) \> dx + \int^b_c f(x) \> dx\]
981
982\[\int^a_a f(x) \> dx = 0\]
983
984\[\int^b_a k \cdot f(x) \> dx = k \int^b_a f(x) \> dx\]
985
986\[\int^b_a f(x) \pm g(x) \> dx = \int^b_a f(x) \> dx \pm \int^b_a g(x) \> dx\]
987
988\[\int^b_a f(x) \> dx = - \int^a_b f(x) \> dx\]
989
990\subsubsection{Integration by substitution}
991
992\[\int f(u) {\frac{du}{dx}} \cdot dx = \int f(u) \cdot du\]
993
994Note \(f(u)\) must be one-to-one \(\implies\) one \(x\) value for each
995\(y\) value
996
997e.g.~for \(y=\int(2x+1)\sqrt{x+4} \cdot dx\):\\
998let \(u=x+4\)\\
999\(\implies {\frac{du}{dx}} = 1\)\\
1000\(\implies x = u - 4\)\\
1001then \(y=\int (2(u-4)+1)u^{1 \over 2} \cdot du\)\\
1002Solve as a normal integral
1003
1004\subsubsection*{Definite integrals by substitution}
1005
1006For \(\int^b_a f(x) {\frac{du}{dx}} \cdot dx\), evaluate new \(a\) and
1007\(b\) for \(f(u) \cdot du\).
1008
1009\subsubsection{Trigonometric integration}
1010
1011\[\sin^m x \cos^n x \cdot dx\]
1012
1013\textbf{\(m\) is odd:}\\
1014\(m=2k+1\) where \(k \in \mathbb{Z}\)\\
1015\(\implies \sin^{2k+1} x = (\sin^2 z)^k \sin x = (1 - \cos^2 x)^k \sin x\)\\
1016Substitute \(u=\cos x\)
1017
1018\textbf{\(n\) is odd:}\\
1019\(n=2k+1\) where \(k \in \mathbb{Z}\)\\
1020\(\implies \cos^{2k+1} x = (\cos^2 x)^k \cos x = (1-\sin^2 x)^k \cos x\)\\
1021Subbstitute \(u=\sin x\)
1022
1023\textbf{\(m\) and \(n\) are even:}\\
1024Use identities:
1025
1026\begin{itemize}
1027\tightlist
1028\item
1029 \(\sin^2x={1 \over 2}(1-\cos 2x)\)
1030\item
1031 \(\cos^2x={1 \over 2}(1+\cos 2x)\)
1032\item
1033 \(\sin 2x = 2 \sin x \cos x\)
1034\end{itemize}
1035
1036\subsection*{Partial fractions}
1037
1038On CAS: Action \(\rightarrow\) Transformation \(\rightarrow\)
1039\texttt{expand/combine}\\
1040or Interactive \(\rightarrow\) Transformation \(\rightarrow\)
1041\texttt{expand} \(\rightarrow\) Partial
1042
1043\subsection*{Graphing integrals on CAS}
1044
1045In main: Interactive \(\rightarrow\) Calculation \(\rightarrow\)
1046\(\int\) (\(\rightarrow\) Definite)\\
1047Restrictions: \texttt{Define\ f(x)=...} \(\rightarrow\)
1048\texttt{f(x)\textbar{}x\textgreater{}1} (e.g.)
1049
1050\subsection{Applications of antidifferentiation}
1051
1052\begin{itemize}
1053\tightlist
1054\item
1055 \(x\)-intercepts of \(y=f(x)\) identify \(x\)-coordinates of
1056 stationary points on \(y=F(x)\)
1057\item
1058 nature of stationary points is determined by sign of \(y=f(x)\) on
1059 either side of its \(x\)-intercepts
1060\item
1061 if \(f(x)\) is a polynomial of degree \(n\), then \(F(x)\) has degree
1062 \(n+1\)
1063\end{itemize}
1064
1065To find stationary points of a function, substitute \(x\) value of given
1066point into derivative. Solve for \({\frac{dy}{dx}}=0\). Integrate to find
1067original function.
1068
1069\subsection*{Solids of revolution}}
1070
1071Approximate as sum of infinitesimally-thick cylinders
1072
1073\subsubsection{Rotation about \(x\)-axis}
1074
1075\begin{align*}
1076 V &= \int^{x=b}_{x-a} \pi y^2 \> dx \\
1077 &= \pi \int^b_a (f(x))^2 \> dx
1078\end{align*}
1079
1080\subsubsection{Rotation about \(y\)-axis}
1081
1082\begin{align*}
1083 V &= \int^{y=b}_{y=a} \pi x^2 \> dy \\
1084 &= \pi \int^b_a (f(y))^2 \> dy
1085\end{align*}
1086
1087\subsubsection{Regions not bound by\(y=0\)}
1088
1089\[V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx\]\\
1090where \(f(x) > g(x)\)
1091
1092\subsection*{Length of a curve}
1093
1094\[L = \int^b_a \sqrt{1 + ({\frac{dy}{dx}})^2} \> dx \quad \text{(Cartesian)}\]
1095
1096\[L = \int^b_a \sqrt{{\frac{dx}{dt}} + ({\frac{dy}{dt}})^2} \> dt \quad \text{(parametric)}\]
1097
1098Evaluate on CAS. Or use Interactive \(\rightarrow\) Calculation
1099\(\rightarrow\) Line \(\rightarrow\) \texttt{arcLen}.
1100
1101\subsection*{Rates}
1102
1103\subsubsection*{Related rates}
1104
1105\[{\frac{da}{db}} \quad \text{(change in } a \text{ with respect to } b)\]
1106
1107\subsubsection{Gradient at a point on parametric curve}
1108
1109\[{\frac{dy}{dx}} = {{\frac{dy}{dt}} \div {\frac{dx}{dt}}} \> \vert \> {\frac{dx}{dt}} \ne 0\]
1110
1111\[\frac{d^2}{dx^2} = \frac{d(y^\prime)}{dx} = {\frac{dy^\prime}{dt} \div {\frac{dx}{dt}}} \> \vert \> y^\prime = {\frac{dy}{dx}}\]
1112
1113\subsection*{Rational functions}
1114
1115\[f(x) = \frac{P(x)}{Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}\]
1116
1117\subsubsection*{Addition of ordinates}
1118
1119\begin{itemize}
1120\tightlist
1121\item
1122 when two graphs have the same ordinate, \(y\)-coordinate is double the
1123 ordinate
1124\item
1125 when two graphs have opposite ordinates, \(y\)-coordinate is 0 i.e.
1126 (\(x\)-intercept)
1127\item
1128 when one of the ordinates is 0, the resulting ordinate is equal to the
1129 other ordinate
1130\end{itemize}
1131
1132\subsection{Fundamental theorem of calculus}
1133
1134If \(f\) is continuous on \([a, b]\), then
1135
1136\[\int^b_a f(x) \> dx = F(b) - F(a)\]
1137
1138where \(F\) is any antiderivative of \(f\)
1139
1140\subsection*{Differential equations}}
1141
1142One or more derivatives
1143
1144\textbf{Order} - highest power inside derivative\\
1145\textbf{Degree} - highest power of highest derivative\\
1146e.g. \({\left(\frac{dy^2}{d^2} x\right)}^3\): order 2, degree 3
1147
1148\subsubsection*{Verifying solutions}
1149
1150Start with \(y=\dots\), and differentiate. Substitute into original
1151equation.
1152
1153\subsubsection{Function of the dependent
1154variable}\label{function-of-the-dependent-variable}
1155
1156If \({\frac{dy}{dx}}=g(y)\), then
1157\(\frac{{dx}{dy} = 1 \div {\frac{dy}{dx}} = \frac{1}{g(y)}\). Integrate
1158both sides to solve equation. Only add \(c\) on one side. Express
1159\(e^c\) as \(A\).
1160
1161\subsubsection*{Mixing problems}
1162
1163\[\left(\frac{dm}{dt}\right)_\Sigma = \left(\frac{dm}{dt}\right)_{\text{in}} - \left({\frac{dm}{dt}\)_{\text{out}}\]
1164
1165\subsubsection*{Separation of variables}
1166
1167If \({\frac{dy}{dx}}=f(x)g(y)\), then:
1168
1169\[\int f(x) \> dx = \int \frac{1}{g(y)} \> dy\]
1170
1171\subsubsection{Using definite integrals to solve DEs}
1172
1173Used for situations where solutions to \({\frac{dy}{dx}} = f(x)\) is not
1174required.
1175
1176In some cases, it may not be possible to obtain an exact solution.
1177
1178Approximate solutions can be found by numerically evaluating a definite
1179integral.
1180
1181\subsubsection{Using Euler's method to solve a differential equation}
1182
1183\[\frac{f(x+h) - f(x)}{h} \approx f^\prime (x) \quad \text{for small } h\]
1184
1185\[\implies f(x+h) \approx f(x) + hf^\prime(x)\]
1186
1187 \end{multicols}
1188\end{document}