b669b55bad715280a6045bc24a11d05ec2486039
1\documentclass[a4paper]{article}
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3\usepackage{multicol}
4\usepackage{multirow}
5\usepackage{amsmath}
6\usepackage{amssymb}
7\usepackage{harpoon}
8\usepackage{tabularx}
9\usepackage[dvipsnames, table]{xcolor}
10\usepackage{blindtext}
11\usepackage{graphicx}
12\usepackage{wrapfig}
13\usepackage{tikz}
14\usepackage{tikz-3dplot}
15\usepackage{pgfplots}
16\usetikzlibrary{calc}
17\usetikzlibrary{angles}
18\usetikzlibrary{datavisualization.formats.functions}
19\usetikzlibrary{decorations.markings}
20\usepgflibrary{arrows.meta}
21\usepackage{fancyhdr}
22\pagestyle{fancy}
23\fancyhead[LO,LE]{Year 12 Specialist}
24\fancyhead[CO,CE]{Andrew Lorimer}
25
26\usepackage{mathtools}
27\usepackage{xcolor} % used only to show the phantomed stuff
28\renewcommand\hphantom[1]{{\color[gray]{.6}#1}} % comment out!
29\setlength\fboxsep{0pt} \setlength\fboxrule{.2pt} % for the \fboxes
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34\definecolor{cas}{HTML}{e6f0fe}
35\linespread{1.5}
36\newcommand{\midarrow}{\tikz \draw[-triangle 90] (0,0) -- +(.1,0);}
37
38\begin{document}
39
40\begin{multicols}{2}
41
42 \section{Complex numbers}
43
44 \[\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\}\]
45
46 \begin{align*}
47 \text{Cartesian form: } & a+bi\\
48 \text{Polar form: } & r\operatorname{cis}\theta
49 \end{align*}
50
51 \subsection*{Operations}
52
53\definecolor{shade1}{HTML}{ffffff}
54\definecolor{shade2}{HTML}{e6f2ff}
55 \definecolor{shade3}{HTML}{cce2ff}
56 \begin{tabularx}{\columnwidth}{r|X|X}
57 & \textbf{Cartesian} & \textbf{Polar} \\
58 \hline
59 \(z_1 \pm z_2\) & \((a \pm c)(b \pm d)i\) & convert to \(a+bi\)\\
60 \hline
61 \(+k \times z\) & \multirow{2}{*}{\(ka \pm kbi\)} & \(kr\operatorname{cis} \theta\)\\
62 \cline{1-1}\cline{3-3}
63 \(-k \times z\) & & \(kr \operatorname{cis}(\theta\pm \pi)\)\\
64 \hline
65 \(z_1 \cdot z_2\) & \(ac-bd+(ad+bc)i\) & \(r_1r_2 \operatorname{cis}(\theta_1 + \theta_2)\)\\
66 \hline
67 \(z_1 \div z_2\) & \((z_1 \overline{z_2}) \div |z_2|^2\) & \(\left(\frac{r_1}{r_2}\right) \operatorname{cis}(\theta_1 - \theta_2)\)
68 \end{tabularx}
69
70 \subsubsection*{Scalar multiplication in polar form}
71
72 For \(k \in \mathbb{R}^+\):
73 \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\theta\]
74
75 \noindent For \(k \in \mathbb{R}^-\):
76 \[k\left(r \operatorname{cis}\theta\right)=kr \operatorname{cis}\left(\begin{cases}\theta - \pi & |0<\operatorname{Arg}(z)\le \pi \\ \theta + \pi & |-\pi<\operatorname{Arg}(z)\le 0\end{cases}\right)\]
77
78 \subsection*{Conjugate}
79
80 \begin{align*}
81 \overline{z} &= a \mp bi\\
82 &= r \operatorname{cis}(-\theta)
83 \end{align*}
84
85 \noindent \colorbox{cas}{On CAS: \texttt{conjg(a+bi)}}
86
87 \subsubsection*{Properties}
88
89 \begin{align*}
90 \overline{z_1 \pm z_2} &= \overline{z_1}\pm\overline{z_2}\\
91 \overline{z_1 \cdot z_2} &= \overline{z_1}\cdot\overline{z_2}\\
92 \overline{kz} &= k\overline{z} \quad | \quad k \in \mathbb{R}\\
93 z\overline{z} &= (a+bi)(a-bi)\\
94 &= a^2 + b^2\\
95 &= |z|^2
96 \end{align*}
97
98 \subsection*{Modulus}
99
100 \[|z|=|\vec{Oz}|=\sqrt{a^2 + b^2}\]
101
102 \subsubsection*{Properties}
103
104 \begin{align*}
105 |z_1z_2|&=|z_1||z_2|\\
106 \left|\frac{z_1}{z_2}\right|&=\frac{|z_1|}{|z_2|}\\
107 |z_1+z_2|&\le|z_1|+|z_2|
108 \end{align*}
109
110 \subsection*{Multiplicative inverse}
111
112 \begin{align*}
113 z^{-1}&=\frac{a-bi}{a^2+b^2}\\
114 &=\frac{\overline{z}}{|z|^2}a\\
115 &=r \operatorname{cis}(-\theta)
116 \end{align*}
117
118 \subsection*{Dividing over \(\mathbb{C}\)}
119
120 \begin{align*}
121 \frac{z_1}{z_2}&=z_1z_2^{-1}\\
122 &=\frac{z_1\overline{z_2}}{|z_2|^2}\\
123 &=\frac{(a+bi)(c-di)}{c^2+d^2}\\
124 & \qquad \text{(rationalise denominator)}
125 \end{align*}
126
127 \subsection*{Polar form}
128
129 \begin{align*}
130 z&=r\operatorname{cis}\theta\\
131 &=r(\cos \theta + i \sin \theta)
132 \end{align*}
133
134 \begin{itemize}
135 \item{\(r=|z|=\sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}\)}
136 \item{\(\theta = \operatorname{arg}(z)\) \quad \colorbox{cas}{On CAS: \texttt{arg(a+bi)}}}
137 \item{\(\operatorname{Arg}(z) \in (-\pi,\pi)\) \quad \bf{(principal argument)}}
138 \item{\colorbox{cas}{Convert on CAS:}\\ \verb|compToTrig(a+bi)| \(\iff\) \verb|cExpand{r·cisX}|}
139 \item{Multiple representations:\\\(r\operatorname{cis}\theta=r\operatorname{cis}(\theta+2n\pi)\) with \(n \in \mathbb{Z}\) revolutions}
140 \item{\(\operatorname{cis}\pi=-1,\qquad \operatorname{cis}0=1\)}
141 \end{itemize}
142
143 \subsection*{de Moivres' theorem}
144
145 \[(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}\]
146
147 \subsection*{Complex polynomials}
148
149 Include \(\pm\) for all solutions, incl. imaginary
150
151 \begin{tabularx}{\columnwidth}{ R{0.55} X }
152 \hline
153 Sum of squares & \(\begin{aligned}
154 z^2 + a^2 &= z^2-(ai)^2\\
155 &= (z+ai)(z-ai) \end{aligned}\) \\
156 \hline
157 Sum of cubes & \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)\\
158 \hline
159 Division & \(P(z)=D(z)Q(z)+R(z)\) \\
160 \hline
161 Remainder theorem & Let \(\alpha \in \mathbb{C}\). Remainder of \(P(z) \div (z-\alpha)\) is \(P(\alpha)\)\\
162 \hline
163 Factor theorem & \(z-\alpha\) is a factor of \(P(z) \iff P(\alpha)=0\) for \(\alpha \in \mathbb{C}\)\\
164 \hline
165 Conjugate root theorem & \(P(z)=0 \text{ at } z=a\pm bi\) (\(\implies\) both \(z_1\) and \(\overline{z_1}\) are solutions)
166 \end{tabularx}
167
168 \subsection*{Roots}
169
170 \(n\)th roots of \(z=r\operatorname{cis}\theta\) are:
171
172 \[z = r^{\frac{1}{n}} \operatorname{cis}\left(\frac{\theta+2k\pi}{n}\right)\]
173
174 \begin{itemize}
175
176 \item{Same modulus for all solutions}
177 \item{Arguments are separated by \(\frac{2\pi}{n}\)}
178 \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 \(\frac{2\pi}{n}\))}
179 \end{itemize}
180
181 \noindent For \(0=az^2+bz+c\), use quadratic formula:
182
183 \[z=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]
184
185 \subsection*{Fundamental theorem of algebra}
186
187 A polynomial of degree \(n\) can be factorised into \(n\) linear factors in \(\mathbb{C}\):
188
189 \[\implies P(z)=a_n(z-\alpha_1)(z-\alpha_2)(z-\alpha_3)\dots(z-\alpha_n)\]
190 \[\text{ where } \alpha_1,\alpha_2,\alpha_3,\dots,\alpha_n \in \mathbb{C}\]
191
192 \subsection*{Argand planes}
193
194 \begin{center}\begin{tikzpicture}[scale=2]
195 \draw [->] (-0.2,0) -- (1.5,0) node [right] {$\operatorname{Re}(z)$};
196 \draw [->] (0,-0.2) -- (0,1.5) node [above] {$\operatorname{Im}(z)$};
197 \coordinate (P) at (1,1);
198 \coordinate (a) at (1,0);
199 \coordinate (b) at (0,1);
200 \coordinate (O) at (0,0);
201 \draw (0,0) -- (P) node[pos=0.5, above left]{\(r\)} node[pos=1, right]{\(\begin{aligned}z&=a+bi\\&=r\operatorname{cis}\theta\end{aligned}\)};
202 \draw [gray, dashed] (1,1) -- (1,0) node[black, pos=1, below]{\(a\)};
203 \draw [gray, dashed] (1,1) -- (0,1) node[black, pos=1, left]{\(b\)};
204 \begin{scope}
205 \path[clip] (O) -- (P) -- (a);
206 \fill[red, opacity=0.5, draw=black] (O) circle (2mm);
207 \node at ($(O)+(20:3mm)$) {$\theta$};
208 \end{scope}
209 \filldraw (P) circle (0.5pt);
210 \end{tikzpicture}\end{center}
211
212 \begin{itemize}
213 \item{Multiplication by \(i \implies\) CCW rotation of \(\frac{\pi}{2}\)}
214 \item{Addition: \(z_1 + z_2 \equiv\) \overrightharp{\(Oz_1\)} + \overrightharp{\(Oz_2\)}}
215 \end{itemize}
216
217 \subsection*{Sketching complex graphs}
218
219 \subsubsection*{Linear}
220
221 \begin{itemize}
222 \item{\(\operatorname{Re}(z)=c\) or \(\operatorname{Im}(z)=c\) (perpendicular bisector)}
223 \item{\(\operatorname{Im}(z)=m\operatorname{Re}(z)\)}
224 \item{\(|z+a|=|z+b| \implies 2(a-b)x=b^2-a^2\)\\Geometric: equidistant from \(a,b\)}
225 \end{itemize}
226
227 \subsubsection*{Circles}
228
229 \begin{itemize}
230 \item \(|z-z_1|^2=c^2|z_2+2|^2\)
231 \item \(|z-(a+bi)|=c \implies (x-a)^2+_(y-b)^2=c^2\)
232 \end{itemize}
233
234 \noindent \textbf{Loci} \qquad \(\operatorname{Arg}(z)<\theta\)
235
236 \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
237 \draw [->] (0,0) -- (1,0) node [right] {$\operatorname{Re}(z)$};
238 \draw [->] (0,-0.5) -- (0,1) node [above] {$\operatorname{Im}(z)$};
239 \draw [<-, dashed, thick, blue] (-1,0) -- (0,0);
240 \draw [->, thick, blue] (0,0) -- (1,1);
241 \fill [gray, opacity=0.2, domain=-1:1, variable=\x] (-1,-0.5) -- (-1,0) -- (0, 0) -- (1,1) -- (1,-0.5) -- cycle;
242 \begin{scope}
243 \path[clip] (0,0) -- (1,1) -- (1,0);
244 \fill[red, opacity=0.5, draw=black] (0,0) circle (2mm);
245 \node at ($(0,0)+(20:3mm)$) {$\frac{\pi}{4}$};
246 \end{scope}
247 \node [font=\footnotesize] at (0.5,-0.25) {\(\operatorname{Arg}(z)\le\frac{\pi}{4}\)};
248 \node [blue, mydot] {};
249 \end{tikzpicture}\end{center}
250
251 \noindent \textbf{Rays} \qquad \(\operatorname{Arg}(z-b)=\theta\)
252
253 \begin{center}\begin{tikzpicture}[scale=2,mydot/.style={circle, fill=white, draw, outer sep=0pt, inner sep=1.5pt}]
254 \draw [->] (-0.75,0) -- (1.5,0) node [right] {$\operatorname{Re}(z)$};
255 \draw [->] (0,-1) -- (0,1) node [above] {$\operatorname{Im}(z)$};
256 \draw [->, thick, brown] (-0.25,0) -- (-0.75,-1);
257 \node [above, font=\footnotesize] at (-0.25,0) {\(\frac{1}{4}\)};
258 \begin{scope}
259 \path[clip] (-0.25,0) -- (-0.75,-1) -- (0,0);
260 \fill[orange, opacity=0.5, draw=black] (-0.25,0) circle (2mm);
261 \end{scope}
262 \node at (-0.08,-0.3) {\(\frac{\pi}{8}\)};
263 \node [font=\footnotesize, left] at (-0.75,-1) {\(\operatorname{Arg}(z+\frac{1}{4})=\frac{\pi}{8}\)};
264 \node [brown, mydot] at (-0.25,0) {};
265 \draw [<->, thick, green] (0,-1) -- (1.5,0.5) node [pos=0.25, black, font=\footnotesize, right] {\(|z-2|=|z-(1+i)|\)};
266 \node [left, font=\footnotesize] at (0,-1) {\(-1\)};
267 \node [below, font=\footnotesize] at (1,0) {\(1\)};
268 \end{tikzpicture}\end{center}
269
270 \section{Vectors}
271\begin{center}\begin{tikzpicture}
272 \draw [->] (-0.5,0) -- (3,0) node [right] {\(x\)};
273 \draw [->] (0,-0.5) -- (0,3) node [above] {\(y\)};
274 \draw [orange, ->, thick] (0.5,0.5) -- (2.5,2.5) node [pos=0.5, above] {\(\vec{u}\)};
275 \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
276 \draw [gray, dashed, thick] (0.5,0.5) -- (2.5,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, below]{\(x\vec{i}\)};
277 \draw [gray, dashed, thick] (2.5,0.5) -- (2.5,2.5) node [pos=0.5] {\midarrow};
278 \end{scope}
279 \node[black, right] at (2.5,1.5) {\(y\vec{j}\)};
280\end{tikzpicture}\end{center}
281\subsection*{Column notation}
282
283\[\begin{bmatrix}x\\ y \end{bmatrix} \iff x\boldsymbol{i} + y\boldsymbol{j}\]
284\(\begin{bmatrix}x_2-x_1\\ y_2-y_1 \end{bmatrix}\) \quad between \(A(x_1,y_1), \> B(x_2,y_2)\)
285
286\subsection*{Scalar multiplication}
287
288\[k\cdot (x\boldsymbol{i}+y\boldsymbol{j})=kx\boldsymbol{i}+ky\boldsymbol{j}\]
289
290\noindent For \(k \in \mathbb{R}^-\), direction is reversed
291
292\subsection*{Vector addition}
293\begin{center}\begin{tikzpicture}[scale=1]
294 \coordinate (A) at (0,0);
295 \coordinate (B) at (2,2);
296 \draw [->, thick, red] (0,0) -- (2,2) node [pos=0.5, below right] {\(\vec{u}=2\vec{i}+2\vec{j}\)};
297 \draw [->, thick, blue] (2,2) -- (1,4) node [pos=0.5, above right] {\(\vec{v}=-\vec{i}+2\vec{j}\)};
298 \draw [->, thick, orange] (0,0) -- (1,4) node [pos=0.5, left] {\(\vec{u}+\vec{v}=\vec{i}+4\vec{j}\)};
299\end{tikzpicture}\end{center}
300
301\[(x\boldsymbol{i}+y\boldsymbol{j}) \pm (a\boldsymbol{i}+b\boldsymbol{j})=(x \pm a)\boldsymbol{i}+(y \pm b)\boldsymbol{j}\]
302
303\begin{itemize}
304 \item Draw each vector head to tail then join lines
305 \item Addition is commutative (parallelogram)
306 \item \(\boldsymbol{u}-\boldsymbol{v}=\boldsymbol{u}+(-\boldsymbol{v}) \implies \overrightharp{AB}=\boldsymbol{b}-\boldsymbol{a}\)
307\end{itemize}
308
309\subsection*{Magnitude}
310
311\[|(x\boldsymbol{i} + y\boldsymbol{j})|=\sqrt{x^2+y^2}\]
312
313\subsection*{Parallel vectors}
314
315\[\boldsymbol{u} || \boldsymbol{v} \iff \boldsymbol{u} = k \boldsymbol{v} \text{ where } k \in \mathbb{R} \setminus \{0\}\]
316
317For parallel vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\):\\
318\[\boldsymbol{a \cdot b}=\begin{cases}
319|\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction}\\
320-|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions}
321\end{cases}\]
322%\includegraphics[width=0.2,height=\textheight]{graphics/parallelogram-vectors.jpg}
323%\includegraphics[width=1]{graphics/vector-subtraction.jpg}
324
325\subsection*{Perpendicular vectors}
326
327\[\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b} = 0\ \quad \text{(since \(\cos 90 = 0\))}\]
328
329\subsection*{Unit vector \(|\hat{\boldsymbol{a}}|=1\)}
330\[\begin{split}\hat{\boldsymbol{a}} & = {\frac{1}{|\boldsymbol{a}|}}\boldsymbol{a} \\ & = \boldsymbol{a} \cdot {|\boldsymbol{a}|}\end{split}\]
331
332 \subsection*{Scalar product \(\boldsymbol{a} \cdot \boldsymbol{b}\)}
333
334
335\begin{center}\begin{tikzpicture}[scale=2]
336 \draw [->] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{b}\)};
337 \draw [->] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{a}\)};
338 \begin{scope}
339 \path[clip] (1,0.5) -- (1,0) -- (0,0);
340 \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
341 \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
342 \end{scope}
343\end{tikzpicture}\end{center}
344\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*}
345\noindent\colorbox{cas}{On CAS: \texttt{dotP({[}a\ b\ c{]},\ {[}d\ e\ f{]})}}
346
347\subsubsection*{Properties}
348
349\begin{enumerate}
350\item
351 \(k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k\boldsymbol{b})\)
352\item
353 \(\boldsymbol{a \cdot 0}=0\)
354\item
355 \(\boldsymbol{a} \cdot (\boldsymbol{b} + \boldsymbol{c})=\boldsymbol{a} \cdot \boldsymbol{b} + \boldsymbol{a} \cdot \boldsymbol{c}\)
356\item
357 \(\boldsymbol{i \cdot i} = \boldsymbol{j \cdot j} = \boldsymbol{k \cdot k}= 1\)
358\item
359 \(\boldsymbol{a} \cdot \boldsymbol{b} = 0 \quad \implies \quad \boldsymbol{a} \perp \boldsymbol{b}\)
360\item
361 \(\boldsymbol{a \cdot a} = |\boldsymbol{a}|^2 = a^2\)
362\end{enumerate}
363
364\subsection*{Angle between vectors}
365
366\[\cos \theta = \frac{\boldsymbol{a} \cdot \boldsymbol{b}}{|\boldsymbol{a}| |\boldsymbol{b}|} = \frac{a_1 b_1 + a_2 b_2}{|\boldsymbol{a}| |\boldsymbol{b}|}\]
367
368\noindent \colorbox{cas}{On CAS:} \texttt{angle([a b c], [a b c])}
369
370(Action \(\rightarrow\) Vector \(\rightarrow\)Angle)
371
372\subsection*{Angle between vector and axis}
373
374\noindent For\(\boldsymbol{a} = a_1 \boldsymbol{i} + a_2 \boldsymbol{j} + a_3 \boldsymbol{k}\)
375which makes angles \(\alpha, \beta, \gamma\) with positive side of
376\(x, y, z\) axes:
377\[\cos \alpha = \frac{a_1}{|\boldsymbol{a}|}, \quad \cos \beta = \frac{a_2}{|\boldsymbol{a}|}, \quad \cos \gamma = \frac{a_3}{|\boldsymbol{a}|}\]
378
379\noindent \colorbox{cas}{On CAS:} \texttt{angle({[}a\ b\ c{]},\ {[}1\ 0\ 0{]})}\\for angle
380between \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) and
381\(x\)-axis
382
383\subsection*{Projections \& resolutes}
384
385\begin{tikzpicture}[scale=3]
386 \draw [->, purple] (0,0) -- (1,0.5) node [pos=0.5, above left] {\(\boldsymbol{a}\)};
387 \draw [->, orange] (0,0) -- (1,0) node [pos=0.5, below] {\(\boldsymbol{u}\)};
388 \draw [->, blue] (1,0) -- (2,0) node [pos=0.5, below] {\(\boldsymbol{b}\)};
389 \begin{scope}
390 \path[clip] (1,0.5) -- (1,0) -- (0,0);
391 \fill[orange, opacity=0.5, draw=black] (0,0) circle (2mm);
392 \node at ($(0,0)+(15:4mm)$) {\(\theta\)};
393 \end{scope}
394 \begin{scope}[very thick, every node/.style={sloped,allow upside down}]
395 \draw [gray, dashed, thick] (1,0) -- (1,0.5) node [pos=0.5] {\midarrow} node[black, pos=0.5, right, rotate=-90]{\(\boldsymbol{w}\)};
396 \end{scope}
397\draw (0,0) coordinate (O)
398 (1,0) coordinate (A)
399 (1,0.5) coordinate (B)
400 pic [draw,red,angle radius=2mm] {right angle = O--A--B};
401\end{tikzpicture}
402
403\subsubsection*{\(\parallel\boldsymbol{b}\) (vector projection/resolute)}
404
405\begin{align*}
406 \boldsymbol{u} & = \frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|^2}\boldsymbol{b} \\
407 & = \left(\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}\right)\left(\frac{\boldsymbol{b}}{|\boldsymbol{b}|}\right) \\
408 & = (\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}
409\end{align*}
410
411\subsubsection*{\(\perp\boldsymbol{b}\) (perpendicular projection)}
412\[\boldsymbol{w} = \boldsymbol{a} - \boldsymbol{u}\]
413
414\subsubsection*{\(|\boldsymbol{u}|\) (scalar resolute)}
415\begin{align*}
416 r_s &= |\boldsymbol{u}|\\
417 &= \boldsymbol{a} \cdot \hat{\boldsymbol{b}}\\
418 &=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{|\boldsymbol{b}|}
419\end{align*}
420
421\subsubsection*{Rectangular (\(\parallel,\perp\)) components}
422
423\[\boldsymbol{a}=\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}+\left(\boldsymbol{a}-\frac{\boldsymbol{a}\cdot\boldsymbol{b}}{\boldsymbol{b}\cdot\boldsymbol{b}}\boldsymbol{b}\right)\]
424
425
426\subsection*{Vector proofs}
427
428\textbf{Concurrent:} intersection of \(\ge\) 3 lines
429
430\begin{tikzpicture}
431 \draw [blue] (0,0) -- (1,1);
432 \draw [red] (1,0) -- (0,1);
433 \draw [brown] (0.4,0) -- (0.6,1);
434 \filldraw (0.5,0.5) circle (2pt);
435\end{tikzpicture}
436
437\subsubsection*{Collinear points}
438
439\(\ge\) 3 points lie on the same line
440
441\begin{tikzpicture}
442 \draw [purple] (0,0) -- (4,1);
443 \filldraw (2,0.5) circle (2pt) node [above] {\(C\)};
444 \filldraw (1,0.25) circle (2pt) node [above] {\(A\)};
445 \filldraw (3,0.75) circle (2pt) node [above] {\(B\)};
446 \coordinate (O) at (2.8,-0.2);
447 \node at (O) [below] {\(O\)};
448 \begin{scope}[->, orange, thick]
449 \draw (O) -- (2,0.5) node [pos=0.5, above, font=\footnotesize, black] {\(\boldsymbol{c}\)};
450 \draw (O) -- (1,0.25) node [pos=0.5, below, font=\footnotesize, black] {\(\boldsymbol{a}\)};
451 \draw (O) -- (3,0.75) node [pos=0.5, right, font=\footnotesize, black] {\(\boldsymbol{b}\)};
452 \end{scope}
453\end{tikzpicture}
454
455\begin{align*}
456 \text{e.g. Prove that}\\
457 \overrightharp{AC}=m\overrightharp{AB} \iff \boldsymbol{c}&=(1-m)\boldsymbol{a}+m\boldsymbol{b}\\
458 \implies \boldsymbol{c} &= \overrightharp{OA} + \overrightharp{AC}\\
459 &= \overrightharp{OA} + m\overrightharp{AB}\\
460 &=\boldsymbol{a}+m(\boldsymbol{b}-\boldsymbol{a})\\
461 &=\boldsymbol{a}+m\boldsymbol{b}-m\boldsymbol{a}\\
462 &=(1-m)\boldsymbol{a}+m{b}
463\end{align*}
464\begin{align*}
465 \text{Also, } \implies \overrightharp{OC} &= \lambda \vec{OA} + \mu \overrightharp{OB} \\
466 \text{where } \lambda + \mu &= 1\\
467 \text{If } C \text{ lies along } \overrightharp{AB}, & \implies 0 < \mu < 1
468\end{align*}
469
470
471\subsubsection*{Parallelograms}
472
473\begin{center}\begin{tikzpicture}
474 \coordinate (O) at (0,0) node [below left] {\(O\)};
475 \coordinate (A) at (4,0);
476 \coordinate (B) at (6,2);
477 \coordinate (C) at (2,2);
478 \coordinate (D) at (6,0);
479
480 \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (O)--(A) node [below left] {\(A\)};
481 \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (A)--(B) node [above right] {\(B\)};
482 \draw[postaction={decorate}, decoration={markings, mark=at position 0.6 with {\arrow{>>}}}] (B)--(C) node [above left] {\(C\)};
483 \draw[postaction={decorate}, decoration={markings,mark=at position 0.5 with {\arrow{>}}}] (C)--(O);
484
485 \draw [gray, dashed] (O) -- (B) node [pos=0.75] {\(\diagdown\diagdown\)} node [pos=0.25] {\(\diagdown\diagdown\)};
486 \draw [gray, dashed] (A) -- (C) node [pos=0.75] {\(\diagup\)} node [pos=0.25] {\(\diagup\)};
487 \begin{scope}
488 \path[clip] (C) -- (A) -- (O);
489 \fill[orange, opacity=0.5, draw=black] (0,0) circle (4mm);
490 \node at ($(0,0)+(20:8mm)$) {\(\theta\)};
491 \end{scope}
492 \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] {\(|\boldsymbol{c}|\cos\theta\)};
493 \draw pic [draw,thick,red,angle radius=2mm] {right angle=O--D--B};
494\end{tikzpicture}\end{center}
495
496\begin{itemize}
497 \item
498 Diagonals \(\overrightharp{OB}, \overrightharp{AC}\) bisect each other
499 \item
500 If diagonals are equal length, it is a rectangle
501 \item
502 \(|\overrightharp{OB}|^2 + |\overrightharp{CA}|^2 = |\overrightharp{OA}|^2 + |\overrightharp{AB}|^2 + |\overrightharp{CB}|^2 + |\overrightharp{OC}|^2\)
503 \item
504 Area \(=\boldsymbol{c} \cdot \boldsymbol{a}\)
505\end{itemize}
506
507 \subsubsection*{Useful vector properties}
508
509\begin{itemize}
510\item
511 \(\boldsymbol{a} \parallel \boldsymbol{b} \implies \boldsymbol{b}=k\boldsymbol{a}\) for some
512 \(k \in \mathbb{R} \setminus \{0\}\)
513\item
514 If \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are parallel with at
515 least one point in common, then they lie on the same straight line
516\item
517 \(\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b}=0\)
518\item
519 \(\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2\)
520\end{itemize}
521
522\subsection*{Linear dependence}
523
524\(\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\) are linearly dependent if they are \(\nparallel\) and:
525\begin{align*}
526 0&=k\boldsymbol{a}+l\boldsymbol{b}+m\boldsymbol{c}\\
527 \therefore \boldsymbol{c} &= m\boldsymbol{a} + n\boldsymbol{b} \quad \text{(simultaneous)}
528\end{align*}
529
530\noindent \(\boldsymbol{a}, \boldsymbol{b},\) and \(\boldsymbol{c}\) are linearly
531independent if no vector in the set is expressible as a linear
532combination of other vectors in set, or if they are parallel.
533
534\subsection*{Three-dimensional vectors}
535
536Right-hand rule for axes: \(z\) is up or out of page.
537
538\tdplotsetmaincoords{60}{120}
539\begin{center}\begin{tikzpicture} [scale=3, tdplot_main_coords, axis/.style={->,thick},
540vector/.style={-stealth,red,very thick},
541vector guide/.style={dashed,gray,thick}]
542
543%standard tikz coordinate definition using x, y, z coords
544\coordinate (O) at (0,0,0);
545
546%tikz-3dplot coordinate definition using x, y, z coords
547
548\pgfmathsetmacro{\ax}{1}
549\pgfmathsetmacro{\ay}{1}
550\pgfmathsetmacro{\az}{1}
551
552\coordinate (P) at (\ax,\ay,\az);
553
554%draw axes
555\draw[axis] (0,0,0) -- (1,0,0) node[anchor=north east]{$x$};
556\draw[axis] (0,0,0) -- (0,1,0) node[anchor=north west]{$y$};
557\draw[axis] (0,0,0) -- (0,0,1) node[anchor=south]{$z$};
558
559%draw a vector from O to P
560\draw[vector] (O) -- (P);
561
562%draw guide lines to components
563\draw[vector guide] (O) -- (\ax,\ay,0);
564\draw[vector guide] (\ax,\ay,0) -- (P);
565\draw[vector guide] (P) -- (0,0,\az);
566\draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
567\draw[vector guide] (\ax,\ay,0) -- (0,\ay,0);
568\draw[vector guide] (\ax,\ay,0) -- (\ax,0,0);
569\node[tdplot_main_coords,above right]
570at (\ax,\ay,\az){(\ax, \ay, \az)};
571\end{tikzpicture}\end{center}
572
573\subsection*{Parametric vectors}
574
575Parametric equation of line through point \((x_0, y_0, z_0)\) and
576parallel to \(a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}\) is:
577
578\[\begin{cases}x = x_o + a \cdot t \\ y = y_0 + b \cdot t \\ z = z_0 + c \cdot t\end{cases}\]
579
580\section{Circular functions}
581
582\(\sin(bx)\) or \(\cos(bx)\): period \(=\frac{2\pi}{b}\)
583
584\noindent \(\tan(nx)\): period \(=\frac{\pi}{n}\)\\
585\indent\indent\indent asymptotes at \(x=\frac{(2k+1)\pi}{2n} \> \vert \> k \in \mathbb{Z}\)
586
587\subsection*{Reciprocal functions}
588
589\subsubsection*{Cosecant}
590
591\[\operatorname{cosec} \theta = \frac{1}{\sin \theta} \> \vert \> \sin \theta \ne 0\]
592
593\begin{itemize}
594\item
595 \textbf{Domain} \(= \mathbb{R} \setminus {n\pi : n \in \mathbb{Z}}\)
596\item
597 \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
598\item
599 \textbf{Turning points} at
600 \(\theta = \frac{(2n + 1)\pi}{2} \> \vert \> n \in \mathbb{Z}\)
601\item
602 \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
603\end{itemize}
604
605\subsubsection*{Secant}
606
607
608\begin{center}\includegraphics[width=0.7\columnwidth]{graphics/sec.png}\end{center}
609
610\[\operatorname{sec} \theta = \frac{1}{\cos \theta} \> \vert \> \cos \theta \ne 0\]
611
612\begin{itemize}
613
614\item
615 \textbf{Domain}
616 \(= \mathbb{R} \setminus \frac{(2n + 1) \pi}{2} : n \in \mathbb{Z}\}\)
617\item
618 \textbf{Range} \(= \mathbb{R} \setminus (-1, 1)\)
619\item
620 \textbf{Turning points} at
621 \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
622\item
623 \textbf{Asymptotes} at
624 \(\theta = \frac{(2n + 1) \pi}{2} \> \vert \> n \in \mathbb{Z}\)
625\end{itemize}
626
627\subsubsection*{Cotangent}
628
629\begin{center}\includegraphics[width=0.7\columnwidth]{graphics/cot.png}\end{center}
630
631\[\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0\]
632
633\begin{itemize}
634
635\item
636 \textbf{Domain} \(= \mathbb{R} \setminus \{n \pi: n \in \mathbb{Z}\}\)
637\item
638 \textbf{Range} \(= \mathbb{R}\)
639\item
640 \textbf{Asymptotes} at \(\theta = n\pi \> \vert \> n \in \mathbb{Z}\)
641\end{itemize}
642
643\subsubsection*{Symmetry properties}
644
645\[\begin{split}
646 \operatorname{sec} (\pi \pm x) & = -\operatorname{sec} x \\
647 \operatorname{sec} (-x) & = \operatorname{sec} x \\
648 \operatorname{cosec} (\pi \pm x) & = \mp \operatorname{cosec} x \\
649 \operatorname{cosec} (-x) & = - \operatorname{cosec} x \\
650 \operatorname{cot} (\pi \pm x) & = \pm \operatorname{cot} x \\
651 \operatorname{cot} (-x) & = - \operatorname{cot} x
652\end{split}\]
653
654\subsubsection*{Complementary properties}
655
656\[\begin{split}
657 \operatorname{sec} \left({\pi \over 2} - x\right) & = \operatorname{cosec} x \\
658 \operatorname{cosec} \left({\pi \over 2} - x\right) & = \operatorname{sec} x \\
659 \operatorname{cot} \left({\pi \over 2} - x\right) & = \tan x \\
660 \tan \left({\pi \over 2} - x\right) & = \operatorname{cot} x
661\end{split}\]
662
663\subsubsection*{Pythagorean identities}
664
665\[\begin{split}
666 1 + \operatorname{cot}^2 x & = \operatorname{cosec}^2 x, \quad \text{where } \sin x \ne 0 \\
667 1 + \tan^2 x & = \operatorname{sec}^2 x, \quad \text{where } \cos x \ne 0
668\end{split}\]
669
670\subsection*{Compound angle formulas}
671
672\[\cos(x \pm y) = \cos x + \cos y \mp \sin x \sin y\]
673\[\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y\]
674\[\tan(x \pm y) = {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}\]
675
676\subsection*{Double angle formulas}
677
678\[\begin{split}
679 \cos 2x &= \cos^2 x - \sin^2 x \\
680 & = 1 - 2\sin^2 x \\
681 & = 2 \cos^2 x -1
682\end{split}\]
683
684\[\sin 2x = 2 \sin x \cos x\]
685
686\[\tan 2x = {{2 \tan x} \over {1 - \tan^2 x}}\]
687
688\subsection*{Inverse circular functions}
689
690Inverse functions: \(f(f^{-1}(x)) = x, \quad f(f^{-1}(x)) = x\)\\
691Must be 1:1 to find inverse (reflection in \(y=x\)).\\
692Domain is restricted to make functions 1:1.
693
694\[\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y\]
695\hfill where \(\sin y = x, \> y \in [{-\pi \over 2}, {\pi \over 2}]\)
696
697\[\cos^{-1}: [-1,1] \rightarrow \mathbb{R}, \quad \cos^{-1} x = y\]
698\hfill where \(\cos y = x, \> y \in [0, \pi]\)
699
700\[\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y\]
701\hfill where \(\tan y = x, \> y \in \left(-{\pi \over 2}, {\pi \over 2}\right)\)
702
703
704\section{Differential calculus}
705
706\subsection*{Limits}
707
708\[\lim_{x \rightarrow a}f(x)\]
709\(L^-,\quad L^+\) \qquad limit from below/above\\
710\(\lim_{x \to a} f(x)\) \quad limit of a point\\
711
712\noindent For solving \(x\rightarrow\infty\), put all \(x\) terms in denominators\\
713 e.g. \[\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2\]
714
715\subsubsection*{Limit theorems}
716
717\begin{enumerate}
718\item
719 For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\)
720\item
721 \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\)
722\item
723 \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\)
724 \item
725\(\therefore \lim_{x \rightarrow a} c \times f(x)=cF\) where \(c=\) constant
726\item
727 \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\)
728\item
729 \(f(x)\) is continuous \(\iff L^-=L^+=f(x) \> \forall x\)
730\end{enumerate}
731
732\subsection*{Gradients of secants and tangents}
733
734\textbf{Secant (chord)} - line joining two points on curve\\
735\textbf{Tangent} - line that intersects curve at one point
736
737\subsection*{First principles derivative}
738
739\[f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={\frac{dy}{dx}}\]
740
741\subsubsection*{Logarithmic identities}
742
743\(\log_b (xy)=\log_b x + \log_b y\)\\
744\(\log_b x^n = n \log_b x\)\\
745\(\log_b y^{x^n} = x^n \log_b y\)
746
747\subsubsection*{Index identities}
748
749\(b^{m+n}=b^m \cdot b^n\)\\
750\((b^m)^n=b^{m \cdot n}\)\\
751\((b \cdot c)^n = b^n \cdot c^n\)\\
752\({a^m \div a^n} = {a^{m-n}}\)
753
754\subsection*{Derivative rules}
755
756\renewcommand{\arraystretch}{1.4}
757\begin{tabularx}{\columnwidth}{rX}
758 \hline
759\(f(x)\) & \(f^\prime(x)\)\\
760\hline
761\(\sin x\) & \(\cos x\)\\
762\(\sin ax\) & \(a\cos ax\)\\
763\(\cos x\) & \(-\sin x\)\\
764\(\cos ax\) & \(-a \sin ax\)\\
765\(\tan f(x)\) & \(f^2(x) \sec^2f(x)\)\\
766\(e^x\) & \(e^x\)\\
767\(e^{ax}\) & \(ae^{ax}\)\\
768\(ax^{nx}\) & \(an \cdot e^{nx}\)\\
769 \(\log_e x\) & \(\dfrac{1}{x}\)\\
770 \(\log_e {ax}\) & \(\dfrac{1}{x}\)\\
771 \(\log_e f(x)\) & \(\dfrac{f^\prime (x)}{f(x)}\)\\
772\(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\\
773 \(\sin^{-1} x\) & \(\dfrac{1}{\sqrt{1-x^2}}\)\\
774 \(\cos^{-1} x\) & \(\dfrac{-1}{sqrt{1-x^2}}\)\\
775 \(\tan^{-1} x\) & \(\dfrac{1}{1 + x^2}\)\\
776 \hline
777\end{tabularx}
778
779\subsection*{Reciprocal derivatives}
780
781\[\frac{1}{\frac{dy}{dx}} = \frac{dx}{dy}\]
782
783\subsection*{Differentiating \(x=f(y)\)}
784\begin{align*}
785 \text{Find }& \frac{dx}{dy}\\
786 \text{Then, }\frac{dx}{dy} &= \frac{1}{\frac{dy}{dx}} \\
787 \implies {\frac{dy}{dx}} &= \frac{1}{\frac{dx}{dy}}\\
788 \therefore {\frac{dy}{dx}} &= \frac{1}{\frac{dx}{dy}}
789\end{align*}
790
791\subsection*{Second derivative}
792\begin{align*}f(x) \longrightarrow &f^\prime (x) \longrightarrow f^{\prime\prime}(x)\\
793\implies y \longrightarrow &\frac{dy}{dx} \longrightarrow \frac{d^2 y}{dx^2}\end{align*}
794
795\noindent Order of polynomial \(n\)th derivative decrements each time the derivative is taken
796
797\subsubsection*{Points of Inflection}
798
799\emph{Stationary point} - i.e.
800\(f^\prime(x)=0\)\\
801\emph{Point of inflection} - max \(|\)gradient\(|\) (i.e.
802\(f^{\prime\prime} = 0\))
803%\begin{table*}[ht]
804%\centering
805% \begin{tabularx}{\textwidth}{XXXX}
806%\hline
807% \rowcolor{shade2}
808% & \(\dfrac{d^2 y}{dx^2} > 0\) & \(\dfrac{d^2y}{dx^2}<0\) & \(\dfrac{d^2y}{dx^2}=0\) (inflection) \\
809%\hline
810% \(\frac{dy}{dx}>0\) & \begin{tikzpicture} \draw[domain=1:2,smooth,variable=\x,blue] plot ({\x},{(1/10)*\x*\x*\x}) plot ({\x},{0.675*\x-0.677}); \end{tikzpicture} & cell 3\\
811%cell 1 & cell 2 & cell 3\\
812%\hline
813%\end{tabularx}
814%\end{table*}
815\begin{itemize}
816
817\item
818 if \(f^\prime (a) = 0\) and \(f^{\prime\prime}(a) > 0\), then point
819 \((a, f(a))\) is a local min (curve is concave up)
820\item
821 if \(f^\prime (a) = 0\) and \(f^{\prime\prime} (a) < 0\), then point
822 \((a, f(a))\) is local max (curve is concave down)
823\item
824 if \(f^{\prime\prime}(a) = 0\), then point \((a, f(a))\) is a point of
825 inflection
826\item
827 if also \(f^\prime(a)=0\), then it is a stationary point of inflection
828\end{itemize}
829
830\begin{table*}[ht]
831 \centering
832 \includegraphics[width=0.7\textwidth]{graphics/second-derivatives.png}
833\end{table*}
834
835\subsection*{Implicit Differentiation}
836
837\noindent Used for differentiating circles etc.
838
839If \(p\) and \(q\) are expressions in \(x\) and \(y\) such that \(p=q\),
840for all \(x\) and \(y\), then:
841
842\[{\frac{dp}{dx}} = {\frac{dq}{dx}} \quad \text{and} \quad {\frac{dp}{dy}} = {\frac{dq}{dy}}\]
843
844\noindent \colorbox{cas}{\textbf{On CAS:}}\\
845Action \(\rightarrow\) Calculation \(\rightarrow\) \texttt{impDiff(y\^{}2+ax=5,\ x,\ y)}\\
846Returns \(y^\prime= \dots\).
847
848\subsection*{Integration}
849
850\[\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)\]
851
852\subsection*{Integral laws}
853
854\renewcommand{\arraystretch}{1.4}
855\begin{tabularx}{\columnwidth}{rX}
856\hline
857 \(f(x)\) & \(\int f(x) \cdot dx\) \\
858 \hline
859 \(k\) (constant) & \(kx + c\)\\
860 \(x^n\) & \(\dfrac{1}{n+1} x^{n+1}\) \\
861 \(a x^{-n}\) &\(a \cdot \log_e x + c\)\\
862 \(\dfrac{1}{ax+b}\) &\(\dfrac{1}{a} \log_e (ax+b) + c\)\\
863 \((ax+b)^n\) & \(\dfrac{1}{a(n+1)}(ax+b)^{n-1} + c\)\\
864 \(e^{kx}\) & \(\dfrac{1}{k} e^{kx} + c\)\\
865 \(e^k\) & \(e^kx + c\)\\
866 \(\sin kx\) & \(\dfrac{-1}{k} \cos (kx) + c\)\\
867 \(\cos kx\) & \(\frac{1}{k} \sin (kx) + c\)\\
868 \(\sec^2 kx\) & \(\frac{1}{k} \tan(kx) + c\)\\
869 \(\dfrac{1}{\sqrt{a^2-x^2}}\) & \(\sin^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
870 \(\dfrac{-1}{\sqrt{a^2-x^2}}\) & \(\cos^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
871 \(\frac{a}{a^2-x^2}\) & \(\tan^{-1} \frac{x}{a} + c\)\\
872 \(\frac{f^\prime (x)}{f(x)}\) & \(\log_e f(x) + c\)\\
873 \(g^\prime(x)\cdot f^\prime(g(x)\) & \(f(g(x))\) (chain rule)\\
874 \(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\
875 \hline
876\end{tabularx}
877
878Note \(\sin^{-1} {x \over a} + \cos^{-1} {x \over a}\) is constant \(\forall x \in (-a, a)\)
879
880\subsection*{Definite integrals}
881
882\[\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)\]
883
884\begin{itemize}
885
886\item
887 Signed area enclosed by\\
888 \(\> y=f(x), \quad y=0, \quad x=a, \quad x=b\).
889\item
890 \emph{Integrand} is \(f\).
891\end{itemize}
892
893\subsubsection*{Properties}
894
895\[\int^b_a f(x) \> dx = \int^c_a f(x) \> dx + \int^b_c f(x) \> dx\]
896
897\[\int^a_a f(x) \> dx = 0\]
898
899\[\int^b_a k \cdot f(x) \> dx = k \int^b_a f(x) \> dx\]
900
901\[\int^b_a f(x) \pm g(x) \> dx = \int^b_a f(x) \> dx \pm \int^b_a g(x) \> dx\]
902
903\[\int^b_a f(x) \> dx = - \int^a_b f(x) \> dx\]
904
905\subsection*{Integration by substitution}
906
907\[\int f(u) {\frac{du}{dx}} \cdot dx = \int f(u) \cdot du\]
908
909\noindent Note \(f(u)\) must be 1:1 \(\implies\) one \(x\) for each \(y\)
910\begin{align*}\text{e.g. for } y&=\int(2x+1)\sqrt{x+4} \cdot dx\\
911 \text{let } u&=x+4\\
912 \implies& {\frac{du}{dx}} = 1\\
913 \implies& x = u - 4\\
914 \text{then } &y=\int (2(u-4)+1)u^{\frac{1}{2}} \cdot du\\
915 &\text{(solve as normal integral)}
916\end{align*}
917
918\subsubsection*{Definite integrals by substitution}
919
920For \(\int^b_a f(x) {\frac{du}{dx}} \cdot dx\), evaluate new \(a\) and
921\(b\) for \(f(u) \cdot du\).
922
923\subsubsection*{Trigonometric integration}
924
925\[\sin^m x \cos^n x \cdot dx\]
926
927\paragraph{\textbf{\(m\) is odd:}}
928\(m=2k+1\) where \(k \in \mathbb{Z}\)\\
929\(\implies \sin^{2k+1} x = (\sin^2 z)^k \sin x = (1 - \cos^2 x)^k \sin x\)\\
930Substitute \(u=\cos x\)
931
932\paragraph{\textbf{\(n\) is odd:}}
933\(n=2k+1\) where \(k \in \mathbb{Z}\)\\
934\(\implies \cos^{2k+1} x = (\cos^2 x)^k \cos x = (1-\sin^2 x)^k \cos x\)\\
935Substitute \(u=\sin x\)
936
937\paragraph{\textbf{\(m\) and \(n\) are even:}}
938use identities...
939
940\begin{itemize}
941
942\item
943 \(\sin^2x={1 \over 2}(1-\cos 2x)\)
944\item
945 \(\cos^2x={1 \over 2}(1+\cos 2x)\)
946\item
947 \(\sin 2x = 2 \sin x \cos x\)
948\end{itemize}
949
950\subsection*{Partial fractions}
951
952\colorbox{cas}{On CAS:}\\
953\indent Action \(\rightarrow\) Transformation \(\rightarrow\)
954\texttt{expand/combine}\\
955\indent Interactive \(\rightarrow\) Transformation \(\rightarrow\)
956Expand \(\rightarrow\) Partial
957
958\subsection*{Graphing integrals on CAS}
959
960\colorbox{cas}{In main:} Interactive \(\rightarrow\) Calculation \(\rightarrow\)
961\(\int\) (\(\rightarrow\) Definite)\\
962Restrictions: \texttt{Define\ f(x)=..} then \texttt{f(x)\textbar{}x\textgreater{}..}
963
964\subsection*{Applications of antidifferentiation}
965
966\begin{itemize}
967
968\item
969 \(x\)-intercepts of \(y=f(x)\) identify \(x\)-coordinates of
970 stationary points on \(y=F(x)\)
971\item
972 nature of stationary points is determined by sign of \(y=f(x)\) on
973 either side of its \(x\)-intercepts
974\item
975 if \(f(x)\) is a polynomial of degree \(n\), then \(F(x)\) has degree
976 \(n+1\)
977\end{itemize}
978
979To find stationary points of a function, substitute \(x\) value of given
980point into derivative. Solve for \({\frac{dy}{dx}}=0\). Integrate to find
981original function.
982
983\subsection*{Solids of revolution}
984
985Approximate as sum of infinitesimally-thick cylinders
986
987\subsubsection*{Rotation about \(x\)-axis}
988
989\begin{align*}
990 V &= \int^{x=b}_{x-a} \pi y^2 \> dx \\
991 &= \pi \int^b_a (f(x))^2 \> dx
992\end{align*}
993
994\subsubsection*{Rotation about \(y\)-axis}
995
996\begin{align*}
997 V &= \int^{y=b}_{y=a} \pi x^2 \> dy \\
998 &= \pi \int^b_a (f(y))^2 \> dy
999\end{align*}
1000
1001\subsubsection*{Regions not bound by \(y=0\)}
1002
1003\[V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx\]
1004\hfill where \(f(x) > g(x)\)
1005
1006\subsection*{Length of a curve}
1007
1008\[L = \int^b_a \sqrt{1 + ({\frac{dy}{dx}})^2} \> dx \quad \text{(Cartesian)}\]
1009
1010\[L = \int^b_a \sqrt{{\frac{dx}{dt}} + ({\frac{dy}{dt}})^2} \> dt \quad \text{(parametric)}\]
1011
1012\noindent \colorbox{cas}{On CAS:}\\
1013\indent Evaluate formula,\\
1014\indent or Interactive \(\rightarrow\) Calculation
1015\(\rightarrow\) Line \(\rightarrow\) \texttt{arcLen}
1016
1017\subsection*{Rates}
1018
1019\subsubsection*{Gradient at a point on parametric curve}
1020
1021\[{\frac{dy}{dx}} = {{\frac{dy}{dt}} \div {\frac{dx}{dt}}} \> \vert \> {\frac{dx}{dt}} \ne 0\]
1022
1023\[\frac{d^2}{dx^2} = \frac{d(y^\prime)}{dx} = {\frac{dy^\prime}{dt} \div {\frac{dx}{dt}}} \> \vert \> y^\prime = {\frac{dy}{dx}}\]
1024
1025\subsection*{Rational functions}
1026
1027\[f(x) = \frac{P(x)}{Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}\]
1028
1029\subsubsection*{Addition of ordinates}
1030
1031\begin{itemize}
1032
1033\item
1034 when two graphs have the same ordinate, \(y\)-coordinate is double the
1035 ordinate
1036\item
1037 when two graphs have opposite ordinates, \(y\)-coordinate is 0 i.e.
1038 (\(x\)-intercept)
1039\item
1040 when one of the ordinates is 0, the resulting ordinate is equal to the
1041 other ordinate
1042\end{itemize}
1043
1044\subsection*{Fundamental theorem of calculus}
1045
1046If \(f\) is continuous on \([a, b]\), then
1047
1048\[\int^b_a f(x) \> dx = F(b) - F(a)\]
1049\hfill where \(F = \int f \> dx\)
1050
1051\subsection*{Differential equations}
1052
1053\noindent\textbf{Order} - highest power inside derivative\\
1054\textbf{Degree} - highest power of highest derivative\\
1055e.g. \({\left(\dfrac{dy^2}{d^2} x\right)}^3\) \qquad order 2, degree 3
1056
1057\subsubsection*{Verifying solutions}
1058
1059Start with \(y=\dots\), and differentiate. Substitute into original
1060equation.
1061
1062\subsubsection*{Function of the dependent
1063variable}
1064
1065If \({\frac{dy}{dx}}=g(y)\), then
1066\(\frac{dx}{dy} = 1 \div \frac{dy}{dx} = \frac{1}{g(y)}\). Integrate both sides to solve equation. Only add \(c\) on one side. Express
1067\(e^c\) as \(A\).
1068
1069\subsubsection*{Mixing problems}
1070
1071\[\left(\frac{dm}{dt}\right)_\Sigma = \left(\frac{dm}{dt}\right)_{\text{in}} - \left(\frac{dm}{dt}_{\text{out}}\right)\]
1072
1073\subsubsection*{Separation of variables}
1074
1075If \({\frac{dy}{dx}}=f(x)g(y)\), then:
1076
1077\[\int f(x) \> dx = \int \frac{1}{g(y)} \> dy\]
1078
1079\subsubsection*{Euler's method for solving DEs}
1080
1081\[\frac{f(x+h) - f(x)}{h} \approx f^\prime (x) \quad \text{for small } h\]
1082
1083\[\implies f(x+h) \approx f(x) + hf^\prime(x)\]
1084
1085 \end{multicols}
1086\end{document}