spec / spec-collated.mdon commit [spec] clean up (0db1737)
   1---
   2header-includes:
   3- \usepackage{harpoon}
   4- \usepackage{amsmath}
   5- \pagenumbering{gobble}
   6---
   7
   8# Complex & Imaginary Numbers
   9
  10## Imaginary numbers
  11
  12$$i^2 = -1 \quad \therefore i = \sqrt {-1}$$
  13
  14### Simplifying negative surds
  15
  16\begin{equation}\begin{split}\sqrt{-2} & = \sqrt{-1 \times 2} \\ & = \sqrt{2}i\end{split}\end{equation}
  17
  18
  19## Complex numbers
  20
  21$$\mathbb{C} = \{a+bi : a, b \in \mathbb{R} \}$$
  22
  23General form: $z=a+bi$  
  24$\operatorname{Re}(z) = a, \quad \operatorname{Im}(z) = b$
  25
  26### Addition
  27
  28If $z_1 = a+bi$ and $z_2=c+di$, then
  29
  30$$z_1+z_2 = (a+c)+(b+d)i$$
  31
  32### Subtraction
  33
  34If $z_1=a+bi$ and $z_2=c+di$, then
  35
  36$$z_1−z_2=(a−c)+(b−d)i$$
  37
  38### Multiplication by a real constant
  39
  40If $z=a+bi$ and $k \in \mathbb{R}$, then
  41
  42$$kz=ka+kbi$$
  43
  44### Powers of $i$
  45
  46- $i^{4n} = 1$
  47- $i^{4n+1} = i$
  48- $i^{4n+2} = -1$
  49- $i^{4n+3} = -i$
  50
  51For $i^n$, find remainder $r$ when $n \div 4$. Then $i^n = i^r$.
  52
  53### Multiplying complex expressions
  54
  55If $z_1 = a+bi$ and $z_2=c+di$, then
  56
  57$$z_1 \times z_2 = (ac-bd)+(ad+bc)i$$
  58
  59### Conjugates
  60
  61$$\overline{z} = a \mp bi$$
  62
  63##### Properties
  64
  65- $\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}$
  66- $\overline{z_1 z_2} = \overline{z_1} \cdot \overline{z_2}$
  67- $\overline{kz} = k \overline{z}, \text{ for } k \in \mathbb{R}$
  68- $z \overline{z} = = (a+bi)(a-bi) = a^2+b^2 = |z|^2$
  69- $z + \overline{z} = 2 \operatorname{Re}(z)$
  70
  71### Modulus
  72
  73Distance from origin.
  74
  75$$|{z}|=\sqrt{a^2+b^2} \quad  \therefore z \overline{z} = |z|^2$$
  76
  77###### Properties
  78
  79- $|z_1 z_2| = |z_1| |z_2|$
  80- $|{z_1 \over z_2}| = {|z_1| \over |z_2|}$
  81- $|z_1 + z_2| \le |z_1 + |z_2|$
  82
  83### Multiplicative inverse
  84
  85\begin{equation}\begin{split}z^{-1} & = {1 \over z} \\ & = {{a-bi} \over {a^2+B^2}} \\ & = {\overline{z} \over {|z|^2}}\end{split}\end{equation}
  86
  87### Dividing complex numbers
  88
  89$${{z_1} \over {z_2}} = {{z_1\ {z_2}^{-1}}} = {{z_1 \overline{z_2}} \over {{|z_2|}^2}} \quad \text{(multiplicative inverse)}$$
  90
  91In practice, rationalise denominator:
  92
  93$${z_1 \over z_2} = {{(a+bi)(c-di)} \over {c^2+d^2}}$$
  94
  95## Argand planes
  96
  97- Geometric representation of $\mathbb{C}$
  98- horizontal $= \operatorname{Re}(z)$; vertical $= \operatorname{Im}(z)$
  99- Multiplication by $i$ results in an anticlockwise rotation of $\pi \over 2$
 100
 101\vfil \break
 102
 103## Complex polynomials
 104
 105**Include $\pm$ for all solutions, including imaginary**
 106
 107### Sum of two squares (quadratics)
 108
 109$$z^2+a^2=z^2-(ai)^2=(z+ai)(z-ai)$$
 110
 111Complete the square to get to this point.
 112
 113#### Dividing complex polynomials
 114
 115$P(z) \div D(z)$ gives quotient $Q(z)$ and remainder $R(z)$:
 116
 117$$P(z) = D(z)Q(z) + R(z)$$
 118
 119#### Remainder theorem
 120
 121Let $\alpha \in \mathbb{C}$. Remainder of $P(z) \div (z - \alpha)$ is $P(\alpha)$
 122
 123#### Factor theorem
 124
 125If $a+bi$ is a solution to $P(z)=0$, then:
 126
 127- $P(a+bi)=0$
 128- $z-(a+bi)$ is a factor of $P(z)$
 129
 130#### Sum of two cubes
 131
 132$$a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$$
 133
 134## Conjugate root theorem
 135
 136If $a+bi$ is a solution to $P(z)=0$, then the conjugate $\overline{z}=a-bi$ is also a solution.
 137
 138## Polar form
 139
 140\begin{equation}\begin{split}z & =r \operatorname{cis} \theta \\ & = r(\operatorname{cos}\theta+i \operatorname{sin}\theta) \\ & = a + bi \end{split}\end{equation}
 141
 142- $r=|z|=\sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}$
 143- $\theta=\operatorname{arg}(z)$ (on CAS: `arg(a+bi)`)
 144- **principal argument** is $\operatorname{Arg}(z) \in (-\pi, \pi]$ (note capital $\operatorname{Arg}$)
 145
 146Each complex number has multiple polar representations:  
 147$z=r \operatorname{cis} \theta = r \operatorname{cis} (\theta+2 n\pi$) with $n \in \mathbb{Z}$ revolutions
 148
 149### Conjugate in polar form
 150
 151$$(r \operatorname{cis} \theta)^{-1} = r\operatorname{cis} (- \theta)$$
 152
 153Reflection of $z$ across horizontal axis.
 154
 155### Multiplication and division in polar form
 156
 157$$z_1z_2=r_1r_2\operatorname{cis}(\theta_1+\theta_2)$$
 158
 159$${z_1 \over z_2} = {r_1 \over r_2} \operatorname{cis}(\theta_1-\theta_2)$$
 160
 161## de Moivres' Theorem
 162
 163$$(r\operatorname{cis}\theta)^n=r^n\operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}$$
 164
 165## Roots of complex numbers
 166
 167$n$th roots of $z = r \operatorname{cis} \theta$ are
 168
 169$$z={r^{1 \over n}} \operatorname{cis}({{\theta + 2 k \pi} \over n})$$
 170
 171Same modulus for all solutions. Arguments are separated by ${2 \pi} \over n$
 172
 173The solutions of $z^n=a \text{ where } a \in \mathbb{C}$ lie on circle
 174
 175$$x^2 + y^2 = (|a|^{1 \over n})^2$$
 176
 177## Sketching complex graphs
 178
 179### Straight line
 180
 181- $\operatorname{Re}(z) = c$ or $\operatorname{Im}(z) = c$ (perpendicular bisector)
 182- $\operatorname{Arg}(z) = \theta$
 183- $|z+a|=|z+bi|$ where $m={a \over b}$
 184- $|z+a|=|z+b| \longrightarrow 2(a-b)x=b^2-a^2$
 185
 186### Circle
 187
 188$|z-z_1|^2 = c^2 |z_2+2|^2$ or $|z-(a + bi)| = c$
 189
 190### Locus
 191
 192$\operatorname{Arg}(z) < \theta$
 193
 194# Vectors
 195
 196- **vector:** a directed line segment  
 197- arrow indicates direction
 198- length indicates magnitude
 199- notated as $\vec{a}, \widetilde{A}, \overrightharp{a}$
 200- column notation: $\begin{bmatrix}
 201       x \\ y
 202     \end{bmatrix}$
 203- vectors with equal magnitude and direction are equivalent
 204
 205
 206![](graphics/vectors-intro.png){#id .class width=20%} 
 207
 208## Vector addition
 209
 210$\boldsymbol{u} + \boldsymbol{v}$ can be represented by drawing each vector head to tail then joining the lines.  
 211Addition is commutative (parallelogram)
 212
 213## Scalar multiplication
 214
 215For $k \in \mathbb{R}^+$, $k\boldsymbol{u}$ has the same direction as $\boldsymbol{u}$ but length is multiplied by a factor of $k$.
 216
 217When multiplied by $k < 0$, direction is reversed and length is multplied by $k$.
 218
 219## Vector subtraction
 220
 221To find $\boldsymbol{u} - \boldsymbol{v}$, add $\boldsymbol{-v}$ to $\boldsymbol{u}$
 222
 223## Parallel vectors
 224
 225Same or opposite direction
 226
 227$$\boldsymbol{u} || \boldsymbol{v} \iff \boldsymbol{u} = k \boldsymbol{v} \text{ where } k \in \mathbb{R} \setminus \{0\}$$ 
 228
 229## Position vectors
 230
 231Vectors may describe a position relative to $O$.
 232
 233For a point $A$, the position vector is $\overrightharp{OA}$
 234
 235\vfill\eject
 236
 237## Linear combinations of non-parallel vectors
 238
 239If two non-zero vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ are not parallel, then:
 240
 241$$m\boldsymbol{a} + n\boldsymbol{b} = p \boldsymbol{a} + q \boldsymbol{b}\quad \therefore \quad m = p, \> n = q$$
 242
 243![](graphics/parallelogram-vectors.jpg){#id .class width=20%}
 244![](graphics/vector-subtraction.jpg){#id .class width=10%}
 245
 246## Column vector notation
 247
 248A vector between points $A(x_1,y_1), \> B(x_2,y_2)$ can be represented as $\begin{bmatrix}x_2-x_1\\ y_2-y_1 \end{bmatrix}$
 249
 250## Component notation
 251
 252A vector $\boldsymbol{u} = \begin{bmatrix}x\\ y \end{bmatrix}$ can be written as $\boldsymbol{u} = x\boldsymbol{i} + y\boldsymbol{j}$.  
 253$\boldsymbol{u}$ is the sum of two components $x\boldsymbol{i}$ and $y\boldsymbol{j}$  
 254Magnitude of vector $\boldsymbol{u} = x\boldsymbol{i} + y\boldsymbol{j}$ is denoted by $|u|=\sqrt{x^2+y^2}$
 255
 256Basic algebra applies:  
 257$(x\boldsymbol{i} + y\boldsymbol{j}) + (m\boldsymbol{i} + n\boldsymbol{j}) = (x + m)\boldsymbol{i} + (y+n)\boldsymbol{j}$  
 258Two vectors equal if and only if their components are equal.
 259
 260## Unit vector $|\hat{\boldsymbol{a}}|=1$
 261
 262\begin{equation}\begin{split}\hat{\boldsymbol{a}} & = {1 \over {|\boldsymbol{a}|}}\boldsymbol{a} \\ & = \boldsymbol{a} \cdot {|\boldsymbol{a}|}\end{split}\end{equation}
 263
 264## Scalar/dot product $\boldsymbol{a} \cdot \boldsymbol{b}$
 265
 266$$\boldsymbol{a} \cdot \boldsymbol{b} = a_1 b_1 + a_2 b_2$$
 267
 268**on CAS:** `dotP([a b c], [d e f])`
 269
 270## Scalar product properties
 271
 2721. $k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k{b})$
 2732. $\boldsymbol{a \cdot 0}=0$
 2743. $\boldsymbol{a \cdot (b + c)}=\boldsymbol{a \cdot b + a \cdot c}$
 2754. $\boldsymbol{i \cdot i} = \boldsymbol{j \cdot j} = \boldsymbol{k \cdot k}= 1$
 2765. If $\boldsymbol{a} \cdot \boldsymbol{b} = 0$, $\boldsymbol{a}$ and $\boldsymbol{b}$ are perpendicular
 2776. $\boldsymbol{a \cdot a} = |\boldsymbol{a}|^2 = a^2$
 278
 279For parallel vectors $\boldsymbol{a}$ and $\boldsymbol{b}$:  
 280$$\boldsymbol{a \cdot b}=\begin{cases}
 281|\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction}\\
 282-|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions}
 283\end{cases}$$
 284
 285## Geometric scalar products
 286
 287$$\boldsymbol{a} \cdot \boldsymbol{b} = |\boldsymbol{a}| |\boldsymbol{b}| \cos \theta$$
 288
 289where $0 \le \theta \le \pi$
 290
 291## Perpendicular vectors
 292
 293If $\boldsymbol{a} \cdot \boldsymbol{b} = 0$, then $\boldsymbol{a} \perp \boldsymbol{b}$ (since $\cos 90 = 0$)
 294
 295## Finding angle between vectors
 296
 297**positive direction**
 298
 299$$\cos \theta = {{\boldsymbol{a} \cdot \boldsymbol{b}} \over {|\boldsymbol{a}| |\boldsymbol{b}|}} = {{a_1 b_1 + a_2 b_2} \over {|\boldsymbol{a}| |\boldsymbol{b}|}}$$
 300
 301**on CAS:** `angle([a b c], [a b c])` (Action -> Vector -> Angle)
 302
 303## Angle between vector and axis
 304
 305Direction of a vector can be given by the angles it makes with $\boldsymbol{i}, \boldsymbol{j}, \boldsymbol{k}$ directions.
 306
 307For $\boldsymbol{a} = a_1 \boldsymbol{i} + a_2 \boldsymbol{j} + a_3 \boldsymbol{k}$ which makes angles $\alpha, \beta, \gamma$ with positive direction of $x, y, z$ axes:
 308$$\cos \alpha = {a_1 \over |\boldsymbol{a}|}, \quad \cos \beta = {a_2 \over |\boldsymbol{a}|}, \quad \cos \gamma = {a_3 \over |\boldsymbol{a}|}$$
 309
 310**on CAS:** `angle([a b c], [1 0 0])` for angle between $a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}$ and $x$-axis
 311
 312## Vector projections
 313
 314Vector resolute of $\boldsymbol{a}$ in direction of $\boldsymbol{b}$ is magnitude of $\boldsymbol{a}$ in direction of $\boldsymbol{b}$:
 315
 316$$\boldsymbol{u}={{\boldsymbol{a}\cdot\boldsymbol{b}}\over |\boldsymbol{b}|^2}\boldsymbol{b}=\left({\boldsymbol{a}\cdot{\boldsymbol{b} \over |\boldsymbol{b}|}}\right)\left({\boldsymbol{b} \over |\boldsymbol{b}|}\right)=(\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}$$
 317
 318## Scalar resolute of $\boldsymbol{a}$ on $\boldsymbol{b}$
 319
 320$$r_s = |\boldsymbol{u}| = \boldsymbol{a} \cdot \hat{\boldsymbol{b}}$$
 321
 322## Vector resolute of $\boldsymbol{a} \perp \boldsymbol{b}$
 323
 324$$\boldsymbol{w} = \boldsymbol{a} - \boldsymbol{u} \> \text{ where } \boldsymbol{u} \text{ is projection } \boldsymbol{a} \text{ on } \boldsymbol{b}$$
 325
 326## Vector proofs
 327
 328### Concurrent lines
 329
 330$\ge$ 3 lines intersect at a single point  
 331
 332### Collinear points
 333
 334$\ge$ 3 points lie on the same line  
 335$\implies \vec{OC} = \lambda \vec{OA} + \mu \vec{OB}$ where $\lambda + \mu = 1$. If $C$ is between $\vec{AB}$, then $0 < \mu < 1$  
 336Points $A, B, C$ are collinear iff $\vec{AC}=m\vec{AB} \text{ where } m \ne 0$
 337
 338### Useful vector properties
 339
 340- If $\boldsymbol{a}$ and $\boldsymbol{b}$ are parallel, then $\boldsymbol{b}=k\boldsymbol{a}$ for some $k \in \mathbb{R} \setminus \{0\}$
 341- If $\boldsymbol{a}$ and $\boldsymbol{b}$ are parallel with at least one point in common, then they lie on the same straight line
 342- Two vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ are perpendicular if $\boldsymbol{a} \cdot \boldsymbol{b}=0$
 343- $\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2$
 344
 345## Linear dependence
 346
 347Vectors $\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}$ are linearly dependent if they are non-parallel and:
 348
 349$$k\boldsymbol{a}+l\boldsymbol{b}+m\boldsymbol{c} = 0$$
 350$$\therefore \boldsymbol{c} = m\boldsymbol{a} + n\boldsymbol{b} \quad \text{(simultaneous)}$$
 351
 352$\boldsymbol{a}, \boldsymbol{b},$ and $\boldsymbol{c}$ are linearly independent if no vector in the set is expressible as a linear combination of other vectors in set, or if they are parallel.
 353
 354Vector $\boldsymbol{w}$ is a linear combination of vectors $\boldsymbol{v_1}, \boldsymbol{v_2}, \boldsymbol{v_3}$
 355
 356## Three-dimensional vectors
 357
 358Right-hand rule for axes: $z$ is up or out of page.
 359
 360i![](graphics/vectors-3d.png)
 361
 362## Parametric vectors
 363
 364Parametric equation of line through point $(x_0, y_0, z_0)$ and parallel to $a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}$ is:
 365
 366\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}
 367
 368# Circular functions
 369
 370Period of $a\sin(bx)$ is ${2\pi} \over b$
 371
 372Period of $a\tan(nx)$ is $\pi \over n$  
 373Asymptotes at $x={2k+1)\pi \over 2n} \> \vert \> k \in \mathbb{Z}$
 374
 375## Reciprocal functions
 376
 377### Cosecant
 378
 379![](graphics/csc.png)
 380
 381$$\operatorname{cosec} \theta = {1 \over \sin \theta} \> \vert \> \sin \theta \ne 0$$
 382
 383- **Domain** $= \mathbb{R} \setminus {n\pi : n \in \mathbb{Z}}$
 384- **Range** $= \mathbb{R} \setminus (-1, 1)$
 385- **Turning points** at $\theta = {{(2n + 1)\pi} \over 2} \> \vert \> n \in \mathbb{Z}$
 386- **Asymptotes** at $\theta = n\pi \> \vert \> n \in \mathbb{Z}$
 387
 388### Secant
 389
 390![](graphics/sec.png)
 391
 392$$\operatorname{sec} \theta = {1 \over \cos \theta} \> \vert \> \cos \theta \ne 0$$
 393
 394- **Domain** $= \mathbb{R} \setminus \{{{(2n + 1) \pi} \over 2 } : n \in \mathbb{Z}\}$
 395- **Range** $= \mathbb{R} \setminus (-1, 1)$
 396- **Turning points** at $\theta = n\pi \> \vert \> n \in \mathbb{Z}$
 397- **Asymptotes** at $\theta = {{(2n + 1) \pi} \over 2} \> \vert \> n \in \mathbb{Z}$
 398
 399### Cotangent
 400
 401![](graphics/cot.png)
 402
 403$$\operatorname{cot} \theta = {{\cos \theta} \over {\sin \theta}} \> \vert \> \sin \theta \ne 0$$
 404
 405- **Domain** $= \mathbb{R} \setminus \{n \pi: n \in \mathbb{Z}\}$
 406- **Range** $= \mathbb{R}$
 407- **Asymptotes** at $\theta = n\pi \> \vert \> n \in \mathbb{Z}$
 408
 409### Symmetry properties
 410
 411\begin{equation}\begin{split}
 412  \operatorname{sec} (\pi \pm x) & = -\operatorname{sec} x \\
 413  \operatorname{sec} (-x) & = \operatorname{sec} x \\
 414  \operatorname{cosec} (\pi \pm x) & = \mp \operatorname{cosec} x \\
 415  \operatorname{cosec} (-x) & = - \operatorname{cosec} x \\
 416  \operatorname{cot} (\pi \pm x) & = \pm \operatorname{cot} x \\
 417  \operatorname{cot} (-x) & = - \operatorname{cot} x
 418\end{split}\end{equation}
 419
 420### Complementary properties
 421
 422\begin{equation}\begin{split}
 423  \operatorname{sec} \left({\pi \over 2} - x\right) & = \operatorname{cosec} x \\
 424  \operatorname{cosec} \left({\pi \over 2} - x\right) & = \operatorname{sec} x \\
 425  \operatorname{cot} \left({\pi \over 2} - x\right) & = \tan x \\
 426  \tan \left({\pi \over 2} - x\right) & = \operatorname{cot} x
 427\end{split}\end{equation}
 428
 429### Pythagorean identities
 430
 431\begin{equation}\begin{split}
 432  1 + \operatorname{cot}^2 x & = \operatorname{cosec}^2 x, \quad \text{where } \sin x \ne 0 \\
 433  1 + \tan^2 x & = \operatorname{sec}^2 x, \quad \text{where } \cos x \ne 0
 434\end{split}\end{equation}
 435
 436## Compound angle formulas
 437
 438$$\cos(x \pm y) = \cos x + \cos y \mp \sin x \sin y$$
 439$$\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y$$
 440$$\tan(x \pm y) = {{\tan x \pm \tan y} \over {1 \mp \tan x \tan y}}$$
 441
 442## Double angle formulas
 443
 444\begin{equation}\begin{split}
 445  \cos 2x &= \cos^2 x - \sin^2 x \\
 446  & = 1 - 2\sin^2 x \\
 447  & = 2 \cos^2 x -1
 448\end{split}\end{equation}
 449
 450$$\sin 2x = 2 \sin x \cos x$$
 451
 452$$\tan 2x = {{2 \tan x} \over {1 - \tan^2 x}}$$
 453
 454## Inverse circular functions
 455
 456Inverse functions: $f(f^{-1}(x)) = x, \quad f(f^{-1}(x)) = x$  
 457Must be 1:1 to find inverse (reflection in $y=x$
 458
 459Domain is restricted to make functions 1:1.
 460
 461### $\arcsin$
 462
 463$$\sin^{-1}: [-1, 1] \rightarrow \mathbb{R}, \quad \sin^{-1} x = y, \quad \text{where } \sin y = x \text{ and } y \in [{-\pi \over 2}, {\pi \over 2}]$$
 464
 465### $\arcos$
 466
 467$$\cos^{-1} \rightarrow \mathbb{R}, \quad \cos^{-1} x = y, \quad \text{where } \cos y = x \text{ and } y \in [0, \pi]$$
 468
 469### $\arctan$
 470
 471$$\tan^{-1}: \mathbb{R} \rightarrow \mathbb{R}, \quad \tan^{-1} x = y, \quad \text{where } \tan y = x \text{ and } y \in \left(-{\pi \over 2}, {\pi \over 2}\right)$$
 472# Differential calculus
 473
 474## Limits
 475
 476$$\lim_{x \rightarrow a}f(x)$$
 477
 478$L^-$ - limit from below
 479
 480$L^+$ - limit from above
 481
 482$\lim_{x \to a} f(x)$ - limit of a point  
 483
 484- Limit exists if $L^-=L^+$
 485- If limit exists, point does not.
 486
 487Limits can be solved using normal techniques (if div 0, factorise)
 488
 489## Limit theorems
 490
 4911. For constant function $f(x)=k$, $\lim_{x \rightarrow a} f(x) = k$
 4922. $\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G$
 4933. $\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G$
 4944. ${\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0$
 495
 496Corollary: $\lim_{x \rightarrow a} c \times f(x)=cF$ where $c=$ constant
 497
 498## Solving limits for $x\rightarrow\infty$
 499
 500Factorise so that all values of $x$ are in denominators.
 501
 502e.g.
 503
 504$$\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2$$
 505
 506
 507## Continuous functions
 508
 509A function is continuous if $L^-=L^+=f(x)$ for all values of $x$.
 510
 511## Gradients of secants and tangents
 512
 513Secant (chord) - line joining two points on curve
 514
 515Tangent - line that intersects curve at one point
 516
 517given $P(x,y) \quad Q(x+\delta x, y + \delta y)$:
 518gradient of chord joining $P$ and $Q$ is ${m_{PQ}}={\operatorname{rise} \over \operatorname{run}} = {\delta y \over \delta x}$
 519
 520As $Q \rightarrow P, \delta x \rightarrow 0$. Chord becomes tangent (two infinitesimal points are equal).
 521
 522Can also be used with functions, where $h=\delta x$.
 523
 524## First principles derivative
 525
 526$$f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={dy \over dx}$$
 527
 528$$m_{\tan}=\lim_{h \rightarrow 0}f^\prime(x)$$
 529
 530
 531
 532$$m_{\vec{PQ}}=f^\prime(x)$$
 533
 534first principles derivative:
 535$${m_{\text{tangent at }P} =\lim_{h \rightarrow 0}}{{f(x+h)-f(x)}\over h}$$
 536
 537## Gradient at a point
 538
 539Given point $P(a, b)$ and function $f(x)$, the gradient is $f^\prime(a)$
 540
 541
 542## Derivatives of $x^n$
 543
 544$${d(ax^n) \over dx}=anx^{n-1}$$
 545
 546If $x=$ constant, derivative is $0$
 547
 548If $y=ax^n$, derivative is $a\times nx^{n-1}$
 549
 550If $f(x)={1 \over x}=x^{-1}, \quad f^\prime(x)=-1x^{-2}={-1 \over x^2}$
 551
 552If $f(x)=^5\sqrt{x}=x^{1 \over 5}, \quad f^\prime(x)={1 \over 5}x^{-4/5}={1 \over 5 \times ^5\sqrt{x^4}}$
 553
 554If $f(x)=(x-b)^2, \quad f^\prime(x)=2(x-b)$
 555
 556$$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$
 557
 558## Derivatives of $u \pm v$
 559
 560$${dy \over dx}={du \over dx} \pm {dv \over dx}$$
 561where $u$ and $v$ are functions of $x$
 562
 563## Euler's number as a limit
 564
 565$$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$
 566
 567## Chain rule for $(f\circ g)$
 568
 569If $f(x) = h(g(x)) = (h \circ g)(x)$:
 570
 571$$f^\prime(x) = h^\prime(g(x)) \cdot g^\prime(x)$$
 572
 573If $y=h(u)$ and $u=g(x)$:
 574
 575$${dy \over dx} = {dy \over du} \cdot {du \over dx}$$
 576$${d((ax+b)^n) \over dx} = {d(ax+b) \over dx} \cdot n \cdot (ax+b)^{n-1}$$
 577
 578Used with only one expression.
 579
 580e.g. $y=(x^2+5)^7$ - Cannot reasonably expand  
 581Let $u-x^2+5$ (inner expression)  
 582${du \over dx} = 2x$  
 583$y=u^7$  
 584${dy \over du} = 7u^6$  
 585
 586## Product rule for $y=uv$
 587
 588$${dy \over dx} = u{dv \over dx} + v{du \over dx}$$
 589
 590## Quotient rule for $y={u \over v}$
 591
 592$${dy \over dx} = {{v{du \over dx} - u{dv \over dx}} \over v^2}$$
 593
 594$$f^\prime(x)={{v(x)u^\prime(x)-u(x)v^\prime(x)} \over [v(x)]^2}$$
 595
 596## Logarithms
 597
 598$$\log_b (x) = n \quad \operatorname{where} \hspace{0.5em} b^n=x$$
 599
 600Wikipedia:
 601
 602> the logarithm of a given number $x$ is the exponent to which another fixed number, the base $b$, must be raised, to produce that number $x$
 603
 604### Logarithmic identities
 605
 606$\log_b (xy)=\log_b x + \log_b y$  
 607$\log_b x^n = n \log_b x$  
 608$\log_b y^{x^n} = x^n \log_b y$
 609
 610### Index identities
 611
 612$b^{m+n}=b^m \cdot b^n$  
 613$(b^m)^n=b^{m \cdot n}$  
 614$(b \cdot c)^n = b^n \cdot c^n$  
 615${a^m \div a^n} = {a^{m-n}}$
 616
 617### $e$ as a logarithm
 618
 619$$\operatorname{if} y=e^x, \quad \operatorname{then} x=\log_e y$$
 620$$\ln x = \log_e x$$
 621
 622### Differentiating logarithms
 623$${d(\log_e x)\over dx} = x^{-1} = {1 \over x}$$
 624
 625## Derivative rules
 626
 627| $f(x)$ | $f^\prime(x)$ |
 628| ------ | ------------- |
 629| $\sin x$ | $\cos x$ |
 630| $\sin ax$ | $a\cos ax$ |
 631| $\cos x$ | $-\sin x$ |
 632| $\cos ax$ | $-a \sin ax$ |
 633| $\tan f(x)$ | $f^2(x) \sec^2f(x)$ |
 634| $e^x$ | $e^x$ |
 635| $e^{ax}$ | $ae^{ax}$ |
 636| $ax^{nx}$ | $an \cdot e^{nx}$ |
 637| $\log_e x$ | $1 \over x$ |
 638| $\log_e {ax}$ | $1 \over x$ |
 639| $\log_e f(x)$ | $f^\prime (x) \over f(x)$ |
 640| $\sin(f(x))$ | $f^\prime(x) \cdot \cos(f(x))$ |
 641| $\sin^{-1} x$ | $1 \over {\sqrt{1-x^2}}$ |
 642| $\cos^{-1} x$ | $-1 \over {sqrt{1-x^2}}$ |
 643| $\tan^{-1} x$ | $1 \over {1 + x^2}$ |
 644
 645## Reciprocal derivatives
 646
 647$${1 \over {dy \over dx}} = {dx \over dy}$$
 648
 649## Differentiating $x=f(y)$
 650
 651Find $dx \over dy$. Then ${dx \over dy} = {1 \over {dy \over dx}} \implies {dy \over dx} = {1 \over {dx \over dy}}$.
 652
 653$${dy \over dx} = {1 \over {dx \over dy}}$$
 654
 655## Second derivative
 656
 657$$f(x) \longrightarrow f^\prime (x) \longrightarrow f^{\prime\prime}(x)$$
 658
 659$$\therefore y \longrightarrow {dy \over dx} \longrightarrow {d({dy \over dx}) \over dx} \longrightarrow {d^2 y \over dx^2}$$
 660
 661Order of polynomial $n$th derivative decrements each time the derivative is taken
 662
 663### Points of Inflection
 664
 665*Stationary point* - point of zero gradient (i.e. $f^\prime(x)=0$)  
 666*Point of inflection* - point of maximum $|$gradient$|$ (i.e.  $f^{\prime\prime} = 0$)
 667
 668* if $f^\prime (a) = 0$ and $f^{\prime\prime}(a) > 0$, then point $(a, f(a))$ is a local min (curve is concave up)
 669* if $f^\prime (a) = 0$ and $f^{\prime\prime} (a) < 0$, then point $(a, f(a))$ is local max (curve is concave down)
 670* if $f^{\prime\prime}(a) = 0$, then point $(a, f(a))$ is a point of inflection
 671  + if also $f^\prime(a)=0$, then it is a stationary point of inflection
 672
 673![](graphics/second-derivatives.png)
 674
 675## Implicit Differentiation
 676
 677**On CAS:** Action $\rightarrow$ Calculation $\rightarrow$ `impDiff(y^2+ax=5, x, y)`. Returns $y^\prime= \dots$.
 678
 679Used for differentiating circles etc.
 680
 681If $p$ and $q$ are expressions in $x$ and $y$ such that $p=q$, for all $x$ nd $y$, then:
 682
 683$${dp \over dx} = {dq \over dx} \quad \text{and} \quad {dp \over dy} = {dq \over dy}$$
 684
 685## Integration
 686
 687$$\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)$$
 688
 689$$\int x^n \cdot dx = {x^{n+1} \over n+1} + c$$
 690
 691- area enclosed by curves
 692- $+c$ should be shown on each step without $\int$
 693
 694### Integral laws
 695
 696$\int f(x) + g(x) dx = \int f(x) dx + \int g(x) dx$  
 697$\int k f(x) dx = k \int f(x) dx$  
 698
 699| $f(x)$                          | $\int f(x) \cdot dx$         |
 700| ------------------------------- | ---------------------------- |
 701| $k$ (constant) | $kx + c$ |
 702| $x^n$ | ${x^{n+1} \over {n+1}} + c$ |
 703| $a x^{-n}$ | $a \cdot \log_e x + c$ |
 704| ${1 \over {ax+b}}$ | ${1 \over a} \log_e (ax+b) + c$ |
 705| $(ax+b)^n$ | ${1 \over {a(n+1)}}(ax+b)^{n-1} + c$ |
 706| $e^{kx}$ | ${1 \over k} e^{kx} + c$ |
 707| $e^k$ | $e^kx + c$ |
 708| $\sin kx$ | $-{1 \over k} \cos (kx) + c$ |
 709| $\cos kx$ | ${1 \over k} \sin (kx) + c$ |
 710| $\sec^2 kx$ | ${1 \over k} \tan(kx) + c$ |
 711| $1 \over \sqrt{a^2-x^2}$ | $\sin^{-1} {x \over a} + c \>\vert\> a>0$ |
 712| $-1 \over \sqrt{a^2-x^2}$ | $\cos^{-1} {x \over a} + c \>\vert\> a>0$ |
 713| $a \over {a^2-x^2}$ | $\tan^{-1} {x \over a} + c$ |
 714| ${f^\prime (x)} \over {f(x)}$ | $\log_e f(x) + c$ |
 715| $g^\prime(x)\cdot f^\prime(g(x)$ | $f(g(x))$ (chain rule)|
 716| $f(x) \cdot g(x)$ | $\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx$ |
 717
 718Note $\sin^{-1} {x \over a} + \cos^{-1} {x \over a}$ is constant for all $x \in (-a, a)$.
 719
 720### Definite integrals
 721
 722$$\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)$$
 723
 724- Signed area enclosed by: $\> y=f(x), \quad y=0, \quad x=a, \quad x=b$.
 725- *Integrand* is $f$.
 726- $F(x)$ may be any integral, i.e. $c$ is inconsequential
 727
 728#### Properties
 729
 730$$\int^b_a f(x) \> dx = \int^c_a f(x) \> dx + \int^b_c f(x) \> dx$$
 731
 732$$\int^a_a f(x) \> dx = 0$$
 733
 734$$\int^b_a k \cdot f(x) \> dx = k \int^b_a f(x) \> dx$$
 735
 736$$\int^b_a f(x) \pm g(x) \> dx = \int^b_a f(x) \> dx \pm \int^b_a g(x) \> dx$$
 737
 738$$\int^b_a f(x) \> dx = - \int^a_b f(x) \> dx$$
 739
 740### Integration by substitution
 741
 742$$\int f(u) {du \over dx} \cdot dx = \int f(u) \cdot du$$
 743
 744Note $f(u)$ must be one-to-one $\implies$ one $x$ value for each $y$ value
 745
 746e.g. for $y=\int(2x+1)\sqrt{x+4} \cdot dx$:  
 747let $u=x+4$  
 748$\implies {du \over dx} = 1$  
 749$\implies x = u - 4$  
 750then $y=\int (2(u-4)+1)u^{1 \over 2} \cdot du$  
 751Solve as a normal integral
 752
 753#### Definite integrals by substitution
 754
 755For $\int^b_a f(x) {du \over dx} \cdot dx$, evaluate new $a$ and $b$ for $f(u) \cdot du$.
 756
 757### Trigonometric integration
 758
 759$$\sin^m x \cos^n x \cdot dx$$
 760
 761**$m$ is odd:**  
 762$m=2k+1$ where $k \in \mathbb{Z}$  
 763$\implies \sin^{2k+1} x = (\sin^2 z)^k \sin x = (1 - \cos^2 x)^k \sin x$  
 764Substitute $u=\cos x$
 765
 766**$n$ is odd:**  
 767$n=2k+1$ where $k \in \mathbb{Z}$  
 768$\implies \cos^{2k+1} x = (\cos^2 x)^k \cos x = (1-\sin^2 x)^k \cos x$  
 769Subbstitute $u=\sin x$
 770
 771**$m$ and $n$ are even:**  
 772Use identities:
 773
 774- $\sin^2x={1 \over 2}(1-\cos 2x)$
 775- $\cos^2x={1 \over 2}(1+\cos 2x)$
 776- $\sin 2x = 2 \sin x \cos x$
 777
 778## Partial fractions
 779
 780On CAS: Action $\rightarrow$ Transformation $\rightarrow$ `expand/combine`  
 781or Interactive $\rightarrow$ Transformation $\rightarrow$ `expand` $\rightarrow$ Partial
 782
 783## Graphing integrals on CAS
 784
 785In main: Interactive $\rightarrow$ Calculation $\rightarrow$ $\int$ ($\rightarrow$ Definite)  
 786Restrictions: `Define f(x)=...` $\rightarrow$ `f(x)|x>1` (e.g.)
 787
 788## Applications of antidifferentiation
 789
 790- $x$-intercepts of $y=f(x)$ identify $x$-coordinates of stationary points on $y=F(x)$
 791- nature of stationary points is determined by sign of $y=f(x)$ on either side of its $x$-intercepts
 792- if $f(x)$ is a polynomial of degree $n$, then $F(x)$ has degree $n+1$
 793
 794To find stationary points of a function, substitute $x$ value of given point into derivative. Solve for ${dy \over dx}=0$. Integrate to find original function.
 795
 796## Solids of revolution
 797
 798Approximate as sum of infinitesimally-thick cylinders
 799
 800### Rotation about $x$-axis
 801
 802\begin{align*}
 803  V &= \int^{x=b}_{x-a} \pi y^2 \> dx \\
 804    &= \pi \int^b_a (f(x))^2 \> dx
 805\end{align*}
 806
 807### Rotation about $y$-axis
 808
 809\begin{align*}
 810  V &= \int^{y=b}_{y=a} \pi x^2 \> dy \\
 811    &= \pi \int^b_a (f(y))^2 \> dy
 812\end{align*}
 813
 814### Regions not bound by $y=0$
 815
 816$$V = \pi \int^b_a f(x)^2 - g(x)^2 \> dx$$  
 817where $f(x) > g(x)$
 818
 819## Length of a curve
 820
 821$$L = \int^b_a \sqrt{1 + ({dy \over dx})^2} \> dx \quad \text{(Cartesian)}$$
 822
 823$$L = \int^b_a \sqrt{{dx \over dt} + ({dy \over dt})^2} \> dt \quad \text{(parametric)}$$
 824
 825Evaluate on CAS. Or use Interactive $\rightarrow$ Calculation $\rightarrow$ Line $\rightarrow$ `arcLen`.
 826
 827## Rates
 828
 829### Related rates
 830
 831$${da \over db} \quad \text{(change in } a \text{ with respect to } b)$$
 832
 833### Gradient at a point on parametric curve
 834
 835$${dy \over dx} = {{dy \over dt} \div {dx \over dt}} \> \vert \> {dx \over dt} \ne 0$$
 836
 837$${d^2 \over dx^2} = {d(y^\prime) \over dx} = {{dy^\prime \over dt} \div {dx \over dt}} \> \vert \> y^\prime = {dy \over dx}$$
 838
 839## Rational functions
 840
 841$$f(x) = {P(x) \over Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}$$
 842
 843### Addition of ordinates
 844
 845- when two graphs have the same ordinate, $y$-coordinate is double the ordinate
 846- when two graphs have opposite ordinates, $y$-coordinate is 0 i.e. ($x$-intercept)
 847- when one of the ordinates is 0, the resulting ordinate is equal to the other ordinate
 848
 849## Fundamental theorem of calculus
 850
 851If $f$ is continuous on $[a, b]$, then
 852
 853$$\int^b_a f(x) \> dx = F(b) - F(a)$$
 854
 855where $F$ is any antiderivative of $f$
 856
 857## Differential equations
 858
 859One or more derivatives
 860
 861**Order** - highest power inside derivative  
 862**Degree** - highest power of highest derivative  
 863e.g. ${\left(dy^2 \over d^2 x\right)}^3$: order 2, degree 3
 864
 865### Verifying solutions
 866
 867Start with $y=\dots$, and differentiate. Substitute into original equation.
 868
 869### Function of the dependent variable
 870
 871If ${dy \over dx}=g(y)$, then ${dx \over dy} = 1 \div {dy \over dx} = {1 \over g(y)}$. Integrate both sides to solve equation. Only add $c$ on one side. Express $e^c$ as $A$.
 872
 873### Mixing problems
 874
 875$$\left({dm \over dt}\right)_\Sigma = \left({dm \over dt}\right)_{\text{in}} - \left({dm \over dt}\)_{\text{out}}$$
 876
 877### Separation of variables
 878
 879If ${dy \over dx}=f(x)g(y)$, then:
 880
 881$$\int f(x) \> dx = \int {1 \over g(y)} \> dy$$
 882
 883### Using definite integrals to solve DEs
 884
 885Used for situations where solutions to ${dy \over dx} = f(x)$ is not required.
 886
 887In some cases, it may not be possible to obtain an exact solution.
 888
 889Approximate solutions can be found by numerically evaluating a definite integral.
 890
 891### Using Euler's method to solve a differential equation
 892
 893$${{f(x+h) - f(x)} \over h } \approx f^\prime (x) \quad \text{for small } h$$
 894
 895$$\implies f(x+h) \approx f(x) + hf^\prime(x)$$
 896