spec / calculus.mdon commit [spec] add to integral laws (b997829)
   1---
   2geometry: margin=2cm
   3columns: 2
   4graphics: yes
   5tables: yes
   6author: Andrew Lorimer
   7classoption: twocolumn
   8---
   9
  10# Differential calculus
  11
  12## Limits
  13
  14$$\lim_{x \rightarrow a}f(x)$$
  15
  16$L^-$ - limit from below
  17
  18$L^+$ - limit from above
  19
  20$\lim_{x \to a} f(x)$ - limit of a point  
  21
  22- Limit exists if $L^-=L^+$
  23- If limit exists, point does not.
  24
  25Limits can be solved using normal techniques (if div 0, factorise)
  26
  27## Limit theorems
  28
  291. For constant function $f(x)=k$, $\lim_{x \rightarrow a} f(x) = k$
  302. $\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G$
  313. $\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G$
  324. ${\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0$
  33
  34Corollary: $\lim_{x \rightarrow a} c \times f(x)=cF$ where $c=$ constant
  35
  36## Solving limits for $x\rightarrow\infty$
  37
  38Factorise so that all values of $x$ are in denominators.
  39
  40e.g.
  41
  42$$\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2$$
  43
  44
  45## Continuous functions
  46
  47A function is continuous if $L^-=L^+=f(x)$ for all values of $x$.
  48
  49## Gradients of secants and tangents
  50
  51Secant (chord) - line joining two points on curve
  52
  53Tangent - line that intersects curve at one point
  54
  55given $P(x,y) \quad Q(x+\delta x, y + \delta y)$:
  56gradient of chord joining $P$ and $Q$ is ${m_{PQ}}={\operatorname{rise} \over \operatorname{run}} = {\delta y \over \delta x}$
  57
  58As $Q \rightarrow P, \delta x \rightarrow 0$. Chord becomes tangent (two infinitesimal points are equal).
  59
  60Can also be used with functions, where $h=\delta x$.
  61
  62## First principles derivative
  63
  64$$f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={dy \over dx}$$
  65
  66$$m_{\operatorname{tangent}}=\lim_{h \rightarrow 0}f^\prime(x)$$
  67
  68
  69
  70$$m_{\operatorname{chord PQ}}=f^\prime(x)$$
  71
  72first principles derivative:
  73$${m_{\operatorname{tangent at P}} =\lim_{h \rightarrow 0}}{{f(x+h)-f(x)}\over h}$$
  74
  75## Gradient at a point
  76
  77Given point $P(a, b)$ and function $f(x)$, the gradient is $f^\prime(a)$
  78
  79
  80## Derivatives of $x^n$
  81
  82$${d(ax^n) \over dx}=anx^{n-1}$$
  83
  84If $x=$ constant, derivative is $0$
  85
  86If $y=ax^n$, derivative is $a\times nx^{n-1}$
  87
  88If $f(x)={1 \over x}=x^{-1}, \quad f^\prime(x)=-1x^{-2}={-1 \over x^2}$
  89
  90If $f(x)=^5\sqrt{x}=x^{1 \over 5}, \quad f^\prime(x)={1 \over 5}x^{-4/5}={1 \over 5 \times ^5\sqrt{x^4}}$
  91
  92If $f(x)=(x-b)^2, \quad f^\prime(x)=2(x-b)$
  93
  94$$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$
  95
  96## Derivatives of $u \pm v$
  97
  98$${dy \over dx}={du \over dx} \pm {dv \over dx}$$
  99where $u$ and $v$ are functions of $x$
 100
 101## Euler's number as a limit
 102
 103$$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$
 104
 105## Chain rule for $(f\circ g)$
 106
 107If $f(x) = h(g(x)) = (h \circ g)(x)$:
 108
 109$$f^\prime(x) = h^\prime(g(x)) \cdot g^\prime(x)$$
 110
 111If $y=h(u)$ and $u=g(x)$:
 112
 113$${dy \over dx} = {dy \over du} \cdot {du \over dx}$$
 114$${d((ax+b)^n) \over dx} = {d(ax+b) \over dx} \cdot n \cdot (ax+b)^{n-1}$$
 115
 116Used with only one expression.
 117
 118e.g. $y=(x^2+5)^7$ - Cannot reasonably expand  
 119Let $u-x^2+5$ (inner expression)  
 120${du \over dx} = 2x$  
 121$y=u^7$  
 122${dy \over du} = 7u^6$  
 123
 124## Product rule for $y=uv$
 125
 126$${dy \over dx} = u{dv \over dx} + v{du \over dx}$$
 127
 128Surds can be left on denomintaors.
 129
 130## Quotient rule for $y={u \over v}$
 131
 132$${dy \over dx} = {{v{du \over dx} - u{dv \over dx}} \over v^2}$$
 133
 134If $f(x)={u(x) \over v(x)}$, then $f^\prime(x)={{v(x)u^\prime(x)-u(x)v^\prime(x)} \over [v(x)]^2}$
 135
 136If $y={u(x) \over v(x)}$, then derivative ${dy \over dx} = {{v{du \over dx} - u{dv \over dx}} \over v^2}$
 137
 138## Logarithms
 139
 140$$\log_b (x) = n \quad \operatorname{where} \hspace{0.5em} b^n=x$$
 141
 142Wikipedia:
 143
 144> 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 $x$
 145
 146### Logarithmic identities
 147
 148$\log_b (xy)=\log_b x + \log_b y$  
 149$\log_b x^n = n \log_b x$  
 150$\log_b y^{x^n} = x^n \log_b y$
 151
 152### Index identities
 153
 154$b^{m+n}=b^m \cdot b^n$  
 155$(b^m)^n=b^{m \cdot n}$  
 156$(b \cdot c)^n = b^n \cdot c^n$  
 157${a^m \div a^n} = {a^{m-n}}$
 158
 159### $e$ as a logarithm
 160
 161$$\operatorname{if} y=e^x, \quad \operatorname{then} x=\log_e y$$
 162$$\ln x = \log_e x$$
 163
 164### Differentiating logarithms
 165$${d(\log_e x)\over dx} = x^{-1} = {1 \over x}$$
 166
 167## Derivative rules
 168
 169| $f(x)$ | $f^\prime(x)$ |xs
 170| ------ | ------------- |
 171| $\sin x$ | $\cos x$ |
 172| $\sin ax$ | $a\cos ax$ |
 173| $\cos x$ | $-\sin x$ |
 174| $\cos ax$ | $-a \sin ax$ |
 175| $\tan f(x)$ | $f^2(x) \sec^2f(x)$ |
 176| $e^x$ | $e^x$ |
 177| $e^{ax}$ | $ae^{ax}$ |
 178| $ax^{nx}$ | $an \cdot e^{nx}$ |
 179| $\log_e x$ | $1 \over x$ |
 180| $\log_e {ax}$ | $1 \over x$ |
 181| $\log_e f(x)$ | $f^\prime (x) \over f(x)$ |
 182| $\sin(f(x))$ | $f^\prime(x) \cdot \cos(f(x))$ |
 183| $\sin^{-1} x$ | $1 \over {\sqrt{1-x^2}}$ |
 184| $\cos^{-1} x$ | $-1 \over {sqrt{1-x^2}}$ |
 185| $\tan^{-1} x$ | $1 \over {1 + x^2}$ |
 186
 187<!-- $${d(ax^{nx}) \over dx} = an \cdot e^nx$$ -->
 188
 189Reciprocal derivatives:
 190
 191$${{dy \over dx} \over 1} = dx \over dy$$
 192
 193## Differentiating $x=f(y)$
 194
 195Find $dx \over dy$. Then $dx \over dy = {1 \over {dy \over dx}} \therefore {dy \over dx} = {1 \over {dx \over dy}}$.
 196
 197$${dy \over dx} = {1 \over {dx \over dy}}$$
 198
 199## Second derivative
 200
 201$$f(x) \implies f^\prime (x) \implies f^{\prime\prime}(x)$$
 202
 203$$\therefore y \implies {dy \over dx} \implies {d({dy \over dx}) \over dx} \implies {d^2 y \over dx^2}$$
 204
 205Order of polynomial $n$th derivative decrements each time the derivative is taken
 206
 207### Points of Inflection
 208
 209*Stationary point* - point of zero gradient (i.e. $f^\prime(x)=0$)  
 210*Point of inflection* - point of maximum $|$gradient$|$ (i.e.  $f^{\prime\prime} = 0$)
 211
 212- if $f^\prime (a) = 0$ and $f^{\prime\prime}(a) > 0$, then point $(a, f(a))$ is a local min (curve is concave up)
 213- if $f^\prime (a) = 0$ and $f^{\prime\prime} (a) < 0$, then point $(a, f(a))$ is local max (curve is concave down)
 214- if $f^{\prime\prime}(a) = 0$, then point $(a, f(a))$ is a point of inflection
 215- - if also $f^\prime(a)=0$, then it is a stationary point of inflection
 216
 217![](graphics/second-derivatives.png)
 218
 219## Antidifferentiation
 220
 221$$y={x^{n+1} \over n+1} + c$$
 222
 223## Integration
 224
 225$$\int f(x) dx = F(x) + c$$
 226
 227- area enclosed by curves
 228- $+c$ should be shown on each step without $\int$
 229
 230$$\int x^n = {x^{n+1} \over n+1} + c$$
 231
 232### Integral laws
 233
 234$\int f(x) + g(x) dx = \int f(x) dx + \int g(x) dx$  
 235$\int k f(x) dx = k \int f(x) dx$  
 236
 237| $f(x)$                          | $\int f(x) \cdot dx$         |
 238| ------------------------------- | ---------------------------- |
 239| $k$ (constant) | $kx + c$ |
 240| $x^n$ | ${x^{n+1} \over {n+1}} + c$ |
 241| $a x^{-n}$ | $a \cdot \log_e x + c$ |
 242| ${1 \over {ax+b}}$ | ${1 \over a} \log_e (ax+b) + c$ |
 243| $(ax+b)^n$ | ${1 \over {a(n+1)}}(ax+b)^{n-1} + c$ |
 244| $e^{kx}$ | ${1 \over k} e^{kx} + c$ |
 245| $e^k$ | $e^kx + c$ |
 246| $\sin kx$ | $-{1 \over k} \cos (kx) + c$ |
 247| $\cos kx$ | ${1 \over k} \sin (kx) + c$ |
 248| ${f^\prime (x)} \over {f(x)}$ | $\log_e f(x) + c$ |
 249| $g^\prime(x)\cdot f^\prime(g(x)$ | $f(g(x))$ (chain rule)|
 250| $f(x) \cdot g(x)$ | $\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx$ |
 251
 252## Applications of antidifferentiation
 253
 254- $x$-intercepts of $y=f(x)$ identify $x$-coordinates of stationary points on $y=F(x)$
 255- the nature of any stationary point of $y=F(x)$ is determined by the way the sign of the graph of $y=f(x)$ changes about its $x$-intercepts
 256- if $f(x)$ is a polynomial of degree $n$, then $F(x)$ has degree $n+1$
 257
 258To 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.
 259
 260## Rates
 261
 262### Related rates
 263
 264$${da \over db} \quad \text{change in } a \text{ with respect to } b$$
 265
 266#### Gradient at a point on parametric curve
 267
 268$${dy \over dx} = {{dy \over dt} \over {dx \over dt}} \> \vert \> {dx \over dt} \ne 0$$
 269
 270$${d^2 \over dx^2} = {d(y^\prime) \over dx} = {{dy^\prime \over dt} \over {dx \over dt}} \> \vert \> y^\prime = {dy \over dx}$$
 271
 272# Rational functions
 273
 274$$f(x) = {P(x) \over Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}$$
 275
 276## Addition of ordinates
 277
 278- when two graphs have the same ordinate, $y$-coordinate is double the ordinate
 279- when two graphs have opposite ordinates, $y$-coordinate is 0 i.e. ($x$-intercept)
 280- when one of the ordinates is 0, the resulting ordinate is equal to the other ordinate
 281