Andrew's git - notes.git/blob - spec/calculus-ref.md

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   4graphics: yes
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   6author: Andrew Lorimer
   7classoption: twocolumn
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   9---
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  11<!-- \renewcommand{\arraystretch}{2} -->
  12# Spec - Calculus
  13
  14## Gradients
  15
  16$$m \operatorname{of} x \in [a,b] = {{f(b)-f(a)}\over {b - a}} = {dy \over dx}$$
  17
  18## Limit theorems
  19
  201. For constant function $f(x)=k$, $\lim_{x \rightarrow a} f(x) = k$
  212. $\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G$
  223. $\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G$
  234. ${\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0$
  24
  25
  26## First principles derivative
  27
  28$$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$
  29
  30
  31## Tangents & gradients
  32
  33**Tangent line** - defined by $y=mx+c$ where $m={dy \over dx}$  
  34**Normal line** - $\perp$ tangent ($m_{\operatorname{tan}} \cdot m_{\operatorname{norm}} = -1$)  
  35**Secant** $={{f(x+h)-f(x)} \over h}$
  36
  37## Derivatives
  38
  39
  40
  41| $f(x)$ | $f^\prime(x)$ |
  42| --- | --- |
  43| $kx^n$ | $knx^{n-1}$ |
  44| $g(x) + h(x)$ | $g^\prime (x) + h^\prime (x)$ |
  45| $c$ | $0$ |
  46| ${u \over v}$ | ${{v{du \over dx} - u{dv \over dx}} \over v^2}$ |
  47| $uv$ | $u{dv \over dx} + v{du \over dx}$ |
  48| $f \circ g$ | ${dy \over du} \cdot {du \over dx}$ |
  49| $\sin ax$ | $a\cos ax$ |
  50| $\sin(f(x))$ | $f^\prime(x) \cdot \cos(f(x))$ |
  51| $\cos ax$ | $-a \sin ax$ |
  52| $e^{ax}$ | $ae^{ax}$ |
  53| $\log_e {ax}$ | $1 \over x$ |
  54| $\log_e f(x)$ | $f^\prime (x) \over f(x)$ |
  55
  56
  57
  58
  59
  60## Product rule for $y=uv$
  61
  62$${dy \over dx} = u{dv \over dx} + v{du \over dx}$$
  63
  64## Logarithms
  65
  66$$\log_b (x) = n \quad \operatorname{where} \hspace{0.5em} b^n=x$$
  67
  68## Integration
  69
  70$$\int f(x) dx = F(x) + c$$
  71
  72- area enclosed by curves
  73
  74| $f(x)$ | $\int f(x) \cdot dx$ |
  75| ----|--- |
  76| $k$ (constant) | $kx + c$ |
  77| $x^n$ | ${1 \over {n+1}}x^{n+1} + c$ |
  78| $a x^{-n}$ | $a \cdot \log_e x + c$ |
  79| $e^{kx}$ | ${1 \over k} e^{kx} + c$ |
  80| $e^k$ | $e^kx + c$ |
  81| $\sin kx$ | $-{1 \over k} \cos (kx) + c$ |
  82| $\cos kx$ | ${1 \over k} \sin (kx) + c$ |
  83| ${f^\prime (x)} \over {f(x)}$ | $\log_e f(x) + c$ |
  84| $g^\prime(x)\cdot f^\prime(g(x)$ | $f(g(x))$ (chain rule)|
  85| $f(x) \cdot g(x)$ | $\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx$ |
  86| ${1 \over {ax+b}}$ | ${1 \over a} \log_e (ax+b) + c$ |
  87| $(ax+b)^n$ | ${1 \over {a(n+1)}}(ax+b)^{n-1} + c$ |
  88
  89
  90## Definite integrals
  91
  92$$\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)_{}$$
  93
  94## Kinematics
  95
  96**position $x$** - distance from origin or fixed point  
  97**displacement $s$** - change in position from starting point (vector)  
  98**velocity $v$** - change in position with respect to time  
  99**acceleration $a$** - change in velocity  
 100**speed** - magnitude of velocity  
 101
 102
 103
 104| | no |
 105| - | -- |
 106| $v=u+at$ | $s$ |
 107| $s=ut + {1 \over 2} at^2$ | $v$ |
 108| $v^2 = u^2 + 2as$ | $t$ |
 109| $s= {1 \over 2}(u+v)t$ | $a$ |
 110