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

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   2geometry: margin=2cm
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   4graphics: yes
   5tables: yes
   6author: Andrew Lorimer
   7---
   8
   9# Spec - Calculus
  10
  11## Gradients
  12
  13$$m \operatorname{of} x \in [a,b] = {{f(b)-f(a)}\over {b - a}} = {dy \over dx}$$
  14
  15## Limit theorems
  16
  171. For constant function $f(x)=k$, $\lim_{x \rightarrow a} f(x) = k$
  182. $\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G$
  193. $\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G$
  204. ${\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0$
  21
  22
  23## First principles derivative
  24
  25$$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$
  26
  27
  28## Tangents & gradients
  29
  30**Tangent line** - defined by $y=mx+c$ where $m={dy \over dx}$  
  31**Normal line** - $\perp$ tangent ($m_{\operatorname{tan}} \cdot m_{\operatorname{norm}} = -1$)  
  32**Secant** $={{f(x+h)-f(x)} \over h}$
  33
  34## Derivatives
  35
  36| $f(x)$ | $f^\prime(x)$ |
  37| ------ | ------------- |
  38| $kx^n$ | $knx^{n-1}$ |
  39| $g(x) + h(x)$ | $g^\prime (x) + h^\prime (x)$ |
  40| $c$    | $0$ |
  41| ${u \over v}$ | ${{v{du \over dx} - u{dv \over dx}} \over v^2}$ |
  42| $uv$ | $u{dv \over dx} + v{du \over dx}$ |
  43| $f \circ g$ | ${dy \over du} \cdot {du \over dx}$ |
  44| $\sin ax$ | $a\cos ax$ |
  45| $\sin(f(x))$ | $f^\prime(x) \cdot \cos(f(x))$ |
  46| $\cos ax$ | $-a \sin ax$ |
  47| $e^{ax}$ | $ae^{ax}$ |
  48| $\log_e {ax}$ | $1 \over x$ |
  49| $\log_e f(x)$ | $f^\prime (x) \over f(x)$ |
  50
  51
  52
  53## Product rule for $y=uv$
  54
  55$${dy \over dx} = u{dv \over dx} + v{du \over dx}$$
  56
  57## Logarithms
  58
  59$$\log_b (x) = n \quad \operatorname{where} \hspace{0.5em} b^n=x$$
  60
  61## Integration
  62
  63$$\int f(x) dx = F(x) + c$$
  64
  65- area enclosed by curves
  66
  67| $f(x)$ | $\int f(x) \cdot dx$ |
  68| --------------- | ------------------ |
  69| $k$ (constant) | $kx + c$ |
  70| $x^n$ | ${1 \over {n+1}}x^{n+1} + c$ |
  71| $a x^{-n}$ | $a \cdot \log_e x + c$ |
  72| $e^{kx}$ | ${1 \over k} e^{kx} + c$ |
  73| $e^k$ | $e^kx + c$ |
  74| $\sin kx$ | $-{1 \over k} \cos (kx) + c$ |
  75| $\cos kx$ | ${1 \over k} \sin (kx) + c$ |
  76| ${f^\prime (x)} \over {f(x)}$ | $\log_e f(x) + c$ |
  77| $g^\prime(x)\cdot f^\prime(g(x)$ | $f(g(x))$ (chain rule)|
  78| $f(x) \cdot g(x)$ | $\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx$ |
  79| ${1 \over {ax+b}}$ | ${1 \over a} \log_e (ax+b) + c$ |
  80| $(ax+b)^n$ | ${1 \over {a(n+1)}}(ax+b)^{n-1} + c$ |
  81
  82
  83## Definite integrals
  84
  85$$\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)_{}$$
  86
  87## Kinematics
  88
  89**position $x$** - distance from origin or fixed point  
  90**displacement $s$** - change in position from starting point (vector)  
  91**velocity $v$** - change in position with respect to time  
  92**acceleration $a$** - change in velocity  
  93**speed** - magnitude of velocity  
  94
  95| | no |
  96| - | -- |
  97| $v=u+at$ | $s$ |
  98| $s=ut + {1 \over 2} at^2$ | $v$ |
  99| $v^2 = u^2 + 2as$ | $t$ |
 100| $s= {1 \over 2}(u+v)t$ | $a$ |