methods / stuff.mdon commit [methods] graphing exponential functions (a83b38f)
   1---
   2geometry: margin=2cm
   3<!-- columns: 2 -->
   4graphics: yes
   5tables: yes
   6author: Andrew Lorimer
   7classoption: twocolumn
   8header-includes: \pagenumbering{gobble}
   9---
  10
  11# Exponential and Index Functions
  12
  13## Index laws
  14
  15$a^m \times a^n = a^{m+n}$  
  16$a^m \div a^n = a^{m-n}4$  
  17$(a^m)^n = a^{_mn}$  
  18$(ab)^m = a^m b^m$  
  19${({a \over b})}^m = {a^m \over b^m}$
  20
  21## Fractional indices
  22
  23$^n\sqrt{x}=x^{1/n}$
  24
  25## Logarithms
  26
  27$$\log_b (x) = n \quad \operatorname{where} \hspace{0.5em} b^n=x$$
  28
  29## Using logs to solve index eq's
  30
  31Used for equations without common base exponent
  32
  33Or change base:  
  34$$\log_b c = {{\log_a c} \over {\log_a b}}$$
  35
  36If $a<1, \quad \log_{b} a < 0$ (flip inequality operator)
  37
  38## Exponential functions
  39
  40$e^x$ - natural exponential function
  41
  42
  43$$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$
  44
  45## Logarithm laws
  46
  47$\log_a(mn) = \log_am + \log_an$  
  48$\log_a({m \over n}) = \log_am - \log_an$  
  49$\log_a(m^p) = p\log_am$  
  50$\log_a(m^{-1}) = -\log_am$  
  51$\log_a1 = 0$ and $\log_aa = 1$
  52
  53
  54## Inverse functions
  55
  56Inverse of $f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=a^x$ is $f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=log_ax$
  57
  58## Euler's number
  59
  60$$e= \lim_{n \rightarrow \infty} (1 + {1 \over n})^n$$
  61
  62## Literal equations
  63
  64_Literal equation_ - no numerical solutions
  65
  66## Exponential and logarithmic modelling
  67
  68$$A = A_0 e^{kt}$$
  69
  70where  
  71$A_0$ is initial value  
  72$t$ is time taken  
  73$k$ is a constant  
  74For continuous growth, $k > 0$  
  75For continuous decay, $k < 0$
  76m
  77## Graphing expomnential functions
  78
  79$$f(x)=Aa^{k(x-b)} + c, \quad \vert a > 1$$
  80
  81- **$y$-intercept** at $(0, {{1+c} \over {a^b}})$
  82- **horizontal asymptote** at $y=c$
  83- **domain** is $\mathbb{R}$
  84- **range** is $(c, \infty)$
  85- dilation of factor $A$ from $x$-axis
  86- dilation of factor $1 \over k$ from $y$-axis
  87
  88