Andrew's git / notes.git / methods/stuff.md
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methods / stuff.mdon commit [methods] re-render methods notes (1dadb9e)
   1---
   2geometry: a4paper, margin=2cm
   3columns: 2
   4author: Andrew Lorimer
   5header-includes:
   6- \usepackage{fancyhdr}
   7- \usepackage{setspace}
   8- \pagestyle{fancy}
   9- \fancyhead[LO,LE]{Year 12 Methods}
  10- \fancyhead[CO,CE]{Andrew Lorimer}
  11- \usepackage{graphicx}
  12- \usepackage{tabularx}
  13- \usepackage[dvipsnames]{xcolor}
  14---
  15
  16\pagenumbering{gobble}
  17\setstretch{1.5}
  18\definecolor{cas}{HTML}{e6f0fe}
  19
  20# Exponentials & Logarithms
  21
  22## Index laws
  23
  24\begin{equation*}\begin{split}
  25  a^m \times a^n & = a^{m+n} \\
  26  a^m \div a^n & = a^{m-n} \\
  27  (a^m)^n & = a^{_mn} \\
  28  (ab)^m & = a^m b^m \\
  29  {({a \over b})}^m & = {a^m \over b^m} \\
  30  ^n\sqrt{x} &=x^{1/n}
  31\end{split}\end{equation*}
  32
  33## Logarithm laws
  34
  35\begin{equation*}\begin{split}
  36  \log_a(mn) & = \log_am + \log_an \\
  37  \log_a({m \over n}) & = \log_am - \log_a \\
  38  \log_a(m^p) & = p\log_am \\
  39  \log_a(m^{-1}) & = -\log_am \\
  40  \log_a1 = 0 & \text{ and } \log_aa = 1 \\
  41  \log_b c &= {{\log_a c} \over {\log_a b}}
  42\end{split}\end{equation*}
  43
  44## Inverse functions
  45
  46For $f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=a^x$, inverse is:
  47
  48$$f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=\log_ax$$
  49
  50## Exponentials
  51
  52$$e^x \quad \text{natural exponential function}$$
  53
  54$$e= \lim_{n \rightarrow \infty} (1 + {1 \over n})^n$$
  55
  56## Modelling
  57
  58$$A = A_0 e^{kt}$$
  59
  60- $A_0$ is initial value
  61- $t$ is time taken
  62- $k$ is a constant
  63- For continuous growth, $k > 0$
  64- For continuous decay, $k < 0$
  65
  66\columnbreak
  67
  68## Graphing exponential functions
  69
  70$$f(x)=Aa^{k(x-b)} + c, \quad \vert \> a > 1$$
  71
  72- **$y$-intercept** at $(0, A \cdot a^{-kb}+c)$ as $x \rightarrow \infty$
  73- **horizontal asymptote** at $y=c$
  74- **domain** is $\mathbb{R}$
  75- **range** is $(c, \infty)$
  76- dilation of factor $|A|$ from $x$-axis
  77- dilation of factor $1 \over k$ from $y$-axis
  78
  79![](graphics/exponential-graphs.png){#id .class width=30%} 
  80
  81## Graphing logarithmic functions
  82
  83$\log_e x$ is the inverse of $e^x$ (reflection across $y=x$)
  84
  85$$f(x)=A \log_a k(x-b) + c$$
  86
  87where
  88
  89- **domain** is $(b, \infty)$
  90- **range** is $\mathbb{R}$
  91- **vertical asymptote** at $x=b$
  92- $y$-intercept exists if $b<0$
  93- dilation of factor $|A|$ from $x$-axis
  94- dilation of factor $1 \over k$ from $y$-axis
  95
  96![](graphics/log-graphs.png){#id .class width=30%} 
  97
  98## Finding equations
  99
 100\colorbox{cas}{On CAS:} ![](graphics/cas-simultaneous.png){#id .class width=75px}