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

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   2geometry: a4paper, margin=2cm
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   4author: Andrew Lorimer
   5header-includes:
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   8- \fancyhead[LO,LE]{Year 12 Methods}
   9- \fancyhead[CO,CE]{Andrew Lorimer}
  10- \usepackage{graphicx}
  11- \usepackage{tabularx}
  12- \usepackage[dvipsnames, table]{xcolor}
  13---
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  18\definecolor{cas}{HTML}{e6f0fe}
  19
  20# Calculus
  21
  22## Average rate of change
  23
  24$$m \operatorname{of} x \in [a,b] = {{f(b)-f(a)}\over {b - a}} = {dy \over dx}$$
  25
  26\colorbox{cas}{On CAS:} Action $\rightarrow$ Calculation $\rightarrow$ `diff`
  27
  28## Instantaneous rate of change
  29
  30**Secant** - line passing through two points on a curve  
  31**Chord** - line segment joining two points on a curve
  32
  33## Limit theorems
  34
  351. For constant function $f(x)=k$, $\lim_{x \rightarrow a} f(x) = k$
  362. $\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G$
  373. $\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G$
  384. ${\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0$
  39
  40A function is continuous if $L^-=L^+=f(x)$ for all values of $x$.
  41
  42## First principles derivative
  43
  44$$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$
  45
  46Not differentiable at:
  47
  48- discontinuous points
  49- sharp point/cusp
  50- vertical tangents ($\infty$ gradient)
  51
  52## Tangents & gradients
  53
  54**Tangent line** - defined by $y=mx+c$ where $m={dy \over dx}$  
  55**Normal line** - $\perp$ tangent ($m_{{tan}} \cdot m_{\operatorname{norm}} = -1$)  
  56**Secant** $={{f(x+h)-f(x)} \over h}$
  57
  58## Strictly increasing/decreasing
  59
  60For $x_2$ and $x_1$ where $x_2 > x_1$:
  61
  62- **strictly increasing** where $f(x_2) > f(x_1)$  
  63or $f^\prime(x)>0$
  64- **strictly decreasing** where $f(x_2) < f(x_1)$  
  65or $f^\prime(x)<0$
  66- Endpoints are included, even where gradient $=0$
  67
  68\columnbreak
  69
  70### Solving on CAS
  71
  72\colorbox{cas}{\textbf{In main}}: type function. Interactive $\rightarrow$ Calculation $\rightarrow$ Line $\rightarrow$ (Normal | Tan line)  
  73\colorbox{cas}{\textbf{In graph}}: define function. Analysis $\rightarrow$ Sketch $\rightarrow$ (Normal | Tan line). Type $x$ value to solve for a point. Return to show equation for line.
  74
  75## Stationary points
  76
  77Stationary where $m=0$.  
  78Find derivative, solve for ${dy \over dx} = 0$
  79
  80\begin{center}
  81  \includegraphics[height=3cm]{graphics/stationary-points.png}
  82\end{center}
  83
  84**Local maximum at point $A$**
  85
  86- $f^\prime (x) > 0$ left of $A$
  87- $f^\prime (x) < 0$ right of $A$
  88
  89**Local minimum at point $B$**
  90
  91- $f^\prime (x) < 0$ left of $B$
  92- $f^\prime (x) > 0$ right of $B$
  93
  94**Stationary** point of inflection at $C$
  95
  96## Function derivatives
  97
  98\definecolor{shade1}{HTML}{ffffff}
  99\definecolor{shade2}{HTML}{F0F9E4}
 100\rowcolors{1}{shade1}{shade2}
 101\begin{tabularx}{\columnwidth}{rX}
 102  
 103  \hline \(f(x)\) & \(f^\prime(x)\) \\ \hline
 104
 105  \hspace{6em} \(kx^n\) & \(knx^{n-1}\)\tabularnewline
 106  \(g(x) \pm h(x)\) & \(g^\prime (x) \pm h^\prime (x)\)\tabularnewline
 107  \(c\) & \(0\)\tabularnewline
 108  \({u \over v}\) &
 109  \({{(v{du \over dx} - u{dv \over dx}}) \div v^2}\)\tabularnewline
 110  \(uv\) & \(u{dv \over dx} + v{du \over dx}\)\tabularnewline
 111  \(f \circ g\) & \({dy \over du} \cdot {du \over dx}\)\tabularnewline
 112  \(\sin ax\) & \(a\cos ax\)\tabularnewline
 113  \(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\tabularnewline
 114  \(\cos ax\) & \(-a \sin ax\)\tabularnewline
 115  \(\cos(f(x))\) & \(f^\prime(x)(-\sin(f(x)))\) \\
 116  \(e^{ax}\) & \(ae^{ax}\)\tabularnewline
 117  \(\log_e {ax}\) & \(1 \over x\)\tabularnewline
 118  \(\log_e f(x)\) & \(f^\prime (x) \over f(x)\)\tabularnewline
 119  
 120  \hline
 121
 122\end{tabularx}