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

   1---
   2geometry: a4paper, margin=2cm
   3columns: 2
   4author: Andrew Lorimer
   5header-includes:
   6- \usepackage{fancyhdr}
   7- \pagestyle{fancy}
   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---
  14
  15\linespread{3}
  16\pagenumbering{gobble}
  17\renewcommand{\arraystretch}{1.4}
  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\colorbox{cas}{On CAS:} Action $\rightarrow$ Calculation $\rightarrow$ Line $\rightarrow$ `tanLine` or `normal`  
  59
  60## Strictly increasing/decreasing
  61
  62For $x_2$ and $x_1$ where $x_2 > x_1$:
  63
  64- **strictly increasing** where $f(x_2) > f(x_1)$  
  65or $f^\prime(x)>0$
  66- **strictly decreasing** where $f(x_2) < f(x_1)$  
  67or $f^\prime(x)<0$
  68- Endpoints are included, even where gradient $=0$
  69
  70\columnbreak
  71
  72### Solving on CAS
  73
  74\colorbox{cas}{\textbf{In main}}: type function. Interactive $\rightarrow$ Calculation $\rightarrow$ Line $\rightarrow$ (Normal | Tan line)  
  75\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.
  76
  77## Stationary points
  78
  79Stationary where $m=0$.  
  80Find derivative, solve for ${dy \over dx} = 0$
  81
  82\begin{center}
  83  \includegraphics[height=3cm]{graphics/stationary-points.png}
  84\end{center}
  85
  86**Local maximum at point $A$**
  87
  88- $f^\prime (x) > 0$ left of $A$
  89- $f^\prime (x) < 0$ right of $A$
  90
  91**Local minimum at point $B$**
  92
  93- $f^\prime (x) < 0$ left of $B$
  94- $f^\prime (x) > 0$ right of $B$
  95
  96**Stationary** point of inflection at $C$
  97
  98## Function derivatives
  99
 100\definecolor{shade1}{HTML}{ffffff}
 101\definecolor{shade2}{HTML}{F0F9E4}
 102\rowcolors{1}{shade1}{shade2}
 103\begin{tabularx}{\columnwidth}{rX}
 104  
 105  \hline \(f(x)\) & \(f^\prime(x)\) \\ \hline
 106
 107  \hspace{6em} \(kx^n\) & \(knx^{n-1}\)\tabularnewline
 108  \(g(x) \pm h(x)\) & \(g^\prime (x) \pm h^\prime (x)\)\tabularnewline
 109  \(c\) & \(0\)\tabularnewline
 110  \({u \over v}\) &
 111  \({{(v{du \over dx} - u{dv \over dx}}) \div v^2}\)\tabularnewline
 112  \(uv\) & \(u{dv \over dx} + v{du \over dx}\)\tabularnewline
 113  \(f \circ g\) & \({dy \over du} \cdot {du \over dx}\)\tabularnewline
 114  \(\sin ax\) & \(a\cos ax\)\tabularnewline
 115  \(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\tabularnewline
 116  \(\cos ax\) & \(-a \sin ax\)\tabularnewline
 117  \(\cos(f(x))\) & \(f^\prime(x)(-\sin(f(x)))\) \\
 118  \(e^{ax}\) & \(ae^{ax}\)\tabularnewline
 119  \(\log_e {ax}\) & \(1 \over x\)\tabularnewline
 120  \(\log_e f(x)\) & \(f^\prime (x) \over f(x)\)\tabularnewline
 121  
 122  \hline
 123
 124\end{tabularx}