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

   1---
   2geometry: margin=1cm
   3columns: 2
   4graphics: yes
   5tables: yes
   6author: Andrew Lorimer
   7header-includes:
   8- \usepackage{tabularx}
   9---
  10
  11
  12\pagenumbering{gobble}
  13\renewcommand{\arraystretch}{1.4}
  14
  15
  16# Methods - Calculus
  17
  18## Average rate of change
  19
  20$$m \operatorname{of} x \in [a,b] = {{f(b)-f(a)}\over {b - a}} = {dy \over dx}$$
  21
  22On CAS: Action $\rightarrow$ Calculation $\rightarrow$ Diff $\rightarrow$ ($f(x)$ | $y$) $=\dots$
  23
  24## Instantaneous rate of change
  25
  26**Secant** - line passing through two points on a curve  
  27**Chord** - line segment joining two points on a curve
  28
  29## Limit theorems
  30
  311. For constant function $f(x)=k$, $\lim_{x \rightarrow a} f(x) = k$
  322. $\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G$
  333. $\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G$
  344. ${\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0$
  35
  36A function is continuous if $L^-=L^+=f(x)$ for all values of $x$.
  37
  38## First principles derivative
  39
  40$$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$
  41
  42Not differentiable at:
  43
  44- discontinuous points
  45- sharp point/cusp
  46- vertical tangents ($\infty$ gradient)
  47
  48## Tangents & gradients
  49
  50**Tangent line** - defined by $y=mx+c$ where $m={dy \over dx}$  
  51**Normal line** - $\perp$ tangent ($m_{{tan}} \cdot m_{\operatorname{norm}} = -1$)  
  52**Secant** $={{f(x+h)-f(x)} \over h}$
  53
  54## Strictly increasing
  55
  56- **strictly increasing** where $f(x_2) > f(x_1)$ and $x_2 > x_1$
  57- **strictly decreasing** where $f(x_2) < f(x_1)$ and $x_2 > x_1$
  58- If $f^\prime (x) > 0$ for all $x$ in interval, then $f$ is **strictly increasing**
  59- If $f^\prime(x) < 0$ for all $x$ in interval, then $f$ is **strictly decreasing**
  60- Endpoints are included, even where gradient $=0$
  61
  62### Solving on CAS
  63
  64**In main**: type function. Interactive $\rightarrow$ Calculation $\rightarrow$ Line $\rightarrow$ (Normal | Tan line)  
  65**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.
  66
  67## Stationary points
  68
  69Stationary where $m=0$.  
  70Find derivative, solve for ${dy \over dx} = 0$
  71
  72\begin{center}
  73  \includegraphics[height=3cm]{graphics/stationary-points.png}
  74\end{center}
  75
  76**Local maximum at point $A$**
  77
  78- $f^\prime (x) > 0$ left of $A$
  79- $f^\prime (x) < 0$ right of $A$
  80
  81**Local minimum at point $B$**
  82
  83- $f^\prime (x) < 0$ left of $B$
  84- $f^\prime (x) > 0$ right of $B$
  85
  86**Stationary** point of inflection at $C$
  87
  88## Function derivatives
  89
  90\begin{tabularx}{\columnwidth}{rl}
  91  
  92  \hline \(f(x)\) & \(f^\prime(x)\) \\ \hline
  93
  94  \(kx^n\) & \(knx^{n-1}\)\tabularnewline
  95  \(g(x) \pm h(x)\) & \(g^\prime (x) \pm h^\prime (x)\)\tabularnewline
  96  \(c\) & \(0\)\tabularnewline
  97  \({u \over v}\) &
  98  \({{(v{du \over dx} - u{dv \over dx}}) \div v^2}\)\tabularnewline
  99  \(uv\) & \(u{dv \over dx} + v{du \over dx}\)\tabularnewline
 100  \(f \circ g\) & \({dy \over du} \cdot {du \over dx}\)\tabularnewline
 101  \(\sin ax\) & \(a\cos ax\)\tabularnewline
 102  \(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\tabularnewline
 103  \(\cos ax\) & \(-a \sin ax\)\tabularnewline
 104  \(\cos(f(x))\) & \(f^\prime(x)(-\sin(f(x)))\) \\
 105  \(e^{ax}\) & \(ae^{ax}\)\tabularnewline
 106  \(\log_e {ax}\) & \(1 \over x\)\tabularnewline
 107  \(\log_e f(x)\) & \(f^\prime (x) \over f(x)\)\tabularnewline
 108  
 109  \hline
 110
 111\end{tabularx}