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

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   6author: Andrew Lorimer
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   9
  10\pagenumbering{gobble}
  11
  12
  13# Methods - Calculus
  14
  15## Average rate of change
  16
  17$$m \operatorname{of} x \in [a,b] = {{f(b)-f(a)}\over {b - a}} = {dy \over dx}$$
  18
  19On CAS: (Action|Interactive) $\rightarrow$ Calculation $\rightarrow$ Diff $\rightarrow$ $f(x)$ or $y=\dots$
  20
  21## Instantaneous rate of change
  22
  23**Secant** - line passing through two points on a curve  
  24**Chord** - line segment joining two points on a curve
  25
  26Estimated by using two given points on each side of the concerned point.
  27
  28## Limits & continuity
  29
  30### Limit theorems
  31
  321. For constant function $f(x)=k$, $\lim_{x \rightarrow a} f(x) = k$
  332. $\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G$
  343. $\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G$
  354. ${\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0$
  36
  37A function is continuous if $L^-=L^+=f(x)$ for all values of $x$.
  38
  39## First principles derivative
  40
  41$$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$
  42
  43Not differentiable at:
  44
  45- discontinuous points
  46- sharp point/cusp
  47- vertical tangents ($\infty$ gradient)
  48
  49## Tangents & gradients
  50
  51**Tangent line** - defined by $y=mx+c$ where $m={dy \over dx}$  
  52**Normal line** - $\perp$ tangent ($m_{{tan}} \cdot m_{\operatorname{norm}} = -1$)  
  53**Secant** $={{f(x+h)-f(x)} \over h}$
  54
  55$$\tan \theta = m = f^\prime (x)$$
  56
  57where $\theta$ is the angle that tangent line makes with +ve direction of $x$-axis
  58
  59## Strictly increasing
  60
  61- $f$ is **strictly increasing** where $f(x_2) > f(x_1)$ and $x_2 > x_1$
  62- $f$ is **strictly decreasing** where $f(x_2) < f(x_1)$ and $x_2 > x_1$
  63- If $f^\prime (x) > 0$ for all $x$ in interval, then $f$ is **strictly increasing**
  64- If $f^\prime(x) < 0$ for all $x$ in interval, then $f$ is **strictly decreasing**
  65- Endpoints are included, even where gradient $=0$
  66
  67### Solving on CAS
  68
  69**In main**: type function. Interactive $\rightarrow$ Calculation $\rightarrow$ Line $\rightarrow$ (Normal | Tan line)  
  70**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.
  71
  72## Stationary points
  73
  74Stationary where $m=0$.  
  75Find derivative, solve for ${dy \over dx} = 0$
  76
  77![](graphics/stationary-points.png){#id .class width=50%}
  78
  79**Local maximum at point $A$**  
  80- $f^\prime (x) > 0$ left of $A$
  81- $f^\prime (x) < 0$ right of $A$
  82
  83**Local minimum at point $B$**  
  84- $f^\prime (x) < 0$ left of $B$
  85- $f^\prime (x) > 0$ right of $B$
  86
  87**Stationary** point of inflection at $C$
  88
  89## Function derivatives
  90
  91
  92| $f(x)$ | $f^\prime(x)$ |
  93| ------ | ------------- |
  94| $x^n$  | $nx^{n-1}$ |
  95| $kx^n$ | $knx^{n-1}$ |
  96| $g(x) + h(x)$ | $g^\prime (x) + h^\prime (x)$ |
  97| $c$    | $0$ |
  98| ${u \over v}$ | ${{v{du \over dx} - u{dv \over dx}} \over v^2}$ |
  99| $uv$ | $u{dv \over dx} + v{du \over dx}$ |
 100| $f \circ g$ | ${dy \over du} \cdot {du \over dx}$ |
 101