From 2d7ed436d425e98f540b71df27a12f5fd80fc609 Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Wed, 29 Aug 2018 19:30:34 +1000 Subject: [PATCH 1/1] calculus ref for methods --- methods/calculus-ref.md | 85 +++++++++++++++++++++++++++++++++++++++ methods/calculus-ref.pdf | Bin 0 -> 159642 bytes methods/calculus.md | 2 + 3 files changed, 87 insertions(+) create mode 100644 methods/calculus-ref.md create mode 100644 methods/calculus-ref.pdf diff --git a/methods/calculus-ref.md b/methods/calculus-ref.md new file mode 100644 index 0000000..5eb48c9 --- /dev/null +++ b/methods/calculus-ref.md @@ -0,0 +1,85 @@ +--- +geometry: margin=2cm + +graphics: yes +tables: yes +author: Andrew Lorimer +--- + + +\pagenumbering{gobble} + + +# Methods - Calculus + +## Average rate of change + +$$m \operatorname{of} x \in [a,b] = {{f(b)-f(a)}\over {b - a}} = {dy \over dx}$$ + +Average rate of change between $x=[a,b]$ given two points $P(a, f(a))$ and $Q(b, f(b))$ is the gradient $m$ of line $\overleftrightarrow{PQ}$ + +On CAS: (Action|Interactive) -> Calculation -> Diff -> $f(x)$ or $y=\dots$ + +## Instantaneous rate of change + +Secant - line passing through two points on a curve +Chord - line segment joining two points on a curve + +Estimated by using two given points on each side of the concerned point. Evaluate as in average rate of change. + +## Limits & continuity + +### Limit theorems + +1. For constant function $f(x)=k$, $\lim_{x \rightarrow a} f(x) = k$ +2. $\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G$ +3. $\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G$ +4. ${\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0$ + +A function is continuous if $L^-=L^+=f(x)$ for all values of $x$. + +## First principles derivative + +$$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$ + +## Tangents & gradients + +**Tangent line** - defined by $y=mx+c$ where $m={dy \over dx}$ +**Normal line** - $\perp$ tangent ($m_{\operatorname{tan}} \cdot m_{\operatorname{norm}} = -1$) +**Secant** $={{f(x+h)-f(x)} \over h}$ + +### Solving on CAS + +**In main**: type function. Interactive -> Calculation -> Line -> (Normal | Tan line) +**In graph**: define function. Analysis -> Sketch -> (Normal | Tan line). Type $x$ value to solve for a point. Return to show equation for line. + +## Stationary points + +Stationary where $m=0$. +Find derivative, solve for ${dy \over dx} = 0$ + +![](https://cdn.edjin.com/upload/RESOURCE/IMAGE/78444.png){#id .class width=20%} + +**Local maximum at point $A$** +- $f^\prime (x) > 0$ left of $A$ +- $f^\prime (x) < 0$ right of $A$ + +**Local minimum at point $B$** +- $f^\prime (x) < 0$ left of $B$ +- $f^\prime (x) > 0$ right of $B$ + +**Stationary** point of inflection at $C$ + +## Function derivatives + + +| $f(x)$ | $f^\prime(x)$ | +| ------ | ------------- | +| $x^n$ | $nx^{n-1}$ | +| $kx^n$ | $knx^{n-1}$ | +| $g(x) + h(x)$ | $g^\prime (x) + h^\prime (x)$ | +| $c$ | $0$ | +| ${u \over v}$ | ${{v{du \over dx} - u{dv \over dx}} \over v^2}$ | +| $uv$ | $u{dv \over dx} + v{du \over dx}$ | +| $f \circ g$ | ${dy \over du} \cdot {du \over dx}$ | + diff --git a/methods/calculus-ref.pdf b/methods/calculus-ref.pdf new file mode 100644 index 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\rightarrow 0}{{f(x+h)-f(x)} \over h}$$ **Tangent line** of function $f$ at point $M(a, f(a))$ is the line through $M$ with gradient $f^\prime(a)$. +For $f(x)=x^n, \hspace{0.5em} f^\prime (x) = nx^{n-1}$ + ## Tangents and gradients -- 2.43.2