1--- 2geometry: margin=2cm 3<!-- columns: 2 --> 4graphics: yes 5tables: yes 6author: Andrew Lorimer 7--- 8 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 19Average 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 $\overleftrightarrow{PQ}$ 20 21On CAS: (Action|Interactive) -> Calculation -> Diff -> $f(x)$ or $y=\dots$ 22 23## Instantaneous rate of change 24 25Secant - line passing through two points on a curve 26Chord - line segment joining two points on a curve 27 28Estimated by using two given points on each side of the concerned point. Evaluate as in average rate of change. 29 30## Limits & continuity 31 32### Limit theorems 33 341. For constant function $f(x)=k$, $\lim_{x \rightarrow a} f(x) = k$ 352. $\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G$ 363. $\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G$ 374. ${\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0$ 38 39A function is continuous if $L^-=L^+=f(x)$ for all values of $x$. 40 41## First principles derivative 42 43$$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$ 44 45Not differentiable at: 46 47- discontinuous points 48- sharp point/cusp 49- vertical tangents ($\infty$ gradient) 50 51## Tangents & gradients 52 53**Tangent line** - defined by $y=mx+c$ where $m={dy \over dx}$ 54**Normal line** - $\perp$ tangent ($m_{\operatorname{tan}} \cdot m_{\operatorname{norm}} = -1$) 55**Secant** $={{f(x+h)-f(x)} \over h}$ 56 57$$\tan \Theta = m = f^\prime x$$ 58 59where $\Theta$ is the angle that tangent line makes with +ve direction of $x$-axis 60 61## Strictly increasing 62 63- Function $f$ is **strictly increasing** where $f(x_2) > f(x_1)$ and $x_2 > x_1$ 64- Function $f$ is **strictly decreasing** where $f(x_2) < f(x_1)$ and $x_2 > x_1$ 65- If $f^\prime (x) > 0$ for all $x$ in interval, then $f$ is **strictly increasing** 66- If $f^\prime(x) < 0$ for all $x$ in interval, then $f$ is **strictly decreasing** 67- Endpoints are included, even where gradient $=0$ 68 69### Solving on CAS 70 71**In main**: type function. Interactive -> Calculation -> Line -> (Normal | Tan line) 72**In graph**: define function. Analysis -> Sketch -> (Normal | Tan line). Type $x$ value to solve for a point. Return to show equation for line. 73 74## Stationary points 75 76Stationary where $m=0$. 77Find derivative, solve for ${dy \over dx} = 0$ 78 79{#id .class width=20%} 80 81**Local maximum at point $A$** 82- $f^\prime (x) > 0$ left of $A$ 83- $f^\prime (x) < 0$ right of $A$ 84 85**Local minimum at point $B$** 86- $f^\prime (x) < 0$ left of $B$ 87- $f^\prime (x) > 0$ right of $B$ 88 89**Stationary** point of inflection at $C$ 90 91## Function derivatives 92 93 94| $f(x)$ | $f^\prime(x)$ | 95| ------ | ------------- | 96| $x^n$ | $nx^{n-1}$ | 97| $kx^n$ | $knx^{n-1}$ | 98| $g(x) + h(x)$ | $g^\prime (x) + h^\prime (x)$ | 99| $c$ | $0$ | 100| ${u \over v}$ | ${{v{du \over dx} - u{dv \over dx}} \over v^2}$ | 101| $uv$ | $u{dv \over dx} + v{du \over dx}$ | 102| $f \circ g$ | ${dy \over du} \cdot {du \over dx}$ | 103