e12b37e76611a8ec1714f7877ef26ab7f979510e
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## Strictly increasing/decreasing
59
60For $x_2$ and $x_1$ where $x_2 > x_1$:
61
62- **strictly increasing** where $f(x_2) > f(x_1)$
63or $f^\prime(x)>0$
64- **strictly decreasing** where $f(x_2) < f(x_1)$
65or $f^\prime(x)<0$
66- Endpoints are included, even where gradient $=0$
67
68\columnbreak
69
70### Solving on CAS
71
72\colorbox{cas}{\textbf{In main}}: type function. Interactive $\rightarrow$ Calculation $\rightarrow$ Line $\rightarrow$ (Normal | Tan line)
73\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.
74
75## Stationary points
76
77Stationary where $m=0$.
78Find derivative, solve for ${dy \over dx} = 0$
79
80\begin{center}
81 \includegraphics[height=3cm]{graphics/stationary-points.png}
82\end{center}
83
84**Local maximum at point $A$**
85
86- $f^\prime (x) > 0$ left of $A$
87- $f^\prime (x) < 0$ right of $A$
88
89**Local minimum at point $B$**
90
91- $f^\prime (x) < 0$ left of $B$
92- $f^\prime (x) > 0$ right of $B$
93
94**Stationary** point of inflection at $C$
95
96## Function derivatives
97
98\definecolor{shade1}{HTML}{ffffff}
99\definecolor{shade2}{HTML}{F0F9E4}
100\rowcolors{1}{shade1}{shade2}
101\begin{tabularx}{\columnwidth}{rX}
102
103 \hline \(f(x)\) & \(f^\prime(x)\) \\ \hline
104
105 \hspace{6em} \(kx^n\) & \(knx^{n-1}\)\tabularnewline
106 \(g(x) \pm h(x)\) & \(g^\prime (x) \pm h^\prime (x)\)\tabularnewline
107 \(c\) & \(0\)\tabularnewline
108 \({u \over v}\) &
109 \({{(v{du \over dx} - u{dv \over dx}}) \div v^2}\)\tabularnewline
110 \(uv\) & \(u{dv \over dx} + v{du \over dx}\)\tabularnewline
111 \(f \circ g\) & \({dy \over du} \cdot {du \over dx}\)\tabularnewline
112 \(\sin ax\) & \(a\cos ax\)\tabularnewline
113 \(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\tabularnewline
114 \(\cos ax\) & \(-a \sin ax\)\tabularnewline
115 \(\cos(f(x))\) & \(f^\prime(x)(-\sin(f(x)))\) \\
116 \(e^{ax}\) & \(ae^{ax}\)\tabularnewline
117 \(\log_e {ax}\) & \(1 \over x\)\tabularnewline
118 \(\log_e f(x)\) & \(f^\prime (x) \over f(x)\)\tabularnewline
119
120 \hline
121
122\end{tabularx}