4478faaa9c9f621770eb0938838f4f8d1bc470f0
1\section{Calculus}
2
3\subsection*{Average rate of change}
4
5\[m \operatorname{of} x \in [a,b] = \dfrac{f(b)-f(a)}{b - a} = \frac{dy}{dx}\]
6
7\colorbox{cas}{On CAS:} Action \(\rightarrow\) Calculation
8\(\rightarrow\) \texttt{diff}
9
10\subsection*{Average value}
11
12\[ f_{\text{avg}} = \dfrac{1}{b-a} \int^b_a f(x) \> dx \]
13
14\subsection*{Instantaneous rate of change}
15
16\textbf{Secant} - line passing through two points on a curve\\
17\textbf{Chord} - line segment joining two points on a curve
18
19\subsection*{Limit theorems}
20
21\begin{enumerate}
22\def\labelenumi{\arabic{enumi}.}
23\tightlist
24\item
25 For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\)
26\item
27 \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\)
28\item
29 \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\)
30\item
31 \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\)
32\end{enumerate}
33
34A function is continuous if \(L^-=L^+=f(x)\) for all values of \(x\).
35
36\subsection*{First principles derivative}
37
38\[f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}\]
39
40Not differentiable at:
41\begin{itemize}
42\tightlist
43\item
44 discontinuous points
45\item
46 sharp point/cusp
47\item
48 vertical tangents (\(\infty\) gradient)
49\end{itemize}
50
51\subsection*{Tangents \& gradients}
52
53\textbf{Tangent line} - defined by \(y=mx+c\) where
54\(m={dy \over dx}\)\\
55\textbf{Normal line} - \(\perp\) tangent
56(\(m_{{tan}} \cdot m_{\operatorname{norm}} = -1\))\\
57\textbf{Secant} \(={{f(x+h)-f(x)} \over h}\)
58
59\colorbox{cas}{On CAS:} \\ Action \(\rightarrow\) Calculation
60\(\rightarrow\) Line \(\rightarrow\) \texttt{tanLine} or \texttt{normal}
61
62\subsection*{Strictly increasing/decreasing}
63
64For \(x_2\) and \(x_1\) where \(x_2 > x_1\):
65
66\begin{itemize}
67\tightlist
68\item
69 \textbf{strictly increasing}\\ where \(f(x_2) > f(x_1)\) or \(f^\prime(x)>0\)
70\item
71 \textbf{strictly decreasing}\\ where \(f(x_2) < f(x_1)\) or \(f^\prime(x)<0\)
72\item
73 Endpoints are included, even where gradient \(=0\)
74\end{itemize}
75
76\columnbreak
77
78\subsubsection*{Solving on CAS}
79
80\colorbox{cas}{\textbf{In main}}: type function. Interactive
81\(\rightarrow\) Calculation \(\rightarrow\) Line \(\rightarrow\) (Normal
82\textbar{} Tan line)\\
83\colorbox{cas}{\textbf{In graph}}: define function. Analysis
84\(\rightarrow\) Sketch \(\rightarrow\) (Normal \textbar{} Tan line).
85Type \(x\) value to solve for a point. Return to show equation for line.
86
87\subsection*{Stationary points}
88
89\begin{align*}
90 \textbf{Stationary point:} && f^\prime(x) &= 0 \\
91 \textbf{Point of inflection:} && f^{\prime\prime} &= 0
92\end{align*}
93
94 \begin{tikzpicture}
95 \begin{axis}[xmin=-21, xmax=21, ymax=1400, ymin=-1000, ticks=none, axis lines=middle]
96 \addplot[color=red, smooth, thick] gnuplot [domain=-15:15,unbounded coords=jump,samples=500] {x^3-3*x^2-144*x+432} node [black, pos=1, right] {\(f(x)\)};
97 \addplot[color=darkgray, dashed, smooth, thick] gnuplot [domain=-15:15,unbounded coords=jump,samples=500] {3*x^2-6*x-144} node [black, pos=1, right] {\(f^\prime(x)\)};
98 \addplot[mark=*, blue] coordinates {(1,286)} node[above right, align=left, font=\footnotesize]{inflection \\ (falling)} ;
99 \addplot[mark=*, orange] coordinates {(-6,972)} node[above left, align=right, font=\footnotesize]{stationary \\ (local max)} ;
100 \addplot[mark=*, orange] coordinates {(8,-400)} node[below, align=left, font=\footnotesize]{stationary \\ (local min)} ;
101 \end{axis}
102 \end{tikzpicture}\\
103 \begin{tikzpicture}
104 \begin{axis}[enlargelimits=true, xmax=3.5, ticks=none, axis lines=middle]
105 \addplot[color=blue, smooth, thick] gnuplot [domain=0.74:3,unbounded coords=jump,samples=500] {(x-2)^3+2} node [black, pos=0.9, left] {\(f(x)\)};
106 \addplot[color=darkgray, dashed, smooth, thick] gnuplot [domain=1:3,unbounded coords=jump,samples=500] {3*(x-2)^2} node [black, pos=0.9, right] {\(f^\prime(x)\)};
107 \addplot[mark=*, purple] coordinates {(2,2)} node[below right, align=left, font=\footnotesize]{stationary \\ inflection} ;
108 \end{axis}
109 \end{tikzpicture}\\
110\pagebreak
111\subsection*{Derivatives}
112
113\definecolor{shade1}{HTML}{ffffff}
114\definecolor{shade2}{HTML}{F0F9E4}
115\rowcolors{1}{shade1}{shade2}
116 \renewcommand{\arraystretch}{1.4}
117 \begin{tabularx}{\columnwidth}{rX}
118 \hline
119 \hspace{6em}\(f(x)\) & \(f^\prime(x)\)\\
120 \hline
121 \(\sin x\) & \(\cos x\)\\
122 \(\sin ax\) & \(a\cos ax\)\\
123 \(\cos x\) & \(-\sin x\)\\
124 \(\cos ax\) & \(-a \sin ax\)\\
125 \(\tan f(x)\) & \(f^2(x) \sec^2f(x)\)\\
126 \(e^x\) & \(e^x\)\\
127 \(e^{ax}\) & \(ae^{ax}\)\\
128 \(ax^{nx}\) & \(an \cdot e^{nx}\)\\
129 \(\log_e x\) & \(\dfrac{1}{x}\)\\
130 \(\log_e {ax}\) & \(\dfrac{1}{x}\)\\
131 \(\log_e f(x)\) & \(\dfrac{f^\prime (x)}{f(x)}\)\\
132 \(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\\
133 \(\sin^{-1} x\) & \(\dfrac{1}{\sqrt{1-x^2}}\)\\
134 \(\cos^{-1} x\) & \(\dfrac{-1}{\sqrt{1-x^2}}\)\\
135 \(\tan^{-1} x\) & \(\dfrac{1}{1 + x^2}\)\\
136 \(\frac{d}{dy}f(y)\) & \(\dfrac{1}{\frac{dx}{dy}}\) \hfill(reciprocal)\\
137 \(uv\) & \(u \frac{dv}{dx}+v\frac{du}{dx}\) \hfill(product rule)\\
138 \(\dfrac{u}{v}\) & \(\dfrac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\) \hfill(quotient rule)\\
139 \(f(g(x))\) & \(f^\prime(g(x))\cdot g^\prime(x)\)\\
140 \hline
141 \end{tabularx}
142 \columnbreak
143\subsection*{Antiderivatives}
144\rowcolors{1}{shade1}{cas}
145 \renewcommand{\arraystretch}{1.4}
146 \begin{tabularx}{\columnwidth}{rX}
147 \hline
148 \(f(x)\) & \(\int f(x) \cdot dx\) \\
149 \hline
150 \(k\) (constant) & \(kx + c\)\\
151 \(x^n\) & \(\dfrac{1}{n+1} x^{n+1}\) \\
152 \(a x^{-n}\) &\(a \cdot \log_e |x| + c\)\\
153 \(\dfrac{1}{ax+b}\) &\(\dfrac{1}{a} \log_e (ax+b) + c\)\\
154 \((ax+b)^n\) & \(\dfrac{1}{a(n+1)}(ax+b)^{n-1} + c\>|\>n\ne 1\)\\
155 \((ax+b)^{-1}\) & \(\dfrac{1}{a}\log_e |ax+b|+c\)\\
156 \(e^{kx}\) & \(\dfrac{1}{k} e^{kx} + c\)\\
157 \(e^k\) & \(e^kx + c\)\\
158 \(\sin kx\) & \(\dfrac{-1}{k} \cos (kx) + c\)\\
159 \(\cos kx\) & \(\dfrac{1}{k} \sin (kx) + c\)\\
160 \(\sec^2 kx\) & \(\dfrac{1}{k} \tan(kx) + c\)\\
161 \(\dfrac{1}{\sqrt{a^2-x^2}}\) & \(\sin^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
162 \(\dfrac{-1}{\sqrt{a^2-x^2}}\) & \(\cos^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\
163 \(\frac{a}{a^2-x^2}\) & \(\tan^{-1} \frac{x}{a} + c\)\\
164 \(\frac{f^\prime (x)}{f(x)}\) & \(\log_e f(x) + c\)\\
165 \(\int f(u) \cdot \frac{du}{dx} \cdot dx\) & \(\int f(u) \cdot du\) \hfill(substitution)\\
166 \(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\
167 \hline
168 \end{tabularx}
169