mp10610.tex (2732B)
1 \documentclass{article} 2 \usepackage[margin=0.25in]{geometry} 3 \usepackage{pgfplots} 4 \pgfplotsset{width=10cm,compat=1.9} 5 6 \newcommand{\Plot}[2] { 7 \begin{center} 8 \textbf{#2} 9 \end{center} 10 11 \begin{center} 12 \begin{tikzpicture} 13 \begin{axis}[ 14 axis lines = middle, 15 xmin = -5, 16 xmax = 5, 17 ymin = -5, 18 ymax = 5, 19 ] 20 21 \addplot[color=red]{ #1 }; 22 \end{axis} 23 \end{tikzpicture} 24 \end{center} 25 } 26 \newcommand{\Cmpplot}[3] { 27 \begin{center} 28 \textbf{#3} 29 \end{center} 30 31 \begin{center} 32 \begin{tikzpicture} 33 \begin{axis}[ 34 axis lines = middle, 35 xmin = -5, 36 xmax = 5, 37 ymin = -5, 38 ymax = 5, 39 ] 40 41 \addplot[color=red, samples=100]{ #1 }; 42 \addplot[color=blue, samples=100]{ #2 }; 43 \end{axis} 44 \end{tikzpicture} 45 \end{center} 46 } 47 48 49 50 \begin{document} 51 \begin{center} 52 53 \section{Exponential functions} 54 Exponential functions are functions that in some way involve \( e^x \). 55 \Plot{exp(x)}{A graph of \(e^x\)} 56 Note the fact that they never cross the X axis. They can be manipulated just 57 like any other graph. 58 \Plot{(exp(x)) + 1}{A graph of \(e^x + 1\)} 59 \(e^x\) is a special function, this is because \(\frac{dy}{dx}\) is still 60 \(e^x\) 61 \newpage 62 63 \section{Logarithmic functions} 64 Logarithmic functions involve log in some way. Log is defined such that 65 \\ 66 \begin{math} 67 \log_2 8 = 3 68 \end{math} 69 \\ 70 is true. 71 72 ln is a special function that defines the log of base e, it can be written 73 like so 74 \\ 75 \begin{math} 76 \log_e x = ln(x) 77 \end{math} 78 \\ 79 The functions \(exp(x)\) and \(ln(x)\) are linked such that \(ln(x)\) 80 is a reflection of \(exp(x)\) on the line \(y = x\) 81 \Cmpplot{exp(x)}{ln(x)}{A graph comparing \(exp(x)\) and \(ln(x)\)} 82 83 \section{Even and odd functions} 84 An even function is a function that satisfies the equation 85 \\ 86 \begin{math} 87 f(x) = f(-x) 88 \end{math} 89 \\ 90 An example of this is the function \(x^2\), as a value of 91 \(-x\) has the same output as its corresponding \(x\). 92 93 An odd function is a function that satisfies the equation 94 \\ 95 \begin{math} 96 f(-x) = -f(x) 97 \end{math} 98 \\ 99 An example of this is \(y = x\), lets say \(x = 1\), 100 this means that \(-x = -1\); if we substitute \(x\) into \(y = f(-x)\) 101 then \(y = -1\). If we then substitute \(x\) into \(y = -f(x)\) then 102 \(y = -1\); we can see that y is the same either way and is thus an odd 103 function. 104 105 Some functions may not be odd or even. 106 107 A function can be split into its odd and even components using the following 108 formula 109 \\ 110 \begin{math} 111 f_o (x) = \frac{f(x) - f(x)}{2} 112 \end{math} \\ 113 \begin{math} 114 f_e (x) = \frac{f(x) + f(x)}{2} 115 \end{math} 116 \\ 117 The expression 118 \begin{math} 119 f_e (x) + f_o (x) = f(x) 120 \end{math} 121 \\ 122 is always true. 123 124 \end{center} 125 \end{document}