Ch14_exponential function

Ch14_exponential function - Chapter 14 Exponential Function...

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Unformatted text preview: Chapter 14 Exponential Function The function e t . The exponential function is denoted mathematically by e t and in Matlab by exp(t) . This function is the solution to the world’s simplest, and perhaps most important, differential equation, ˙ y = ky This equation is the basis for any mathematical model describing the time evolution of a quantity with a rate of production that is proportional to the quantity itself. Such models include populations, investments, feedback, and radioactivity. We are using t for the independent variable, y for the dependent variable, k for the proportionality constant, and ˙ y = dy dt for the rate of growth, or derivative, with respect to t . We are looking for a function that is proportional to its own derivative. Let’s start by examining the function y = 2 t We know what 2 t means if t is an integer, 2 t is the t-th power of 2. 2- 1 = 1 / 2 , 2 = 1 , 2 1 = 1 , 2 2 = 4 ,... We also know what 2 t means if t = p/q is a rational number, the ratio of two integers, 2 p/q is the q-th root of the p-th power of 2. 2 1 / 2 = √ 2 = 1 . 4142 ..., Copyright c 2009 Cleve Moler Matlab R is a registered trademark of The MathWorks, Inc. TM August 8, 2009 1 2 Chapter 14. Exponential Function 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.5 1 1.5 2 2.5 3 3.5 4 Figure 14.1. The blue curve is the graph of y = 2 t . The green curve is the graph of the rate of growth, ˙ y = dy/dt . 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 Figure 14.2. The ratio, ˙ y/y . 2 5 / 3 = 3 √ 2 5 = 3 . 1748 ..., 2 355 / 113 = 113 √ 2 355 = 8 . 8250 ... In principal, for floating point arithmetic, this is all we need to know. All floating point numbers are ratios of two integers. We do not have to be concerned yet about the definition of 2 t for irrational t . If Matlab can compute powers and roots, we can plot the graph of 2 t , the blue curve in figure 14.1 What is the derivative of 2 t ? Maybe you have never considered this question, or don’t remember the answer. (Be careful, it is not t 2 t- 1 .) We can plot the graph of the approximate derivative, using a step size of something like 0 . 0001. The 3 following code produces figure 14.1, the graphs of both y = 2 t and its approximate derivative, ˙ y . t = 0:.01:2; h = .00001; y = 2.^t; ydot = (2.^(t+h) - 2.^t)/h; plot(t,[y; ydot]) The graph of the derivative has the same shape as the graph of the original function. Let’s look at their ratio, ˙ y ( t ) /y ( t ). plot(t,ydot./y) We see that the ratio of the derivative to the function, shown in figure 14.2, has a constant value, ˙ y/y = 0 . 6932 ... , that does not depend upon t . Now, if you are following along with a live Matlab , repeat the preceeding calculations with y = 3 t instead of y = 2 t . You should find that the ratio is again independent of t . This time ˙ y/y = 1 . 0986 ... . Better yet, experiment with expgui ....
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This note was uploaded on 10/11/2011 for the course MTHSC 365 taught by Professor Adams during the Spring '11 term at Clemson.

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Ch14_exponential function - Chapter 14 Exponential Function...

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