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Unformatted text preview: Growth Authors: Bill Davis, Horacio Porta and Jerry Uhl ©19962007 Publisher: Math Everywhere, Inc. Version 6.0 1.07 The Race Track Principle BASICS B.1) The Race Track Principle You've been using implicit versions of the Race Track Principle all along your Calculus& Mathematica odyssey. This problem gives you a chance to toss around this basic calculus idea. · B.1.a) One version of the Race Track Principle: · Horses: If two horses start a race at the same point, then the faster horse is always ahead. · Functions: If f @ a D = g @ a D and f' @ x D ¥ g' @ x D for x ¥ a , then f @ x D ¥ g @ x D for x ¥ a . This version of the Race Track Principle is good for explaining why one function plots out above another function. · B.1.a.i) Here is a plot of f @ x D = x and g @ x D = Sin @ x D for § x § 3 . Clear @ f, g, x D ; f @ x _ D = x; g @ x _ D = Sin @ x D ; Plot @8 f @ x D , g @ x D< , 8 x, 0, 3 < , PlotStyle Ø 88 Red, Thickness @ 0.01 D< , Thickness @ 0.01 D< , AxesLabel Ø 8 "x" , "" <D 0.5 1.0 1.5 2.0 2.5 3.0 x 0.5 1.0 1.5 2.0 2.5 3.0 That's f @ x D = x sailing high above g @ x D = Sin @ x D . Use the Race Track Principle to explain why the plot turned out this way. · Answer: Look at f @ D and g @ D : 8 f @ D , g @ D< 8 0, 0 < The two functions start their race at the same point. Next look at f' @ x D and g' @ x D for x ¥ : 8 f' @ x D , g' @ x D< 8 1, Cos @ x D< Because Cos @ x D spends its miserable life oscillating between  1 and 1 , you know that 1 ¥ Cos @ x D no matter what x is. Consequently, f' @ x D = 1 ¥ Cos @ x D = g' @ x D for x ¥ 0. In other words, f @ x D = x grows faster than g @ x D = Sin @ x D . The Race Track Principle tells you that f @ x D ¥ g @ x D for x ¥ 0. And this explains the plot. · B.1.a.ii) Here's a plot of f @ x D = H 1 + x L 3 ê 2 and g @ x D = 1 + H 3 ê 2 L x for § x § 24 . Clear @ f, g, x D ; f @ x _ D = H 1 + x L 3 ê 2 ; g @ x _ D = 1 + 3 x 2 ; Plot @8 f @ x D , g @ x D< , 8 x, 0, 24 < , PlotStyle Ø 88 Red, Thickness @ 0.01 D< , Thickness @ 0.01 D< , AxesLabel Ø 8 "x" , "" <D 5 10 15 20 x 20 40 60 80 100 120 That's f @ x D = H 1 + x L 3 ê 2 sailing high above g @ x D = 1 + H 3 ê 2 L x. Use the Race Track Principle to explain why the plot turned out this way. · Answer: Look at f @ D and g @ D : 8 f @ D , g @ D< 8 1, 1 < The two functions start their race at the same point. Next look at f' @ x D and g' @ x D for x ¥ : 8 f' @ x D , g' @ x D< : 3 1 + x 2 , 3 2 > When x ¥ , Sqrt @ 1 + x D ¥ Sqrt @ 1 + D = 1. Consequently f' @ x D ¥ g' @ x D for x ¥ 0. In other words, f @ x D grows faster than g @ x D . The Race Track Principle tells you that f @ x D ¥ g @ x D for x ¥ 0. And this explains the plot....
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This note was uploaded on 10/11/2011 for the course MATH 231 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.
 Spring '08
 Staff
 Math

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