1.03.GrowthRates - Growth Authors: Bill Davis, Horacio...

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Unformatted text preview: Growth Authors: Bill Davis, Horacio Porta and Jerry Uhl 1996-2007 Publisher: Math Everywhere, Inc. Version 6.0 1.03 Instantaneous Growth Rates BASICS B.1) Instantaneous growth rates Here is a friendly function f @ x D = 1 + 2 x 3- x 4 and a plot: Clear @ f, x D ; f @ x _ D = 1 + 2 x 3- x 4 ; fplot = Plot @ f @ x D , 8 x,- 1, 2 < , PlotStyle 88 Thickness @ 0.01 D , Blue << , AspectRatio-> 1 GoldenRatio, AxesLabel 8 "x" , "" <D- 1.0- 0.5 0.5 1.0 1.5 2.0 x- 2- 1 1 2 B.1.a.i) Measure the net growth of f @ x D = 1 + 2 x 3- x 4 over the interval @- 1, 2 D . Then measure the average growth rate of f @ x D over the interval @- 1, 2 D . Answer: Here you go: Over the interval @- 1, 2 D , the function starts out at: f @- 1 D- 2 And it ends up at: f @ 2 D 1 Its net growth is: fgrowth = f @ 2 D- f @- 1 D 3 Its average growth rate in units on the y-axis per unit on the x-axis over the interval @- 1, 2 D is: xgrowth = 2- H- 1 L 3 fgrowth xgrowth 1 As x grows from - 1 to 2 , on the average, f @ x D grows at a rate of 1 unit every time x grows by 1 unit. The average growth rate of f @ x D over the interval @- 1, 2 D is 1 . B.1.a.ii) Measure the average growth rate of f @ x D = 1 + 2 x 3- x 4 over the interval @ x, x + 0.5 D . Interpret the result. Answer: Clear @ f, x D ; f @ x _ D = 1 + 2 x 3- x 4 1 + 2 x 3- x 4 Over the interval @ x, x + 0.5 D , the function starts out at: f @ x D 1 + 2 x 3- x 4 And it ends up at: f @ x + 0.5 D 1 + 2 H 0.5 + x L 3- H 0.5 + x L 4 Its net growth over the interval @ x, x + 0.5 D is: fgrowth = Expand @ f @ x + 0.5 D- f @ x DD 0.1875 + 1. x + 1.5 x 2- 2. x 3 Its average growth rate in units on the y-axis per unit on the x-axis over the interval @ x, x + 0.5 D is: xgrowth = 0.5; Clear @ fAverageGrowthRate, x D ; fAverageGrowthRate @ x _ D = Expand B fgrowth xgrowth F 0.375 + 2. x + 3. x 2- 4. x 3 On the interval @ x, x + 0.5 D , the average growth rate of f @ x D is 0.375 + 2 x + 3 x 2- 4 x 3 . For instance, when you look at: fAverageGrowthRate @ D 0.375 then you see that, on the average,, f @ x D grows 0.375 times as fast as x grows as x advances from to + 0.5 = 0.5 . But when you look at: fAverageGrowthRate @ 1.5 D- 3.375 Then you see that, on the average, f @ x D goes down 3.375 times as fast as x grows as x advances from 1.5 to 1.5 + 0.5 = 2 . B.1.a.iii) Given a positive number h , measure the average growth rate of f @ x D = 1 + 2 x 3- x 4 over the interval @ x, x + h D . Interpret the result. Answer: Clear @ f, x D ; f @ x _ D = 1 + 2 x 3- x 4 1 + 2 x 3- x 4 Over the interval @ x, x + h D , the function starts out at: f @ x D 1 + 2 x 3- x 4 And it ends up at: Clear @ h D ; Expand @ f @ x + h DD 1 + 2 h 3- h 4 + 6 h 2 x- 4 h 3 x + 6 h x 2- 6 h 2 x 2 + 2 x 3- 4 h x 3- x 4 Its net growth over the interval @ x, x + h D is: fgrowth = Expand @ f @ x + h D- f @ x DD 2 h 3- h 4 + 6 h 2 x- 4 h 3 x + 6 h x 2- 6 h 2 x 2- 4 h x 3 Its average growth rate in units on the y-axis per unit on the x-axis over the interval...
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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.

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1.03.GrowthRates - Growth Authors: Bill Davis, Horacio...

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