1.05.Tools - Growth Authors: Bill Davis, Horacio Porta and...

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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 1.05 Using the Tools BASICS B.1) Using the derivative for finding maximum values and minimum values You can tell what happens to f @ x D = e- x 2 I x 4- 2 Sin @ x DM as x leaves x = 1.0 looking at f £ @ 1.0 D : Clear @ f, x D ; f @ x _ D = E- x 2 I x 4- 2 Sin @ x DM ; f' @ 1.0 D 1.57647 Positive. This tells you that f @ x D increases as x leaves 1.0 and advances a little bit. This also tells you that f @ x D decreases as x leaves 1.0 and decreases a little bit. Check with a plot: a = 1.0; localfplot = Plot @ f @ x D , 8 x, a- 0.2, a + 0.2 < , PlotStyle-> 88 Red, Thickness @ 0.01 D<<D ; points = 8 Graphics @8 PointSize @ 0.03 D , Point @8 a, f @ a D<D<D , Graphics @8 PointSize @ 0.03 D , Point @8 a, 0 <D<D< ; line = Graphics @ Line @88 a, 0 < , 8 a, f @ a D<<DD ; Show @ localfplot, points, line, AxesLabel-> 8 "x" , "f @ x D " <D 0.9 1.0 1.1 1.2 x- 0.5- 0.4- 0.3- 0.2- 0.1 f @ x D Yep. Ø As x leaves 1.0 and advances a little bit, then f @ x D goes up. Ø But as x leaves 1.0 and decreases a bit, then f @ x D goes down. This was predicted in advance because f' @ 1.0 D is positive. Now see what f @ x D is doing at x = 1.8 : f' @ 1.8 D- 0.27404 Negative. Ø This tells you that f @ x D decreases as x leaves 1.8 and advances a little bit. Ø This also tells you that f @ x D increases as x leaves 1.8 and decreases a little bit. Check it out with a plot: a = 1.8; localfplot = Plot @ f @ x D , 8 x, a- 0.2, a + 0.2 < , PlotStyle-> 88 Red, Thickness @ 0.01 D<<D ; points = 8 Graphics @8 PointSize @ 0.03 D , Point @8 a, f @ a D<D<D , Graphics @8 PointSize @ 0.03 D , Point @8 a, 0 <D<D< ; line = Graphics @ Line @88 a, 0 < , 8 a, f @ a D<<DD ; Show @ localfplot, points, line, AxesLabel-> 8 "x" , "f @ x D " <D 1.7 1.8 1.9 2.0 x 0.28 0.30 0.32 0.34 f @ x D Lookin' good. Ø As x leaves 1.8 and advances a little bit, then f @ x D goes down. Ø But as x leaves 1.8 and decreases a bit, then f @ x D goes up. This was predicted in advance because f' @ 1.8 D is negative. Try some other selections of x . · B.1.a.i) What's the moral? · Answer: When you go with any old function f @ x D , you can be sure that: Ø If f' @ a D > , then you can increase f @ x D by pushing x slightly to the right of a . Ø If f' @ a D > , then you can decrease f @ x D by pushing x slightly to the left of a . Ø If f' @ a D < , then you can increase f @ x D by pushing x slightly to the left of a . Ø If f' @ a D < , then you can decrease f @ x D by pushing x slightly to the right of a . The upshot: If 8 a, f @ a D< sits at the top of a crest or at the bottom of a dip on the plot of f @ x D , then f' @ a D = ....
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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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1.05.Tools - Growth Authors: Bill Davis, Horacio Porta and...

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