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Unformatted text preview: Accumulation Authors: Bill Davis, Horacio Porta and Jerry Uhl ©19962007 Publisher: Math Everywhere, Inc. Version 6.0 2.02 Breaking the Code of the Integral: The Fundamental Formula BASICS B.1) The fundamental theorem: If f @ x D is given by f @ x D = Ÿ a x g @ t D ‚ t , then f' @ x D = g @ x D · B.1.a.i) Experienced integral watchers know how to break the code of the integral. The first step toward this is to learn how to calculate f £ @ x D when f @ x D is given by f @ t D = Ÿ a t g @ x D „ x for some other function g @ x D . Go with the specific case of f @ x D = Ÿ a x g @ t D „ t with g @ x D = xSin @ 3 x D + 1 and a = 1. and look at this plot of g @ x D for a § x § b = 5: a = 1; b = 5; Clear @ f, g, x, t D ; g @ x _ D = x Sin @ 3 x D + 1; gplot = Plot @ g @ x D , 8 x, a, b < , PlotStyle Ø 88 Red, Thickness @ 0.01 D<< , AxesLabel Ø 8 "x" , "g @ x D " <D 2 3 4 5 x 2 2 4 g @ x D Now look at a plot of f @ x D = Ÿ a x g @ t D „ t for a § x § b: f @ x _ D = ‡ a x g @ t D „ t; fplot = Plot @ f @ x D , 8 x, a, b < , PlotStyle Ø 88 Blue, Thickness @ 0.02 D<< , AxesLabel Ø 8 "x" , "f @ x D " <D 2 3 4 5 x 1 2 3 4 5 f @ x D What does f @ x D measure? · Answer: For example, f @ 2.7 D measures the signed area between the xaxis and the g @ x D curve for a § x § 2.7. Take a look: end = 2.7; Plot @ g @ x D , 8 x, a, end < , AxesLabel> 8 "x" , "g @ x D " < , Filling> Axis, AxesOrigin> 8 1, 0 <D The signed area you are looking at measures out to f @ 2.7 D square units: f @ end D 1.68128 On the otherhand, f @ 1.75 D measures the signed area between the xaxis and the g @ x D curve for a § x § 1.75. Take a look: end = 1.75; Plot @ g @ x D , 8 x, a, end < , AxesLabel> 8 "x" , "g @ x D " < , Filling> Axis, AxesOrigin> 8 1, 0 <D The signed area you are looking at measures out to f @ 1.75 D square units: f @ end D 0.0101688 Get it? · B.1.a.ii) The fundamental theorem: If f @ x D = Ÿ a x g @ t D ‚ t , then f ¢ @ x D = g @ x D . Stay with the same f @ x D and g @ x D as in part i) and look at this plot of both g @ x D (red) and f @ x D = Ÿ a x g @ t D „ t (blue): Show @8 fplot, gplot < , AxesLabel> 8 "x" , "" < , PlotRange> All D 2 3 4 5 x 2 2 4 Describe what you see, paying special attention to what f @ x D = Ÿ a x g @ t D „ t (thick) is doing when g @ x D (thin) is positive and to what f @ x D = Ÿ a x g @ t D „ t is doing when g @ x D is negative. What clue does this give you about the relationship between f' @ x D and g @ x D ? · Answer: Take another look: Show @8 fplot, gplot < , AxesLabel> 8 "x" , "" < , PlotRange> All D 2 3 4 5 x 2 2 4 The g @ x D curve is red and thin. The f @ x D curve is blue and thick....
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