2.02.FundamentalFormula

# 2.02.FundamentalFormula - Accumulation Authors Bill Davis...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Accumulation Authors: Bill Davis, Horacio Porta and Jerry Uhl ©1996-2007 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 x-axis 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 x-axis 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....
View Full Document

{[ snackBarMessage ]}

### Page1 / 26

2.02.FundamentalFormula - Accumulation Authors Bill Davis...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online