Section 7.1 Constructing Antiderivatives Analytically Applied Calculus,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved.

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Presentation transcript:

Section 7.1 Constructing Antiderivatives Analytically Applied Calculus,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved

Applied Calculus,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved

Applied Calculus,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved Solution

Applied Calculus,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved

Applied Calculus,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved Example 3 Solution

Applied Calculus,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved

Applied Calculus,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved Solution

Applied Calculus,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved Section 7.2 Integration by Substitution

Applied Calculus,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved Solution (continued on next slide)

Applied Calculus,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved

Applied Calculus,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved (b) Solution for (b)

Applied Calculus,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved Solution

Applied Calculus,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved Section 7.3 Using the Fundamental Theorem to Find Definite Integrals

Applied Calculus,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved Example 3 Solution

Applied Calculus,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved Figure 7.2: Area representation of improper integral Improper Integrals

Applied Calculus,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved Section 7.4 Integration by Parts

Applied Calculus,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved The General Formula for Integration by Parts Integration by Parts Problem 4 Use integration by parts to find Solution Let u = ln y and dv = y dy. Then du = 1/y dy and v = y 2 /2 giving Note that by letting u = ln y, the natural log was eliminated from the second integral. Also, u’ dv was something that could be integrated easily.

Applied Calculus,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved Section 7.5 Analyzing Antiderivatives Graphically and Numerically

Applied Calculus,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved Example

Applied Calculus,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved Example continued

Applied Calculus,4/E, Deborah Hughes-Hallett Copyright 2010 by John Wiley and Sons, All Rights Reserved Problem 2 Figure 7.10 shows the derivative g’. If g (0) = 0, graph g. Give (x, y)-coordinates of all local maxima and minima. Figure 7.10

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