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Homework Homework Assignment #30 Read Section 4.9 Page 282, Exercises: 1 – 13(Odd) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 282 Use Newton’s Method with the given function and initial value x o to calculate x 1, x 2, and x 3. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 282 Use Newton’s Method with the given function and initial value x o to calculate x 1, x 2, and x 3. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 282 5. Use Figure 6 to choose an initial guess x o to the unique real root of x 3 + 2x +5 = 0. Then compute the first three iterates of Newton’s Method. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 282 7. Use Newton’s Method to find two solutions of e x = 5x to three decimal places (Figure 7). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 282 7. Use Newton’s Method to find two solutions of e x = 5x to three decimal places (Figure 7). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 282 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 282 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 282 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Jon Rogawski Calculus, ET First Edition Chapter 4: Applications of the Derivative Section 4.9: Antiderivatives

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company In some cases, we may know the derivative and wish to find the function itself. A function F (x) whose derivative is f (x) is called an antiderivative of f (x). Theorem 1 gives us a method of finding the general antiderivative of a function, namely: Theorem 1 is proven as follows:

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 1 graphically demonstrates how two functions (F (x) and F (x) + C) differing only by a constant C, have the same slope for a given value of x and, thus, the same derivative.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The process of antidifferentiation is denoted by the Leibniz symbol ∫ which specifies finding the indefinite integral of a function. Just as we have rules for differentiation, we also have rules for integration, of which the first is:

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Example, Page 292 Find the general antiderivative of f (x) and check your answer by differentiating. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Example, Page 292 Find the general antiderivative of f (x) and check your answer by differentiating. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company In Figure 2, we see that the slope of the graph of y = ln |x| is x –1 for all x. This leads to Theorem 3, which covers the exception to Theorem 2.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The linearity rules for differentiation have their counterparts for integration as noted in Theorem 4.

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Example, Page 292 Evaluate the indefinite integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Example, Page 292 Evaluate the indefinite integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Example, Page 292 Evaluate the indefinite integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Recalling derivatives of the basic trigonometric functions, we can recognize the basic trigonometric integrals.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Recalling the Chain Rule, we have the following trigonometric integrals.

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Example, Page 292 Evaluate the indefinite integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Example, Page 292 Evaluate the indefinite integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Remembering that the exponential function is its own derivative, we see how the indefinite integral of e x is e x + C. Applying the Chain rule in reverse to the exponential function, we obtain:

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Example, Page 292 Evaluate the indefinite integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework Homework Assignment #31 Review Section 4.9 Page 292, Exercises: 1 – 41(EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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