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Homework Homework Assignment #26 Read Section 4.6 Page 256, Exercises: 1 – 89 (EOO), skip 37, 41 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 1. Determine the sign combinations of f ′ and f ″ for each interval A – G in Figure 20. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 Draw the graph of a function for which f ′ and f ″ take on the given sign combinations. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 9. Sketch the graph of the cubic f (x) = x 3 – 3x 2 + 2. For extra accuracy, plot the zeroes. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 Sketch the graph of the function. Indicate the transition points. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 Sketch the graph of the function. Indicate the transition points. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 Sketch the graph of the function. Indicate the transition points. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 Sketch the graph of the function. Indicate the transition points. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 Sketch the graph of the function. Indicate the transition points. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 29.Continued Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 29.Continued Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 Sketch the graph of the function. Indicate the transition points. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 33.Continued. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 Sketch the graph over the given interval. Indicate transition points. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 49.Suppose f is twice differentiable satisfying (a) f (0) =1, (b) f ' (x) > 0 for all x ≠ 0, and (c) f " (x) 0 for x > 0. Let g (x) = f (x 2 ). (a)Sketch a possible graph of f (x). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 49.Suppose f is twice differentiable satisfying (a) f (0) =1, (b) f ' (x) > 0 for all x ≠ 0, and (c) f " (x) 0 for x > 0. Let g (x) = f (x 2 ). (a)Sketch a possible graph of f (x). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 49.Continued. Let g (x) = f (x 2 ). (b)Prove that g (x) has no point of inflection and a unique local extreme value at x = 0. Sketch a possible graph of g (x). Since g (x) = f (x 2 ) and x 2 > 0 for all x, g′ (x) > 0 for all x and g″ > 0 for all x. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 Calculate the following limits. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 Calculate the following limits. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 Calculate the limit. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 Calculate the limit. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 Calculate the limit. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 73.Match the function with their graphs in Figure 25. (A) (b), (B) (c), (C) (d), (D) (a) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 Sketch the graph of the function. Indicate the asymptotes, local extrema, and points of inflection. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 Sketch the graph of the function. Indicate the asymptotes, local extrema, and points of inflection. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 Sketch the graph of the function. Indicate the asymptotes, local extrema, and points of inflection. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 Sketch the graph of the function. Indicate the asymptotes, local extrema, and points of inflection. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 89.Continued. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 256 89.Continued. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Chapter 4: Applications of the Derivative Section 4.6: Applied Optimization Jon Rogawski Calculus, ET First Edition

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Optimization – the process of finding the “best” solution to a problem, e.g., the largest area that a given length of fence can enclose the path that minimizes travel time the dimensions that maximize the volume of a box made from a rectangle of cardboard The process of optimization usually involves three steps: 1. Choose variables. 2. Determine the function. 3. Optimize the function. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Example: Determine the lengths of the sides of a rectangle, formed by bending a given length wire, which enclose the largest area.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company In deciding the alignment of a driveway, the time of travel to the nearby city (Figure 3) is the primary factor. Figure 4 shows a graph of travel time versus distance x. The optimum value of x is the one that yields the shortest travel time. The critical point for the function gives the optimum distance x.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 5 shows the graph of a function relating bushels per acre of corn to the amount of nitrogen in the fertilizer.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company An infinite number of cylinder dimensions will yield the same volume, but only one will minimize the surface area.

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Example, Page 265 2.A 100-in. piece of wire is cut into two pieces, and each is bent into a square. How should this be done to minimize the sum of the areas of the two squares? (a) Express the sum of the areas of the squares in terms of the lengths x and y of the two pieces. (b) What is the constraint equation relating x and y? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Example, Page 265 2.(c)Does this problem require optimization over an open or closed interval? (d)Solve the optimization problem. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Example, Page 265 4.The legs of a right triangle have lengths a and b satisfying a + b =10. Which values of a and b maximize the area of the triangle? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Example, Page 265 4.Find a good numerical approximation to the coordinates of the point on the graph of y = ln x – x closest to the origin. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Example, Page 265 22.Find the dimensions x and y of the rectangle inscribed in a circle of radius r that maximizes the quantity xy 2. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework Homework Assignment #27 Review Section 4.6 Page 265, Exercises: 1 – 29 (EOO), 19, skip 17 Quiz next time Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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