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Homework Homework Assignment #23 Read Section 4.3 Page 227, Exercises: 1 – 77 (EOO), skip 57, 69 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 227 1. The following questions refer to Figure 15. (a) How many critical points does f (x) have? Three, x = {3, 5, 7} (b) What is the maximum value of f (x) on [0, 8]? 6 is the maximum value of f (x) on [0, 8] (c) What are the local maximum values of f (x)? 6 at x = 0, 5 at x = 5, and 4 at x = 8 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 227 1. (d) Find a closed interval on which both the minimum and maximum occur at critical points. [2, 6] or [4, 8] (e) Find an interval on which the minimum occurs at an endpoint. [0, 3], [3, 4], [4, 7], or [7, 8] Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 227 Find all critical points of the function. 5. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 227 Find all critical points of the function. 9. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 227 Find all critical points of the function. 13. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 227 17. Find the minimum value of. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 227 21. Plot f (x) = ln x – 5 sin x on [0, 2π] and approximate both the critical points and the extreme values. The critical points are x = {0.204, 1.431, 4.754} Relative maximum y(0.204) = –2.603 Absolute minimum y(1.430) = –4.593 Absolute maximum y(4.754) = 6.555 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 227 Find the minimum and maximum values of the function on the given interval. 25. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 227 Find the minimum and maximum values of the function on the given interval. 29. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 227 Find the minimum and maximum values of the function on the given interval. 33. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 227 Find the minimum and maximum values of the function on the given interval. 37. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 227 Find the minimum and maximum values of the function on the given interval. 41. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 227 Find the minimum and maximum values of the function on the given interval. 45. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 227 Find the minimum and maximum values of the function on the given interval. 49. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 227 Find the minimum and maximum values of the function on the given interval. 53. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 227 Find the critical points and the extreme values on [0, 3]. Refer to Figure 18. 61. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 227 Verify Rolle’s Theorem for the given interval. 65. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 227 73. Migrating fish tend to swim at a velocity that minimizes the total expenditure of energy E. According to one model, E is proportional to where v r is the velocity of the river water. (a) Find the critical points of f (v). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 227 73. (b) Choose a value of v r, (say v r = 10) and plot f (v). Confirm that f (v) has a minimum value at the critical point. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 227 Plot the function and find the critical points and extreme values on [–5, 5]. 77. 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 4.3: The Mean Value Theorem and Monotonicity Jon Rogawski Calculus, ET First Edition

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company If a function is defined over a closed interval, then there is some point x = c on the interval such that the slope at point c equals the slope of the secant line joining the end points of the interval. This is known as the Mean Value Theorem. Theorem 1 is illustrated in Figure 1.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The Mean Value Theorem set forth on the previous slide is a generalization of Rolle’s Theorem from the previous section. A corollary to the Mean Value Theorem is:

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Example, Page236 Find a point c satisfying the conclusion of the MVT for the given function and the interval. 2. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company A function f (x) is monotonic if it is strictly increasing or strictly decreasing on some interval (a, b).

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The derivative of the function on the left of Figure 3 would be positive and the derivative of the function on the right of Figure 3 would be negative, as noted in Theorem 2

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company For which intervals is f (x) as graphed in Figure 5 increasing? Decreasing? What happens to the derivative at the point the function changes from decreasing to increasing?

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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 Figure 7 further illustrates the connection between the graphs of f (x) and f ′(x).

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company What we observed in Figures 6 and 7 lead us to Theorem 3. Not stated, but implied in Theorem 3 is that the sign of f ′(x) may change at a critical point, but it may not change anywhere in the interval between two consecutive critical points.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 8 shows how the critical points of y = f ′(x) correspond to the local minimum and maximums of y = f (x).

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The chart below is a variation of a sign chart used to analyze the behavior of a function on different intervals The sign chart might also be drawn as below.

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The sign charts on the previous slide both show that the function has: A relative maximum at π/6 because f ′(x) changes from increasing to decreasing at x = π/6. A relative minimum at π/2 because f ′(x) changes from decreasing to increasing at x = π/2. A relative maximum at 5π/6 because f ′(x) changes from increasing to decreasing at x = 5π/6. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company If we were asked to justify that a relative maximum occurs at x = π/6, we could say: “ A local maximum occurs at x = π/6 by the First Derivative Test since the sign of f ′(x) changes from positive to negative at x = π/6.”

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company A critical point without an accompanying change of sign of the derivative is neither a minimum nor a maximum, as shown in Figure 9.

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Example, Page236 24. Figure 13 shows the graph of the derivative f ′(x) of a function f (x). Find the critical points of f (x) and determine whether they are local minima, maxima, or neither. 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 The table below summarizes the significance of the sign change of f ′(x) at a critical point. We always evaluate the sign change in the direction of increasing values of x.

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Example, Page236 Find the critical points and the intervals on which the function is increasing or decreasing, and apply the First Derivative Test to each critical point.. 30. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Example, Page236 Find the critical points and the intervals on which the function is increasing or decreasing, and apply the First Derivative Test to each critical point.. 38. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework Homework Assignment #24 Read Section 4.4 Page 2236, Exercises: 1 – 61 (EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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