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

Homework, Page 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

Homework, Page (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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Homework, Page 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

Homework, Page (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

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

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

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.

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:

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

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).

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

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?

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

Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 7 further illustrates the connection between the graphs of f (x) and f ′(x).

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.

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).

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.

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.”

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.

Example, Page 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

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

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.

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

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

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