1. Homework Homework Assignment #22 Read Section 4.2 Page 217, Exercises: 1 – 65 (EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

2. Example, Page 217 Use the Linear Approximation to estimate Δf = f (3.02) – f (3) for the given function. 1. f (x) = x 2 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

3. Example, Page 217 Use the Linear Approximation to estimate Δf = f (3.02) – f (3) for the given function. 5. f (x) = e 2x Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

4. Example, Page 217 Estimate Δf using Linear Approximation and use a calculator to compute both the error and the percentage error. 9. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

5. Example, Page 217 Estimate Δf using Linear Approximation and use a calculator to compute both the error and the percentage error. 13. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

6. Example, Page 217 Estimate the quantity using Linear Approximation and find the error using a calculator. 17. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

7. Example, Page 217 Estimate the quantity using Linear Approximation and find the error using a calculator. 21. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

8. Example, Page 217 25. The cube root of 27 is 3. How much larger is the cube root of 27.2? Estimate using Linear Approximation. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

9. Example, Page 217 29. The atmospheric pressure P (kPa) at altitudes h (km) for 11 ≤ h ≤ 15 is approximately P (h) = 128 e –0.157h. (a) Use Linear Approximation to estimate the change in pressure at h = 20 when Δh = 0.5. (b) Compute the actual change and compute the percentage error in the Linear Approximation. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

10. Example, Page 217 33. A stone tossed vertically in the air with initial velocity v ft/s reaches a maximum height h = v 2 /64 ft. (a) Estimate Δh if v is increased from 25 to 26 ft/s. (b) Estimate Δh if v is increased from 30 to 31 ft/s. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

11. Example, Page 217 33. (c) In general, does a 1 ft/s increase in initial velocity cause a greater change in maximum height at low or high initial velocities? Explain. Since the value of the derivative of the height function increases as initial velocity increase, the greater change in maximum height occurs at high initial velocities. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

12. Example, Page 217 37. Estimate the weight loss per mile of altitude gained for a 130-lb pilot. At which altitude would she weigh 129.5 lb? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

13. Example, Page 217 Find the linearization at x = a. 41. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

14. Example, Page 217 Find the linearization at x = a. 45. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

15. Example, Page 217 Find the linearization at x = a. 49. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

16. Example, Page 217 Approximate using linearization and use a calculator to compute percentage error. 53. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

17. Example, Page 217 Approximate using linearization and use a calculator to compute percentage error. 57. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

18. Example, Page 217 Approximate using linearization and use a calculator to compute percentage error. 61. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

19. Example, Page 217 From Newton’s Laws, an object released at angle θ with initial velocity v ft/s, travels a total distance s = 1/32 v 2 sin 2θ ft. 65. Estimate the change in distance s of the shot if the angle changes from 50º to 51º for v = 25 ft/s and v = 30 ft/s. Is the shot more sensitive to the angle when the velocity is large or small? Explain. Work on next slide. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

20. Example, Page 217 65. Continued Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

21. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Jon Rogawski Calculus, ET First Edition Chapter 4: Applications of the Derivative Section 4.2: Extreme Values

22. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The extreme values of a function f (x) on an interval I are the minimum and maximum values of f (x) for If f (x) is continuous on [a, b], then f (x) has a minimum and a maximum value on [a, b]. f (c) is a local maximum if f (x) ≤ f (c) for all x in some open interval around c. Similarly, f (c) is a local minimum if f (x) ≥ f (c) for all x in some open interval around c. c is a critical point of the function f (x) if either f (c) = 0 or f (c) D.N.E.

23. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company A doctor could use a graph like the one in Figure 1 to determine the maximum concentration of a drug in a patient’s bloodstream and the time after dosing at which it occurs. Maximum and minimum values are called extreme values or extrema. The process of finding extrema is optimization.

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

25. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company A discontinuous function may or may not have a min or max. A function defined on an open interval may or may not have a min or max. A continuous function defined on a closed interval will have both a min and a max.

26. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Examples of the local min and max are seen in Figure 3.

27. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company As shown in Figure 4(A), the tangent line at a min or max is horizontal. But in Figure 4(B), there is not tangent at the min, leading to the following definition:

28. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Find the critical points of x 3 – 9x 2 + 24x – 10.

29. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Find the critical points for f (x) =|x|.

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

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

32. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The converse of Fermat’s Theorem, that if c is a critical point, then f (c) is a min or max, is not necessarily true, as illustrated in Figure 8.

33. Example, Page 227 Find all critical points of the function. 6. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

34. Example, Page 227 Find all critical points of the function. 8. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

35. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 9 illustrates a situation where f (6) is a maximum, but 6 is not a critical point. But f (4) is a minimum and 4 is a critical point.

36. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Find the maximum of f (x) = 1 – (x – 1) 2/3 on [–1, 2].

37. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Find the extreme values of f (x) = x 2 – 8lnx on [1, 4].

38. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Rolle’s Theorem is illustrated in Figure 13.

39. Example, Page 227 Find the minimum and maximum values of the function on the given interval. 54. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

40. Example, Page 227 58. Find the critical points of f (x) = 2cos3x +3cos2x. check your answer against a graph of f (x). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

41. Example, Page 227 72. Bees build honeycomb structures out of cells with a hexagonal base and three-rhombus shaped faces on top as in Figure 20. the surface area of the cell is where h, s, and θ are as indicated in the figure. (a) Show that by finding the critical point of A(θ) for 0 ≤ θ ≤ π/2, assuming h and s are constant Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

42. Example, Page 227 72. (b) Confirm by graphing that the critical point indeed minimizes the surface area. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

43. 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