ESSENTIAL CALCULUS CH03 Applications of differentiation

1. ESSENTIAL CALCULUSCH03 Applications of differentiation

2. In this Chapter: • 3.1 Maximum and Minimum Values • 3.2 The Mean Value Theorem • 3.3 Derivatives and the Shapes of Graphs • 3.4 Curve Sketching • 3.5 Optimization Problems • 3.6 Newton’s Method • 3.7 Antiderivatives Review

3. Chapter 3, 3.1, P142

4. 1 DEFINITION A function f has an absolute maximum (or global maximum) at c if f(c)≥f(x) for all x in D, where D is the domain of f. The number f(c) is called the maximum value of f on D. Similarly, f has an absolute minimum at c if f(c)≤f(x) for all x in D and the number f(c) is called the minimum value of f on D. The maximum and minimum values of f are called the extreme values of f. Chapter 3, 3.1, P142

5. 2. DEFINITION A function f has a local maximum (or relative maximum) at c if f(c) ≥f(x) when x is near c. [This means that f(c) ≥f(x) for all x in some open interval containing c.] Similarly, f has a local minimum at c if f(c)≤f(x) when x is near c. Chapter 3, 3.1, P143

6. Chapter 3, 3.1, P143

7. Chapter 3, 3.1, P143

8. Chapter 3, 3.1, P143

9. 3. THE EXTREME VALUE THEOREM If f is continuous on a closed interval [a,b] , then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers and d in [a,b] . Chapter 3, 3.1, P143

10. Chapter 3, 3.1, P143

11. Chapter 3, 3.1, P143

12. Chapter 3, 3.1, P143

13. Chapter 3, 3.1, P144

14. Chapter 3, 3.1, P144

15. 4. FERMAT’S THEOREM If f has a local maximum or minimum at c , and if f’(c) exists, then f’(c)=0. Chapter 3, 3.1, P144

16. Chapter 3, 3.1, P145

17. Chapter 3, 3.1, P145

18. 6. DEFINITION A critical number of a function f is a number c in the domain of f such that either f’(0)=0 or f’(c) does not exist. Chapter 3, 3.1, P146

19. 7. If f has a local maximum or minimum at c, then c is a critical number of f. Chapter 3, 3.1, P146

20. THE CLOSED INTERVAL METHOD To find the absolute maximum and minimum values of a continuous function on a closed interval [a,b]: • Find the values of f at the critical numbers of f • in (a,b) ： • 2. Find the values of f at the endpoints of the interval. • 3. The largest of the values from Steps 1 and 2 is the absolute maximum value; the smallest of these values is the absolute minimum value. Chapter 3, 3.1, P146

21. 5-6 ▓Use the graph to state the absolute and local maximum and minimum values of the function. Chapter 3, 3.1, P147

22. Chapter 3, 3.1, P147

23. ROLLE’S THEOREM Let f be a function that satisfies the following three hypotheses: • f is continuous on the closed interval [a,b]. • 2. f is differentiable on the open interval (a,b). • 3. f(a)=f(b) • Then there is a number in (a,b) such that f’(c)=0. Chapter 3, 3.2, P149

24. Chapter 3, 3.2, P150

25. Chapter 3, 3.2, P150

26. Chapter 3, 3.2, P150

27. Chapter 3, 3.2, P150

28. THE MEAN VALUE THEOREM Let f be a function that satisfies the following hypotheses: 1. f is continuous on the closed interval [a,b]. 2. f is differentiable on the open interval (a,b). Then there is a number in (a,b) such that 1 or, equivalently, 2 Chapter 3, 3.2, P151

29. 5. THEOREM If f’(x)=0 for all x in an interval (a,b) , then f is constant on (a,b). Chapter 3, 3.2, P153

30. 7. COROLLARY If f’(x)=g’(x) for all x in an interval (a,b) , then f-g is constant on (a,b); that is, f(x)=g(x)+c where c is a constant. Chapter 3, 3.2, P154

31. 7. Use the graph of f to estimate the values of c that satisfy the conclusion of the Mean Value Theorem for the interval [0,8]. Chapter 3, 3.2, P154

32. Chapter 3, 3.3, P156

33. INCREASING/DECREASING TEST (a) If f’(x)>0 on an interval, then f is increasing on that interval . (b) If f’(x)<0 on an interval, then f is decreasing on that interval. Chapter 3, 3.3, P156

34. THE FIRST DERIVATIVE TEST Suppose that c is a critical number of a continuous function f. • If f’ changes from positive to negative at c, • then f has a local maximum at c. • (b) If f’ changes from negative to positive at c, then f has a local minimum at c. • (c) If f’ does not change sign at c (that is, f’ is positive on both sides of c or negative on both sides), then f has no local maximum or minimum at c. Chapter 3, 3.3, P157

35. Chapter 3, 3.3, P157

36. Chapter 3, 3.3, P157

37. Chapter 3, 3.3, P157

38. Chapter 3, 3.3, P157

39. Chapter 3, 3.3, P158

40. Chapter 3, 3.3, P158

41. Chapter 3, 3.3, P158

42. Chapter 3, 3.3, P158

43. DEFINITION If the graph of f lies above all of its tangents on an interval I, then it is called concave upward on I. If f the graph of lies below all of its tangents on I, it is called concave downward on I. Chapter 3, 3.32, P158

44. Chapter 3, 3.3, P159

45. DEFINITION A point P on a curve y=f(x) is called an inflection point if f is continuous there and the curve changes from concave upward to concave downward or from concave downward to concave upward at P. Chapter 3, 3.2, P159

46. CONCAVITY TEST • If f”(x)>0 for all x in I, then the graph of f is • concave upward on I. • (b) If f”(x)<0 for all x in I, then the graph of f is • concave downward on I. Chapter 3, 3.3, P159

47. THE SECOND DERIVATIVE TEST Suppose f” is continuous near c. • If f’(c)=0 and f”(c)>0, then f has a local minimum at c. • (b)If f’(c)=0 and f”(c)<0 , then f has a local maximum at c. Chapter 3, 3.3, P160

48. 11. In each part state the x-coordinates of the inflection points of f. Give reasons for your answers. (a) The curve is the graph of f. (b) The curve is the graph of f. (c) The curve is the graph of f. Chapter 3, 3.3, P162

49. 12. The graph of the first derivative f’ of a function f is shown. (a) On what intervals is f increasing? Explain. (b) At what values of x does f have a local maximum or minimum? Explain. (c) On what intervals is f concave upward or concave downward? Explain. (d) What are the x-coordinates of the inflection points of f? Why? Chapter 3, 3.3, P162

50. Chapter 3, 3.3, P162