1 / 64

ESSENTIAL CALCULUS CH03 Applications of differentiation

ESSENTIAL CALCULUS CH03 Applications of differentiation. 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.

kaori
Download Presentation

ESSENTIAL CALCULUS CH03 Applications of differentiation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related