1. Graphs and the Derivative
Chapter 5
Graphs and the Derivative

2. Increasing and Decreasing Functions
Section 5.1
Increasing and Decreasing Functions

3. Figure 2

5. Your Turn 1
Find where the function is increasing and decreasing. Solution: Moving from left to right, the function is decreasing for x-values up to −1, then increasing for x-values from to −1 to 2. For x-values from 2 to 4, decreasing and increasing for all x-values larger than 4. In interval notation, the function is increasing on (−1, 2) and (4, ∞). Function is decreasing on (−∞, −1) and (2,4).

6. Figure 4 - 5

10. Figure 6

11. Figure 7

12. Your Turn 2
Find the intervals in which is increasing or decreasing. Graph the function. Solution: Here To find the critical numbers, set this derivative equal to 0 and solve the resulting equation by factoring. Continued

13. Your Turn 2 Continued
The tangent line is horizontal at −3 or 5/3. Since there are no values of x where derivative fails to exist, the only critical numbers are −3 and 5/3. To determine where the function is increasing or decreasing, locate −3 and 5/3 on a number line. The number line is divided into three intervals namely, (−∞, −3), ( −3, 5/3), and (5/3, ∞). Now choose any value of x in the interval (−∞, −3), choosing − 4 and evaluating which is negative. Therefore, f is decreasing on (−∞, −3). Continued

14. Your Turn 2 Continued
Now choose any value of x in the interval ( −3, 5/3), choosing x = 0 and evaluating which is positive. Therefore, f is increasing on ( −3, 5/3). Now choose any value of x in the interval (5/3, ∞), choosing x = 2 and evaluating which is negative. Therefore, f is decreasing on (5/3, ∞). Continued

15. Your Turn 2 Continued

16. Figure 8

17. Figure 9

18. Figure 10

19. Figure 14

20. Section 5.2
Relative Extrema

22. Your Turn 1
Identify the x-values of all points where the graph has relative extrema. Solution:

23. Figure 16

24. Figure 17

26. Figure 18

29. Figure 19

30. Figure 20

31. Your Turn 2
Find all relative extrema Solution: Here To find the critical numbers, set this derivative equal to 0 and solve the resulting equation by factoring. Continued

32. Your Turn 2 Continued
The tangent line is horizontal at −3 or 5/3. Since there are no values of x where derivative fails to exist, the only critical numbers are −3 and 5/3. The number line is divided into three intervals namely, (−∞, −3), ( −3, 5/3), and (5/3, ∞). Any number from each of the three intervals can be used as a test point to find the sign of in each interval. Using − 4, 0, and 2 gives the following information. Continued

33. Your Turn 2 Continued
Thus derivative is negative on(−∞, −3), positive on ( −3, 5/3), and negative on (5/3, ∞). Using the first derivative test, this means that the function has a relative maximum of f (5/3) = 670/27 and f has a relative minimum of f (−3) = −26.

34. Figure 21

35. Figure

36. Figure

37. Figure 27

38. Higher Derivatives, Concavity, and the Second Derivative Test
Section 5.3
Higher Derivatives, Concavity, and
the Second Derivative Test

40. Your Turn 1
Solution: To find the second derivative of f (x), find the first derivative, and then take its derivative.

41. Your Turn 2(a)
Solution : Here, using the chain rule,

42. Your Turn 2(b)
Solution: Use the product rule,

43. Your Turn 2(c)
Solution: Here, we need the quotient rule.

44. Figure 28

45. Figure 29

46. Figure 30

47. Figure 31

48. Figure 32

49. Figure 33

51. Figure 34

52. Figure

54. Figure 37

55. Figure 38

57. Your Turn 5
Solution: First, find the points where the derivative is 0. Here Now use the second derivative test. The second derivative is Continued

58. Your Turn 5 Continued
so that by second derivative test, −3 leads to a relative minimum of Also, when x = 4, so that by second derivative test, 4 leads to a relative maximum

59. Figure 39

60. Section 5.4
Curve Sketching

63. Figure 40

64. Figure 41

66. Figure 42

68. Figure 43

69. Figure 44

71. Figure 45

72. Figure 46