2. Chapter 1 Section 2 Rate of Change 2

3. Sales of digital video disc (DVD) players have been increasing since they were introduced in early 1998. To measure how fast sales were increasing, we calculate a rate of change of the form: Page 103

4. At the same time, sales of video cassette recorders (VCRs) have been decreasing. See Table 1.11 below:1.11 Year199819992000200120022003 VCR sales (million $) 2409233318691058826407 DVD player sales (million $) 42110991717209724273050 Page 104

5. To calculate the rate of change of DVD players: Average rate of change of DVD player sales (1998-> 2003)= Year199819992000200120022003 VCR sales (million $) 2409233318691058826407 DVD player sales (million $) 42110991717209724273050 Page 105

6. To calculate the rate of change of DVD players: Average rate of change of DVD player sales (1998-> 2003)= Year199819992000200120022003 VCR sales (million $) 2409233318691058826407 DVD player sales (million $) 42110991717209724273050 Page 106

7. To calculate the rate of change of DVD players: Average rate of change of DVD player sales (1998-> 2003)= Year199819992000200120022003 VCR sales (million $) 2409233318691058826407 DVD player sales (million $) 42110991717209724273050 Page 107

8. To calculate the rate of change of DVD players: Average rate of change of DVD player sales (1998-> 2003)= Year199819992000200120022003 VCR sales (million $) 2409233318691058826407 DVD player sales (million $) 42110991717209724273050 Page 108

9. Graphically, here is what we have: Page 109

10. To calculate the rate of change of VCR players: Average rate of change of VCR player sales (1998-> 2003)= Year199819992000200120022003 VCR sales (million $) 2409233318691058826407 DVD player sales (million $) 42110991717209724273050 Page 1010

11. To calculate the rate of change of VCR players: Average rate of change of VCR player sales (1998-> 2003)= Year199819992000200120022003 VCR sales (million $) 2409233318691058826407 DVD player sales (million $) 42110991717209724273050 Page 1011

12. To calculate the rate of change of VCR players: Average rate of change of VCR player sales (1998-> 2003)= Year199819992000200120022003 VCR sales (million $) 2409233318691058826407 DVD player sales (million $) 42110991717209724273050 Page 1012

13. To calculate the rate of change of VCR players: Average rate of change of VCR player sales (1998-> 2003)= Year199819992000200120022003 VCR sales (million $) 2409233318691058826407 DVD player sales (million $) 42110991717209724273050 Page 1013

14. Graphically, here is what we have: Page 1014

15. In general, if Q = f(t), we write ΔQ for a change in Q and Δt for a change in t. We define: The average rate of change, or rate of change, of Q with respect to t over an interval is: Average rate of change over an interval Page 1115

16. The average rate of change of the function Q = f(t) over an interval tells us how much Q changes, on average, for each unit change in t within that interval. On some parts of the interval, Q may be changing rapidly, while on other parts Q may be changing slowly. The average rate of change evens out these variations. Page 1116

17. DVD Player Sales: Average rate of change is POSITIVE on the interval from 1998 to 2003, since sales increased over this interval. An increasing function. VCR Player Sales: Average rate of change is NEGATIVE on the interval from 1998 to 2003, since sales decreased over this interval. A decreasing function. Page 1117

18. In general terms: If Q = f(t) for t in the interval a ≤ t ≤ b: f is an increasing function if the values of f increase as t increases in this interval. f is a decreasing function if the values of f decrease as t increases in this interval. Page 1118

19. And if Q=f(t): If f is an increasing function, then the average rate of change of Q with respect to is positive on every interval. If f is a decreasing function, then the average rate of change of Q with respect to t is negative on every interval. Page 1119

20. The function A = q(r) = πr 2 gives the area, A, of a circle as a function of its radius, r. Graph q. Explain how the fact that q is an increasing function can be seen on the graph. Page 11 (Example 1)20

21. The function A = q(r) = πr 2 gives the area, A, of a circle as a function of its radius, r. Graph q. rA 00 13.14159 212.5664 328.2743 450.2654 578.5398 Graph climbs as we go from left to right. Page 1221

22. rAΔrΔrΔAΔAΔA/Δr 00 13.14159 212.5664 328.2743 450.2654 578.5398 Page 1222

23. rAΔrΔrΔAΔAΔA/Δr 00 1 13.14159 1 212.5664 1 328.2743 1 450.2654 1 578.5398 Page 1223

24. rAΔrΔrΔAΔAΔA/Δr 00 13.14159 1 19.42477 212.5664 115.708 328.2743 121.9911 450.2654 128.2743 578.5398 Page 1224

25. rAΔrΔrΔAΔAΔA/Δr 00 13.14159 1 19.42477 212.5664 115.708 328.2743 121.9911 450.2654 128.2743 578.5398 Page 1225

26. rAΔrΔrΔAΔAΔA/Δr 00 13.14159 1 19.42477 212.5664 115.708 328.2743 121.9911 450.2654 128.2743 578.5398 Note: A increases as r increases, so A=q(r) is an increasing function. Also: Avg rate of change (ΔA/Δr) is positive on every interval. Page 1226

27. Carbon-14 is a radioactive element that exists naturally in the atmosphere and is absorbed by living organisms. When an organism dies, the carbon-14 present at death begins to decay. Let L = g(t) represent the quantity of carbon-14 (in micrograms, μg) in a tree t years after its death. See Table 1.12. Explain why we expect g to be a decreasing function of t. How is this represented on a graph?1.12 Page 12 (Example 2)27

28. Let L = g(t) represent the quantity of carbon-14 (in micrograms, μg) in a tree t years after its death. See Table 1.12. Explain why we expect g to be a decreasing function of t. How is this represented on a graph?1.12 t, time010002000300040005000 L, carbon-14200177157139123109 Page 1228

29. tLΔtΔtΔLΔLΔL/Δt 0200 1000177 2000157 3000139 4000123 5000109 Like in the last example, let’s fill in the table on the right, one column at a time: Page 1229

30. tLΔtΔtΔLΔLΔL/Δt 0200 1000 177 1000 2000157 1000 3000139 1000 4000123 1000 5000109 Like in the last example, let’s fill in the table on the right, one column at a time: Page 1230

31. tLΔtΔtΔLΔLΔL/Δt 0200 1000-23 1000177 1000-20 2000157 1000-18 3000139 1000-16 4000123 1000-14 5000109 Like in the last example, let’s fill in the table on the right, one column at a time: Page 1231

32. tLΔtΔtΔLΔLΔL/Δt 0200 1000-23-.023 1000177 1000-20-.020 2000157 1000-18-.018 3000139 1000-16-.016 4000123 1000-14-.014 5000109 Like in the last example, let’s fill in the table on the right, one column at a time: Page 1232

33. Since the amount of carbon-14 is decaying over time, g is a decreasing function. In Figure 1.10, the graph falls as we move from left to right and the average rate of change in the level of carbon-14 with respect to time, ΔL/Δt, is negative on every interval.1.10 Page 1233

34. tLΔtΔtΔLΔLΔL/Δt 0200 1000-23-.023 1000177 1000-20-.020 2000157 1000-18-.018 3000139 1000-16-.016 4000123 1000-14-.014 5000109 Here you can again see what was said on the last slide. (Lower values of t result in higher values of L, and vice versa. And ΔL/Δt is negative on every interval.) Page 1234

35. In general, we can identify an increasing or decreasing function from its graph as follows: The graph of an increasing function rises when read from left to right. The graph of a decreasing function falls when read from left to right. Page 1235

36. On what intervals is the function graphed in Figure 1.11 increasing? Decreasing? 1.11 Page 1336

37. On what intervals is the function graphed in Figure 1.11 increasing? Decreasing? 1.11 Inc Page 1337

38. On what intervals is the function graphed in Figure 1.11 increasing? Decreasing? 1.11 Inc Dec Page 1338

39. On what intervals is the function graphed in Figure 1.11 increasing? Decreasing? 1.11 Inc Dec Page 1339

40. On what intervals is the function graphed in Figure 1.11 increasing? Decreasing? 1.11 Inc Dec Inc Page 1340

41. On what intervals is the function graphed in Figure 1.11 increasing? Decreasing? 1.11 Inc Dec Inc Dec Page 1341

42. Using inequalities, we say that f is increasing for −3<x<−2, for 0<x<1, and for 2<x<3. Similarly, f is decreasing for −2<x<0 and 1<x<2. Inc Dec Inc Dec Inc Page 1342

43. Function Notation for the Average Rate of Change Suppose we want to find the average rate of change of a function Q = f(t) over the interval a ≤ t ≤ b. On this interval, the change in t is given by: Page 1343

44. At t = a, the value of Q is f(a), and at t = b, the value of Q is f(b). Therefore, the change in Q is given by: Function Notation for the Average Rate of Change Page 1344

45. Using function notation, we express the average rate of change as follows: The average rate of change of Q = f(t) over the interval a ≤ t ≤ b is given by: Function Notation for the Average Rate of Change Page 1345

46. Let’s review: Page 1346

47. Page 1347

48. Page 1448

49. Calculate the avg rate of change of the function f(x) = x 2 between x = 1 and x = 3 and between x = -2 and x = 1. Show results in a graph: Page 14 (Example 4)49

50. Calculate the avg rate of change of the function f(x) = x 2 between x = 1 and x = 3 and between x = -2 and x = 1. Show results in a graph: Between x=1 and x=3: Page 1450

51. Calculate the avg rate of change of the function f(x) = x 2 between x = 1 and x = 3 and between x = -2 and x = 1. Show results in a graph: Between x=1 and x=3: Page 1451

52. Calculate the avg rate of change of the function f(x) = x 2 between x = 1 and x = 3 and between x = -2 and x = 1. Show results in a graph: Between x=-2 and x=1: Page 1452

53. Calculate the avg rate of change of the function f(x) = x 2 between x = 1 and x = 3 and between x = -2 and x = 1. Show results in a graph: Between x=-2 and x=1: Page 1453

54. Calculate the avg rate of change of the function f(x) = x 2 between x = 1 and x = 3 and between x = -2 and x = 1. Show results in a graph: Page 1454

55. End of Section 1.2 55