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Chapter 1 Section 2 Rate of Change

Chapter 1 Section 2 Rate of Change. 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:.

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Chapter 1 Section 2 Rate of Change

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  1. Chapter 1 Section 2 Rate of Change

  2. 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 10

  3. At the same time, sales of video cassette recorders (VCRs) have been decreasing. See Table 1.11 below: Page 10

  4. To calculate the rate of change of DVD players: Average rate of change of DVD player sales (1998-> 2003)= Page 10

  5. To calculate the rate of change of DVD players: Average rate of change of DVD player sales (1998-> 2003)= Page 10

  6. To calculate the rate of change of DVD players: Average rate of change of DVD player sales (1998-> 2003)= Page 10

  7. To calculate the rate of change of DVD players: Average rate of change of DVD player sales (1998-> 2003)= Page 10

  8. Graphically, here is what we have: Page 10

  9. To calculate the rate of change of VCR players: Average rate of change of VCR player sales (1998-> 2003)= Page 10

  10. To calculate the rate of change of VCR players: Average rate of change of VCR player sales (1998-> 2003)= Page 10

  11. To calculate the rate of change of VCR players: Average rate of change of VCR player sales (1998-> 2003)= Page 10

  12. To calculate the rate of change of VCR players: Average rate of change of VCR player sales (1998-> 2003)= Page 10

  13. Graphically, here is what we have: Page 10

  14. 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 11

  15. 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 11

  16. 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 11

  17. 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 11

  18. 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 11

  19. The function A = q(r) = πr2 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. The function A = q(r) = πr2 gives the area, A, of a circle as a function of its radius, r. Graph q. Graph climbs as we go from left to right. Page 12

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  25. 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 12

  26. 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? Page 12 (Example 2)

  27. 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? Page 12

  28. Like in the last example, let’s fill in the table on the right, one column at a time: Page 12

  29. Like in the last example, let’s fill in the table on the right, one column at a time: Page 12

  30. Like in the last example, let’s fill in the table on the right, one column at a time: Page 12

  31. Like in the last example, let’s fill in the table on the right, one column at a time: Page 12

  32. 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. Page 12

  33. 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 12

  34. 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 12

  35. On what intervals is the function graphed in Figure 1.11 increasing? Decreasing? Page 13

  36. On what intervals is the function graphed in Figure 1.11 increasing? Decreasing? Inc Page 13

  37. On what intervals is the function graphed in Figure 1.11 increasing? Decreasing? Inc Dec Page 13

  38. On what intervals is the function graphed in Figure 1.11 increasing? Decreasing? Inc Dec Dec Page 13

  39. On what intervals is the function graphed in Figure 1.11 increasing? Decreasing? Inc Dec Inc Dec Page 13

  40. On what intervals is the function graphed in Figure 1.11 increasing? Decreasing? Inc Dec Dec Inc Dec Page 13

  41. 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 Dec Inc Inc Dec Page 13

  42. 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 13

  43. Function Notation for the Average Rate of Change 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: Page 13

  44. Function Notation for the Average Rate of Change 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: Page 13

  45. Let’s review: Page 13

  46. Page 13

  47. Page 14

  48. Calculate the avg rate of change of the function f(x) = x2between x = 1 and x = 3 and between x = -2 and x = 1. Show results in a graph: Page 14 (Example 4)

  49. Calculate the avg rate of change of the function f(x) = x2between x = 1 and x = 3 and between x = -2 and x = 1. Show results in a graph: Between x=1 and x=3: Page 14

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