Functions Modeling Change A Preparation for Calculus Third Edition

Transcription

1 Powerpoint slides copied from or based upon: Functions Modeling Change A Preparation for Calculus Third Edition Connally, Hughes-Hallett, Gleason, Et Al. Copyright 2007 John Wiley & Sons, Inc.

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 To measure how fast sales were increasing, we calculate a rate of change of the form: Change in sales Change in time Page 10 3

4 At the same time, sales of video cassette recorders (VCRs) have been decreasing. See Table 1.11 below: Year VCR sales (million $) DVD player sales (million $) Page 10 4

5 Year VCR sales (million $) DVD player sales (million $) To calculate the rate of change of DVD players: Average rate of change of DVD player sales (1998-> 2003)= Change in DVD player sales Change in time Page 10 5

6 Year VCR sales (million $) DVD player sales (million $) To calculate the rate of change of DVD players: Average rate of change of DVD player sales (1998-> 2003)= Change in DVD player sales Change in time Page 10 6

7 Year VCR sales (million $) DVD player sales (million $) To calculate the rate of change of DVD players: Average rate of change of DVD player sales (1998-> 2003)= Change in DVD player sales Change in time Page 10 7

8 Year VCR sales (million $) DVD player sales (million $) To calculate the rate of change of DVD players: Average rate of change of DVD player sales (1998-> 2003)= Change in DVD player sales 2629 Change in time 5 $525.8 mill./yr Page 10 8

9 Graphically, here is what we have: Page 10 9

10 Year VCR sales (million $) DVD player sales (million $) To calculate the rate of change of VCR players: Average rate of change of VCR player sales (1998-> 2003)= Change in VCR player sales Change in time Page 10 10

11 Year VCR sales (million $) DVD player sales (million $) To calculate the rate of change of VCR players: Average rate of change of VCR player sales (1998-> 2003)= Change in VCR player sales Change in time Page 10 11

12 Year VCR sales (million $) DVD player sales (million $) To calculate the rate of change of VCR players: Average rate of change of VCR player sales (1998-> 2003)= Change in VCR player sales Change in time Page 10 12

13 Year VCR sales (million $) DVD player sales (million $) To calculate the rate of change of VCR players: Average rate of change of VCR player sales (1998-> 2003)= Change in VCR player sales 2002 Change in time 5 $400.4 mill./yr Page 10 13

14 Graphically, here is what we have: Page 10 14

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 Change in Q Change in t Q t Page 11 15

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 11 16

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 11 17

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 11 18

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 11 19

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. r A Graph climbs as we go from left to right. Page 12 21

22 r A Δr ΔA ΔA/Δr Page 12 22

23 r A Δr ΔA ΔA/Δr Page 12 23

24 r A Δr ΔA ΔA/Δr Page 12 24

25 r A Δr ΔA ΔA/Δr Page 12 25

26 Note: r A Δr ΔA ΔA/Δr 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

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 Explain why we expect g to be a decreasing function of t. How is this represented on a graph? 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 Explain why we expect g to be a decreasing function of t. How is this represented on a graph? t, time L, carbon Page 12 28

29 Like in the last example, let s fill in the table on the right, one column at a time: t L Δt ΔL ΔL/Δt Page 12 29

30 Like in the last example, let s fill in the table on the right, one column at a time: t L Δt ΔL ΔL/Δt Page 12 30

31 Like in the last example, let s fill in the table on the right, one column at a time: t L Δt ΔL ΔL/Δt Page 12 31

32 Like in the last example, let s fill in the table on the right, one column at a time: t L Δt ΔL ΔL/Δt Page 12 32

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. Page 12 33

34 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.) t L Δt ΔL ΔL/Δt Page 12 34

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 12 35

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

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

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

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

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

41 On what intervals is the function graphed in Figure 1.11 increasing? Decreasing? Inc Dec Inc Dec Dec Page 13 41

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 Dec Inc Page 13 42

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: t b a Page 13 43

44 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: Q f ( b) f ( a) Page 13 44

45 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: Change in Q Q f ( b) f ( a) Change in t t b a Page 13 45

46 Let s review: Q f () t Interval: a t b t a Q f ( a) t b Q f ( b) Q f ( b) f ( a) t b a Page 13 46

47 Change in Q Q f ( b) f ( a) Change in t t b a Page 13 47

48 Change in Q Q f ( b) f ( a) Change in t t b a Page 14 48

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 14 50

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: f ( b) f ( a) b a f (3) f(1) Page 14 51

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 14 52

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: f ( b) f ( a) b a f 2 2 (1) f( 2) 1 ( 2) ( 2) 1 ( 2) Page 14 53

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 14 54

55 End of Section