Powerpoint slides copied from or based upon:

Connally,

Hughes-Hallett,

Gleason, Et Al.

Copyright 2007 John Wiley & Sons, Inc.

Functions Modeling Change

A Preparation for Calculus

Third Edition

Chapter 1Section 2

Rate of Change

2

Change in sales

Change in time

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:

Year 1998 1999 2000 2001 2002 2003VCR sales (million $)

2409 2333 1869 1058 826 407

DVD player sales (million

$)

421 1099 1717 2097 2427 3050

Page 10 4

To calculate the rate of change of DVD players:

Average rate of change of DVD player sales (1998-> 2003)=

Year 1998 1999 2000 2001 2002 2003VCR sales (million $)

2409 2333 1869 1058 826 407

DVD player sales (million

$)

421 1099 1717 2097 2427 3050

Change in DVD player sales

Change in timePage 10 5

To calculate the rate of change of DVD players:

Average rate of change of DVD player sales (1998-> 2003)=

Year 1998 1999 2000 2001 2002 2003VCR sales (million $)

2409 2333 1869 1058 826 407

DVD player sales (million

$)

421 1099 1717 2097 2427 3050

Change in DVD player sales 3050 421

Change in time 2003 1998

Page 10 6

To calculate the rate of change of DVD players:

Average rate of change of DVD player sales (1998-> 2003)=

Year 1998 1999 2000 2001 2002 2003VCR sales (million $)

2409 2333 1869 1058 826 407

DVD player sales (million

$)

421 1099 1717 2097 2427 3050

Change in DVD player sales 3050 421 2629

Change in time 2003 1998 5

Page 10 7

To calculate the rate of change of DVD players:

Average rate of change of DVD player sales (1998-> 2003)=

Year 1998 1999 2000 2001 2002 2003VCR sales (million $)

2409 2333 1869 1058 826 407

DVD player sales (million

$)

421 1099 1717 2097 2427 3050

Change in DVD player sales 2629$525.8 mill./yr

Change in time 5

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)=

Year 1998 1999 2000 2001 2002 2003VCR sales (million $)

2409 2333 1869 1058 826 407

DVD player sales (million

$)

421 1099 1717 2097 2427 3050

Change in VCR player sales

Change in timePage 10 10

To calculate the rate of change of VCR players:

Average rate of change of VCR player sales (1998-> 2003)=

Year 1998 1999 2000 2001 2002 2003VCR sales (million $)

2409 2333 1869 1058 826 407

DVD player sales (million

$)

421 1099 1717 2097 2427 3050

Change in VCR player sales 407 2409

Change in time 2003 1998

Page 10 11

To calculate the rate of change of VCR players:

Average rate of change of VCR player sales (1998-> 2003)=

Year 1998 1999 2000 2001 2002 2003VCR sales (million $)

2409 2333 1869 1058 826 407

DVD player sales (million

$)

421 1099 1717 2097 2427 3050

Change in VCR player sales 407 2409 2002

Change in time 2003 1998 5

Page 10 12

To calculate the rate of change of VCR players:

Average rate of change of VCR player sales (1998-> 2003)=

Year 1998 1999 2000 2001 2002 2003VCR sales (million $)

2409 2333 1869 1058 826 407

DVD player sales (million

$)

421 1099 1717 2097 2427 3050

Change in VCR player sales 2002$400.4 mill./yr

Change in time 5

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

Change in Q

Change in t

Q

t

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.

r A0 0

1 3.14159

2 12.5664

3 28.2743

4 50.2654

5 78.5398

Graph climbs as we go from left to right.

Page 12 21

r A Δr ΔA ΔA/Δr

0 0

1 3.14159

2 12.5664

3 28.2743

4 50.2654

5 78.5398

Page 12 22

r A Δr ΔA ΔA/Δr

0 0

1

1 3.14159

1

2 12.5664

1

3 28.2743

1

4 50.2654

1

5 78.5398

Page 12 23

r A Δr ΔA ΔA/Δr

0 0

1 3.14159

1 3.14159

1 9.42477

2 12.5664

1 15.708

3 28.2743

1 21.9911

4 50.2654

1 28.2743

5 78.5398

Page 12 24

r A Δr ΔA ΔA/Δr

0 0

1 3.14159

3.14159

1 3.14159

1 9.42477

9.42477

2 12.5664

1 15.708 15.708

3 28.2743

1 21.9911

21.9911

4 50.2654

1 28.2743

28.2743

5 78.5398

Page 12 25

r A Δr ΔA ΔA/Δr

0 0

1 3.14159

3.14159

1 3.14159

1 9.42477

9.42477

2 12.5664

1 15.708

15.708

3 28.2743

1 21.9911

21.9911

4 50.2654

1 28.2743

28.2743

5 78.5398

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 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?

t, time 0 1000

2000

3000

4000

5000

L, carbon-14

200 177 157 139 123 109

Page 12 28

t L Δt ΔL ΔL/Δt

0 200

1000 177

2000 157

3000 139

4000 123

5000 109

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

Page 12 29

t L Δt ΔL ΔL/Δt

0 200

1000

1000 177

1000

2000 157

1000

3000 139

1000

4000 123

1000

5000 109

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

Page 12 30

t L Δt ΔL ΔL/Δt

0 200

1000 -23

1000 177

1000 -20

2000 157

1000 -18

3000 139

1000 -16

4000 123

1000 -14

5000 109

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

Page 12 31

t L Δt ΔL ΔL/Δt

0 200

1000 -23 -.023

1000 177

1000 -20 -.020

2000 157

1000 -18 -.018

3000 139

1000 -16 -.016

4000 123

1000 -14 -.014

5000 109

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

t L Δt ΔL ΔL/Δt

0 200

1000 -23 -.023

1000 177

1000 -20 -.020

2000 157

1000 -18 -.018

3000 139

1000 -16 -.016

4000 123

1000 -14 -.014

5000 109

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

DecPage 13 39

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

Inc

Dec

Dec

Inc

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

Dec

IncPage 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:

t b a

Page 13 43

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

Function Notation for the Average Rate of Change

Page 13 44

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 ( ) ( )

Change in t

Q f b f a

t b a

Function Notation for the Average Rate of Change

Page 13 45

( )

Interval: a t b

( )

( )

( ) ( )

Q f t

t a Q f a

t b Q f b

Q f b f a

t b a

Let’s review:

Page 13 46

Change in Q ( ) ( )

Change in t

Q f b f a

t b a

Page 13 47

Change in Q ( ) ( )

Change in t

Q f b f a

t b a

Page 14 48

Calculate the avg rate of change of the function f(x) = x2 between 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) = x2 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

Calculate the avg rate of change of the function f(x) = x2 between x = 1 and x = 3 and between x = -2 and x = 1. Show results in a graph:

Between x=1 and x=3:

2 2

( ) ( )

(3) (1) 3 1 9 1

3 1 3 1 28

42

f b f a

b a

f f

Page 14 51

Calculate the avg rate of change of the function f(x) = x2 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

Calculate the avg rate of change of the function f(x) = x2 between x = 1 and x = 3 and between x = -2 and x = 1. Show results in a graph:

Between x=-2 and x=1:

2 2

( ) ( )

(1) ( 2) 1 ( 2) 1 4

1 ( 2) 1 ( 2) 3

31

3

f b f a

b a

f f

Page 14 53

Calculate the avg rate of change of the function f(x) = x2 between x = 1 and x = 3 and between x = -2 and x = 1. Show results in a graph:

Page 14 54

End of Section 1.2

55