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Powerpoint slides copied from or based upon:

Connally,

Hughes-Hallett,

Gleason, Et Al.

Copyright 2007 John Wiley & Sons, Inc.

Functions Modeling ChangeA Preparation for Calculus

Third Edition

Section 1.3 Linear Functions

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Constant Rate of Change

In the previous section, we introduced the average rate of change of a function on an interval. For many functions, the average rate of change is different on different intervals.

For the remainder of this chapter, we consider functions which have the same average rate of change on every interval. Such a function has a graph which is a line and is called linear.

Page 17 3

A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years.

(a)  What is the average rate of change of P over every time interval?

(b)  Make a table that gives the town's population every five years over a 20-year period. Graph the population.

(c)  Find a formula for P as a function of t.Page 18 (Example 1) 4

A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years.

(a)  What is the average rate of change of P over every time interval?

This is given in the problem: 2,000 people / year

Page 18 5

A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years.

(b)  Make a table that gives the town's population every five years over a 20-year period. Graph the population.

Page 18 6

A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years.

(b)  Make a table that gives the town's population every five years over a 20-year period. Graph the population.t, years P, population

05101520Page 18 7

A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years.

(b)  Make a table that gives the town's population every five years over a 20-year period. Graph the population.t, years P, population

0 30,0005 40,00010 50,00015 60,00020 70,000Page 18 8

(b)  Make a table that gives the town's population every five years over a 20-year period. Graph the population.

Page 18 9

A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years.

(c)  Find a formula for P as a function of t.

Page 18 10

A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years.

(c)  Find a formula for P as a function of t.We want: P = f(t)

Page 18 11

A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years.

(c)  Find a formula for P as a function of t.We want: P = f(t)

If we define: P = initial pop + (growth/year)(# of yrs)

Page 18 12

A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years.

(c)  Find a formula for P as a function of t.t P0 30,00

05 40,00

010 50,00

015 60,00

020 70,00

0

If we define:

P = initial pop + (growth/year)(# of yrs)

Page 18 13

A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years.

(c)  Find a formula for P as a function of t.t P0 30,00

05 40,00

010 50,00

015 60,00

020 70,00

0

We substitute the initial value of P:

P = 30,000 + (growth/year)(# of yrs)

Page 18 14

A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years.

(c)  Find a formula for P as a function of t.t P0 30,00

05 40,00

010 50,00

015 60,00

020 70,00

0

And our rate of change:

P = 30,000 + (2,000/year)(# of yrs)

Page 18 15

A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years.

(c)  Find a formula for P as a function of t.t P0 30,00

05 40,00

010 50,00

015 60,00

020 70,00

0

And we substitute in t:

P = 30,000 + (2,000/year)(t)

Page 18 16

A town of 30,000 people grows by 2000 people every year. Since the population, P, is growing at the constant rate of 2000 people per year, P is a linear function of time, t, in years.

(c)  Find a formula for P as a function of t.t P0 30,00

05 40,00

010 50,00

015 60,00

020 70,00

0

Our final answer:

P = 30,000 + 2,000t

Page 18 17

Here again is the graph and the function.

Page 18 18

Any linear function has the same average rate of change over every interval. Thus, we talk about the rate of change of a linear function.

In general:

•A linear function has a constant rate of change.

•The graph of any linear function is a straight line.

Page 19 19

Depreciation Problem

A small business spends $20,000 on new computer equipment and, for tax purposes, chooses to depreciate it to $0 at a constant rate over a five-year period.

(a)  Make a table and a graph showing the value of the equipment over the five-year period.

(b) Give a formula for value as a function of time.

Page 19 (Example 2) 20

Used by economists/accounts: a linear function for straight-line depreciation.

Example: tax purposes-computer equipment depreciates (decreases in value) over time. Straight-line depreciation assumes:

the rate of change of value with respect to time is constant.

Page 19 21

t, years V, value ($)

Let's fill in the table:

Page 19 22

t, years V, value ($)

0

1

2

3

4

5

Let's fill in the table:

Page 19 23

t, years V, value ($)

0 $20,000

1 $16,000

2 $12,000

3 $8,000

4 $4,000

5 $0

Let's fill in the table:

Page 19 24

And our graph:

Page 19 25

Give a formula for value as a function of time:

Page 19 26

Give a formula for value as a function of time:

Change in value ?Change in time

Page 19 27

Give a formula for value as a function of time:

Change in value ?Change in time

Vt

Page 19 28

Give a formula for value as a function of time:

Change in value $20,000 ?Change in time 5 years

Vt

Page 19 29

Give a formula for value as a function of time:

Change in value $20,000 $4,000 per yearChange in time 5 years

Vt

Page 19 30

Give a formula for value as a function of time:

Change in value $20,000 $4,000 per yearChange in time 5 years

Vt

More generally, after t years?

Page 19 31

Give a formula for value as a function of time:

More generally, after t years?

$4,000t

Page 19 32

Give a formula for value as a function of time:

What about the initial value of the equipment?

Page 19 33

Give a formula for value as a function of time:

What about the initial value of the equipment?

$20,000

Page 19 34

Give a formula for value as a function of time:

What about the initial value of the equipment?

$20,000

What is our final answer for the function?

Page 19 35

Give a formula for value as a function of time:

What about the initial value of the equipment?

$20,000

What is our final answer for the function?

V = 20,000 - 4,000tPage 19 36

Let's summarize:

Output = Initial Value + (Rate of Change Input)

y xmb

Page 20 37

Let's summarize:

Output = Initial Value + (Rate of Change Input)

y xmb

b = y intercept (when x=0)

m = slopePage 20 38

Let's summarize:

Output = Initial Value + (Rate of Change Input)

y xmb

y = b + mx

Page 20 39

Let's summarize:

Output = Initial Value + (Rate of Change Input)

y xmbymx

Page 20 40

Let's summarize:

Output = Initial Value + (Rate of Change Input)

y xmb

1 0

1 0

y yymx x x

Page 20 41

Let's recap:

example #1: P = 30,000 + 2,000t

m = ? b = ?

Page 20 42

Let's recap:

example #1: P = 30,000 + 2,000t

m = 2,000 b = 30,000

Page 20 43

Let's recap:

example #2: V = 20,000 - 4,000t

m = ? b = ?

Page 20 44

Let's recap:

example #2: V = 20,000 - 4,000t

m = -4,000 b = 20,000

Page 20 45

Can a table of values represent a linear function?

Page 21 46

Could a table of values represent a linear function?

Yes, it could if:

Page 21 47

Could a table of values represent a linear function?

Yes, it could if:

Rate of change of linear function = Change in output ConstantChange in input

Page 21 48

x p(x) Δx Δp Δp/Δx50 .10

55 .11

60 .12

65 .13

70 .14

Could p(x) be a linear function?

Page 21 49

x p(x) Δx Δp Δp/Δx50 .10

555 .11

560 .12

565 .13

570 .14

Could p(x) be a linear function?

Page 21 50

x p(x) Δx Δp Δp/Δx50 .10

5 .0155 .11

5 .0160 .12

5 .0165 .13

5 .0170 .14

Could p(x) be a linear function?

Page 21 51

x p(x) Δx Δp Δp/Δx50 .10

5 .01 .00255 .11

5 .01 .00260 .12

5 .01 .00265 .13

5 .01 .00270 .14

Could p(x) be a linear function?

Page 21 52

x p(x) Δx Δp Δp/Δx50 .10

5 .01 .00255 .11

5 .01 .00260 .12

5 .01 .00265 .13

5 .01 .00270 .14

Since Δp/Δx is constant, p(x) could represent a linear

function.

Page 21 53

x q(x) Δx Δq Δq/Δx50 .01

55 .03

60 .06

65 .14

70 .15

Could q(x) be a linear function?

Page 21 54

x q(x) Δx Δq Δq/Δx50 .01

555 .03

560 .06

565 .14

570 .15

Could q(x) be a linear function?

Page 21 55

x q(x) Δx Δq Δq/Δx50 .01

5 .0255 .03

5 .0360 .06

5 .0865 .14

5 .0170 .15

Could q(x) be a linear function?

Page 21 56

x q(x) Δx Δq Δq/Δx50 .01

5 .02 .00455 .03

5 .03 .00660 .06

5 .08 .01665 .14

5 .01 .00270 .15

Could q(x) be a linear function?

Page 21 57

x q(x) Δx Δq Δq/Δx50 .01

5 .02 .00455 .03

5 .03 .00660 .06

5 .08 .01665 .14

5 .01 .00270 .15

Since Δq/Δx is NOT constant, q(x) does not represent a linear

function.

Page 21 58

Year p, Price ($)

Q, # sold (cars)

Δp ΔQ ΔQ/Δp

1985 3,990 49,000

1986 4,110 43,000

1987 4,200 38,500

1988 4,330 32,000

What about the following example?

Yugos exported from Yugoslavia to US.

Page 2259

Year p, Price ($)

Q, # sold (cars)

Δp ΔQ ΔQ/Δp

1985 3,990 49,000120

1986 4,110 43,00090

1987 4,200 38,500130

1988 4,330 32,000

What about the following example?

Yugos exported from Yugoslavia to US.

Page 2260

Year p, Price ($)

Q, # sold (cars)

Δp ΔQ ΔQ/Δp

1985 3,990 49,000120 -6,000

1986 4,110 43,00090 -4,500

1987 4,200 38,500130 -6,500

1988 4,330 32,000

What about the following example?

Yugos exported from Yugoslavia to US.

Page 2261

Year p, Price ($)

Q, # sold (cars)

Δp ΔQ ΔQ/Δp

1985 3,990 49,000120 -6,000 -50 cars/$

1986 4,110 43,00090 -4,500 -50 cars/$

1987 4,200 38,500130 -6,500 -50 cars/$

1988 4,330 32,000

What about the following example?

Yugos exported from Yugoslavia to US.

Page 22 62

Δp ΔQ ΔQ/Δp

120 -6,000 -50 cars/$

90 -4,500 -50 cars/$

130 -6,500 -50 cars/$

Although Δp and ΔQ are not constant, ΔQ/Δp is.

Therefore, since the rate of change (ΔQ/Δp) is constant, we could have a linear function here.

Page 22 63

Page 22 64

The function P = 100(1.02)t approximates the population of Mexico in the early 2000's.

Here P is the population (in millions) and t is the number of years since 2000.

Table 1.25 and Figure 1.21 show values of P over a 5-year period. Is P a linear function of t?

Page 23 65

t, years P (mill.) Δt ΔP ΔP/Δt0 100

1 2 21 102

1 2.04 2.042 104.04

1 2.08 2.083 106.12

1 2.12 2.124 108.24

1 2.17 2.175 110.41 Page 23 66

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t, years P (mill.) Δt ΔP ΔP/Δt0 100

10 21.90 2.19010 121.90

10 26.69 2.66920 148.59

10 32.55 3.25530 181.14

10 39.66 3.96640 220.80

10 48.36 4.83650 269.16 Page 24 68

Page 24 69

The formula P = 100(1.02)t is not of the form P = b + mt, so P is not a linear function of t.

Page 24 70

This completes Section 1.3.