1

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

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

0

5

10

15

20Page 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 30,000

5 40,000

10 50,000

15 60,000

20 70,000Page 188

(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 1810

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 1811

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 1812

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

0

5 40,000

10 50,000

15 60,000

20 70,000

If we define:

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

Page 1813

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

0

5 40,000

10 50,000

15 60,000

20 70,000

We substitute the initial value of P:

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

Page 1814

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

0

5 40,000

10 50,000

15 60,000

20 70,000

And our rate of change:

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

Page 1815

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

0

5 40,000

10 50,000

15 60,000

20 70,000

And we substitute in t:

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

Page 1816

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

0

5 40,000

10 50,000

15 60,000

20 70,000

Our final answer:

P = 30,000 + 2,000t

Page 1817

Here again is the graph and the function.

Page 1818

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 1919

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 1921

t, years V, value ($)

Let's fill in the table:

Page 1922

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 1924

And our graph:

Page 19 25

Give a formula for value as a function of time:

Page 1926

Give a formula for value as a function of time:

Change in value?

Change in time

Page 1927

Give a formula for value as a function of time:

Change in value?

Change in time

V

t

Page 1928

Give a formula for value as a function of time:

Change in value $20,000?

Change in time 5 years

V

t

Page 1929

Give a formula for value as a function of time:

Change in value $20,000$4,000 per year

Change in time 5 years

V

t

Page 1930

Give a formula for value as a function of time:

Change in value $20,000$4,000 per year

Change in time 5 years

V

t

More generally, after t years?

Page 1931

Give a formula for value as a function of time:

More generally, after t years?

$4,000t

Page 1932

Give a formula for value as a function of time:

What about the initial value of the equipment?

Page 1933

Give a formula for value as a function of time:

What about the initial value of the equipment?

$20,000

Page 1934

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 1935

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 2037

Let's summarize:

Output = Initial Value + (Rate of Change Input)

y xmb

b = y intercept (when x=0)

m = slopePage 20

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Let's summarize:

Output = Initial Value + (Rate of Change Input)

y xmb

y = b + mx

Page 2039

Let's summarize:

Output = Initial Value + (Rate of Change Input)

y xmb

ym

x

Page 20

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Let's summarize:

Output = Initial Value + (Rate of Change Input)

y xmb

1 0

1 0

y yym

x x x

Page 20

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Let's recap:

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

m = ? b = ?

Page 2042

Let's recap:

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

m = 2,000 b = 30,000

Page 2043

Let's recap:

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

m = ? b = ?

Page 2044

Let's recap:

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

m = -4,000 b = 20,000

Page 2045

Can a table of values represent a linear function?

Page 2146

Could a table of values represent a linear function?

Yes, it could if:

Page 2147

Could a table of values represent a linear function?

Yes, it could if:

Rate of change of linear function =

Change in output Constant

Change in input

Page 2148

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 22

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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 22

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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 22

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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

Page 23

67

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.