1
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.