Download - Section 1.3 Linear Functions
![Page 1: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/1.jpg)
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
![Page 2: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/2.jpg)
Section 1.3 Linear Functions
2
![Page 3: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/3.jpg)
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
![Page 4: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/4.jpg)
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
![Page 5: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/5.jpg)
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
![Page 6: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/6.jpg)
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
![Page 7: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/7.jpg)
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
![Page 8: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/8.jpg)
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
![Page 9: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/9.jpg)
(b) Make a table that gives the town's population every five years over a 20-year period. Graph the population.
Page 18 9
![Page 10: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/10.jpg)
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
![Page 11: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/11.jpg)
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
![Page 12: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/12.jpg)
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
![Page 13: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/13.jpg)
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
![Page 14: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/14.jpg)
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
![Page 15: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/15.jpg)
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
![Page 16: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/16.jpg)
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
![Page 17: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/17.jpg)
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
![Page 18: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/18.jpg)
Here again is the graph and the function.
Page 1818
![Page 19: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/19.jpg)
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
![Page 20: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/20.jpg)
Depreciation Problem
A small business spends $20,000 on new computer equipment and, for tax purposes, chooses to depreciate it to $0 at a constant rate over a five-year period.
(a) Make a table and a graph showing the value of the equipment over the five-year period.
(b) Give a formula for value as a function of time.
Page 19 (Example 2)20
![Page 21: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/21.jpg)
Used by economists/accounts: a linear function for straight-line depreciation.
Example: tax purposes-computer equipment depreciates (decreases in value) over time. Straight-line depreciation assumes:
the rate of change of value with respect to time is constant.
Page 1921
![Page 22: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/22.jpg)
t, years V, value ($)
Let's fill in the table:
Page 1922
![Page 23: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/23.jpg)
t, years V, value ($)
0
1
2
3
4
5
Let's fill in the table:
Page 19 23
![Page 24: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/24.jpg)
t, years V, value ($)
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
![Page 25: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/25.jpg)
And our graph:
Page 19 25
![Page 26: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/26.jpg)
Give a formula for value as a function of time:
Page 1926
![Page 27: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/27.jpg)
Give a formula for value as a function of time:
Change in value?
Change in time
Page 1927
![Page 28: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/28.jpg)
Give a formula for value as a function of time:
Change in value?
Change in time
V
t
Page 1928
![Page 29: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/29.jpg)
Give a formula for value as a function of time:
Change in value $20,000?
Change in time 5 years
V
t
Page 1929
![Page 30: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/30.jpg)
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
![Page 31: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/31.jpg)
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
![Page 32: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/32.jpg)
Give a formula for value as a function of time:
More generally, after t years?
$4,000t
Page 1932
![Page 33: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/33.jpg)
Give a formula for value as a function of time:
What about the initial value of the equipment?
Page 1933
![Page 34: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/34.jpg)
Give a formula for value as a function of time:
What about the initial value of the equipment?
$20,000
Page 1934
![Page 35: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/35.jpg)
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
![Page 36: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/36.jpg)
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
![Page 37: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/37.jpg)
Let's summarize:
Output = Initial Value + (Rate of Change Input)
y xmb
Page 2037
![Page 38: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/38.jpg)
Let's summarize:
Output = Initial Value + (Rate of Change Input)
y xmb
b = y intercept (when x=0)
m = slopePage 20
38
![Page 39: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/39.jpg)
Let's summarize:
Output = Initial Value + (Rate of Change Input)
y xmb
y = b + mx
Page 2039
![Page 40: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/40.jpg)
Let's summarize:
Output = Initial Value + (Rate of Change Input)
y xmb
ym
x
Page 20
40
![Page 41: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/41.jpg)
Let's summarize:
Output = Initial Value + (Rate of Change Input)
y xmb
1 0
1 0
y yym
x x x
Page 20
41
![Page 42: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/42.jpg)
Let's recap:
example #1: P = 30,000 + 2,000t
m = ? b = ?
Page 2042
![Page 43: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/43.jpg)
Let's recap:
example #1: P = 30,000 + 2,000t
m = 2,000 b = 30,000
Page 2043
![Page 44: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/44.jpg)
Let's recap:
example #2: V = 20,000 - 4,000t
m = ? b = ?
Page 2044
![Page 45: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/45.jpg)
Let's recap:
example #2: V = 20,000 - 4,000t
m = -4,000 b = 20,000
Page 2045
![Page 46: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/46.jpg)
Can a table of values represent a linear function?
Page 2146
![Page 47: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/47.jpg)
Could a table of values represent a linear function?
Yes, it could if:
Page 2147
![Page 48: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/48.jpg)
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
![Page 49: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/49.jpg)
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
![Page 50: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/50.jpg)
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
![Page 51: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/51.jpg)
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
![Page 52: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/52.jpg)
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
![Page 53: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/53.jpg)
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
![Page 54: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/54.jpg)
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
![Page 55: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/55.jpg)
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
![Page 56: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/56.jpg)
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
![Page 57: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/57.jpg)
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
![Page 58: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/58.jpg)
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
![Page 59: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/59.jpg)
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
59
![Page 60: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/60.jpg)
Year p, Price ($)
Q, # sold (cars)
Δp ΔQ ΔQ/Δp
1985 3,990 49,000120
1986 4,110 43,00090
1987 4,200 38,500130
1988 4,330 32,000
What about the following example?
Yugos exported from Yugoslavia to US.
Page 22
60
![Page 61: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/61.jpg)
Year p, Price ($)
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
61
![Page 62: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/62.jpg)
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
![Page 63: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/63.jpg)
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 64: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/64.jpg)
Page 22 64
![Page 65: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/65.jpg)
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
![Page 66: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/66.jpg)
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 67: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/67.jpg)
Page 23
67
![Page 68: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/68.jpg)
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 69: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/69.jpg)
Page 24 69
![Page 70: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/70.jpg)
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
![Page 71: Section 1.3 Linear Functions](https://reader036.vdocuments.net/reader036/viewer/2022062517/568137f8550346895d9fb9cc/html5/thumbnails/71.jpg)
This completes Section 1.3.