Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
1.3
Slide 1.3- 1
Average Rates of Change
OBJECTIVES
Compute an average rate of change.
Find a simplified difference quotient.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
1.3 Average Rates of Change
DEFINITION:
The average rate of change of y with respect to x, as
x changes from x1 to x2, is the ratio of the change in
output to the change in input:
y2 − y1
,
x2 − x1
where x2 ≠ x1.
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Slide 1.3- 3
1
1.3 Average Rates of Change
DEFINITION (concluded):
If we look at a graph of
the function, we see that
y2 − y1
f (x2 ) − f (x1 )
=
,
x2 − x1
x2 − x1
which is both the average
rate of change and the
slope of the line from
P(x1, y1) to Q(x2, y2).
suur
The line through P and Q, PQ, is called a secant line.
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Slide 1.3- 4
1.3 Average Rates of Change
Example 3: For y = f (x) = x 2 find the
average rate of change as:
a) x changes from 1 to 3.
b) x changes from 1 to 2.
c) x changes from 2 to 3.
a) When x1 = 1, y = f (x1 ) = f (1) = 12 = 1.
2
When x2 = 3, y = f (x2 ) = f (3) = 3 = 9.
Thus, the average rate of change is
9 −1 8
= = 4.
3−1 2
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Slide 1.3- 5
1.3 Average Rates of Change
Example 3 (concluded):
2
b) When x1 = 1, y = f ( x1 ) = f (1) = 1 = 1.
2
y
=
f
(
x
)
=
f
(2)
=
2
= 4.
When x2 = 2,
1
Thus, the average rate of change is
4 −1 3
= = 3.
2 −1 1
2
c) When x1 = 2, y = f (x1 ) = f (2) = 2 = 4.
2
When x2 = 3, y = f ( x1 ) = f (3) = 3 = 9.
Thus, the average rate of change is
9−4 5
= = 5.
3− 2 1
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Slide 1.3- 6
2
1.3 Average Rates of Change
DEFINITION:
The average rate of change of f with respect to x is also
called the difference quotient. It is given by
f (x + h) − f (x) where h ≠ 0.
,
h
The difference
quotient is equal
to the slope of the
line from (x, f (x))
to (x+h, f (x+h)).
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 1.3- 7
1.3 Average Rates of Change
Example 4: For f ( x ) = x 2 find the
difference quotient when:
a) x = 5 and h = 3.
b) x = 5 and h = 0.1.
a) We substitute x = 5 and h = 3 into the formula:
f (x + h) − f (x)
f (5 + 3) − f (5)
f (8) − f (5)
=
=
h
3
3
=
82 − 52
64 − 25
39
=
=
= 13
3
3
3
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Slide 1.3- 8
1.3 Average Rates of Change
Example 4 (concluded):
b) We substitute x = 5 and h = 0.1 into the formula:
f (x + h) − f (x)
f (5 + 0.1) − f (5)
f (5.1) − f (5)
=
=
h
0.1
0.1
=
5.12 − 5 2
26.01 − 25
1.01
=
=
= 10.1
0.1
0.1
0.1
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Slide 1.3- 9
3
1.3 Average Rates of Change
Example 6: For f (x ) = x 3 find a simplified
form of the difference quotient.
3
f (x + h ) − f (x ) (x + h ) − x 3
=
h
h
x 3 + 3x 2 h + 3xh 2 + h 3 − x 3
=
h
h 3x 2 + 3xh + h 2
=
h
= 3x 2 + 3xh + h 2 , h ≠ 0.
(
)
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Slide 1.3- 10
1.3 Average Rates of Change
3
find a simplified
x
form of the difference quotient.
3
3 3x − 3(x + h )
f (x + h ) − f (x ) x + h − x
x (x + h )
=
=
h
h
h
3x − 3x − 3h
−3h
x (x + h )
x (x + h )
=
=
h
h
−3
=
, h ≠ 0.
x (x + h )
Example 7: For f (x ) =
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Slide 1.3- 11
4