Derivative Worksheet With Answers
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14APPLICATIONS OF
4
THE DERIVATIVE
4.1 Linear Approximation and Applications
Preliminary Questions
1. True or False? The Linear Approximation says that the vertical change in the graph is approximately equal to the vertical
change in the tangent line.
This statement is true. The linear approximation does say that the vertical change in the graph is approximately equal
SOLUTION
to the vertical change in the tangent line.
0
2. Estimate g.1:2/
g.1/ if g
.1/ D 4.
Using the Linear Approximation,
SOLUTION
0
g.1:2/
g.1/
g
.1/.1:2
1/ D 4.0:2/ D 0:8:
0
3. Estimate f .2:1/ if f .2/ D 1 and f
.2/ D 3.
Using the Linearization,
SOLUTION
0
f .2:1/
f .2/ C f
.2/.2:1
2/ D 1 C 3.0:1/ D 1:3
4. Complete the sentence: The Linear Approximation shows that up to a small error, the change in output f is directly propor-
tional to . . . .
The Linear Approximation tells us that up to a small error, the change in output f is directly proportional to the
SOLUTION
change in input x when x is small.
Exercises
In Exercises 1–6, use Eq. (1) to estimate f D f .3:02/
f .3/.
2
1. f .x/ D x
2
0
0
Let f .x/ D x
. Then f
.x/ D 2x and f
f
.3/x D 6.0:02/ D 0:12.
SOLUTION
4
2. f .x/ D x
4
0
3
0
Let f .x/ D x
. Then f
and f
.x/ D 4x
f
.3/x D 4.27/.0:02/ D 2:16:
SOLUTION
1
3. f .x/ D x
1
1
0
2
0
Let f .x/ D x
. Then f
and f
.x/ D x
f
.3/x D
.0:02/ D 0:00222:
SOLUTION
9
1
4. f .x/ D
x C 1
1
1
0
2
0
Let f .x/ D .x C 1/
. Then f
.x/ D .x C 1/
and f
f
.3/x D
.0:02/ D 0:00125:
SOLUTION
16
p
5. f .x/ D
x C 6
p
0
1
1=2
Let f .x/ D
x C 6. Then f
.x/ D
.x C 6/
and
SOLUTION
2
1
0
1=2
f
f
.3/x D
9
.0:02/ D 0:003333:
2
x
6. f .x/ D tan
3
x
x
0
2
Let f .x/ D tan
. Then f
sec
and
.x/ D
SOLUTION
3
3
3
0
f
f
.3/x D
.0:02/ D 0:020944:
3
7. The cube root of 27 is 3. How much larger is the cube root of 27.2? Estimate using the Linear Approximation.
1
1
1=3
0
2=3
0
0
Let f .x/ D x
, a D 27, and x D 0:2. Then f
.x/ D
x
and f
.a/ D f
.27/ D
. The Linear
SOLUTION
3
27
Approximation is
1
0
f
f
.a/x D
.0:2/ D 0:0074074
27
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