Derivative Worksheet With Answers Page 3
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14S E C T I O N 4.1
Linear Approximation and Applications
357
2
10
x
15. y D
, a D 1, dx D 0:01
2
2 C x
2
10
x
Let f .x/ D
. Then
SOLUTION
2
2 C x
2
2
.2 C x
/. 2x/
.10
x
/.2x/
24x
0
f
.x/ D
D
2
2
2
2
.2 C x
/
.2 C x
/
and
24
0
y
dy D f
.a/dx D
.0:01/ D 0:026667:
9
1=3
x 1
16. y D x
, a D 1, dx D 0:1
e
1=3
x 1
0
1
2=3
x 1
0
0
4
Let y D x
, a D 1, and dx D 0:1. Then y
, and
e
.x/ D
x
e
.3x C 1/, y
.a/ D y
.1/ D
SOLUTION
3
3
4
0
y
dy D y
.a/ dx D
.0:1/ D 0:133333.
3
In Exercises 17–24, estimate using the Linear Approximation and find the error using a calculator.
p
p
17.
26
25
p
1
1
0
1=2
0
0
Let f .x/ D
x, a D 25, and x D 1. Then f
and f
.
.x/ D
x
.a/ D f
.25/ D
SOLUTION
2
10
1
0
The Linear Approximation is f
f
.a/x D
.1/ D 0:1.
10
The actual change is f D f .a C x/
f .a/ D f .26/
f .25/
0:0990195.
The error in this estimate is j0:0990195
0:1j D 0:000980486.
1=4
1=4
18. 16:5
16
1=4
0
1
3=4
0
0
1
Let f .x/ D x
, a D 16, and x D :5. Then f
.x/ D
x
and f
.a/ D f
.16/ D
.
SOLUTION
4
32
1
0
The Linear Approximation is f
f
.a/x D
.0:5/ D 0:015625.
32
The actual change is
f D f .a C x/
f .a/ D f .16:5/
f .16/
2:015445
2 D 0:015445
The error in this estimate is j0:015625
0:015445j
0:00018.
1
1
19.
p
10
101
1
d
1
1
1
0
1=2
3=2
0
Let f .x/ D
, a D 100, and x D 1. Then f
.x/ D
.x
/ D
x
and f
.a/ D
.
/ D
p
SOLUTION
2
2
1000
dx
x
0:0005.
0
The Linear Approximation is f
f
.a/x D 0:0005.1/ D 0:0005.
The actual change is
1
1
f D f .a C x/
f .a/ D
p
D 0:000496281:
10
101
6
The error in this estimate is j 0:0005
.
. 0:000496281/j D 3:71902
10
1
1
20.
p
10
98
1
d
1
1
1
0
1=2
3=2
0
Let f .x/ D
, a D 100, and x D
2. Then f
and f
p
.x/ D
.x
/ D
x
.a/ D
.
/ D
SOLUTION
2
2
1000
dx
x
0:0005.
0
The Linear Approximation is f
f
.a/x D 0:0005. 2/ D 0:001:
The actual change is f D f .a C x/
f .a/ D f .98/
f .100/ D 0:00101525.
The error in this estimate is j0:001
0:00101525j
0:00001525.
1=3
21. 9
2
1
1
1=3
0
2=3
0
0
Let f .x/ D x
, a D 8, and x D 1. Then f
.x/ D
x
and f
.a/ D f
.8/ D
.
SOLUTION
3
12
0
1
The Linear Approximation is f
f
.a/x D
.1/ D 0:083333.
12
The actual change is f D f .a C x/
f .a/ D f .9/
f .8/ D 0:080084.
3
The error in this estimate is j0:080084
.
0:083333j
3:25
10
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