Derivative Worksheet With Answers Page 9
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14S E C T I O N 4.1
Linear Approximation and Applications
363
1
52. y D tan
x, a D 1
1
Let f .x/ D tan
x. Then
SOLUTION
1
1
0
0
; and f
f
.x/ D
; f .a/ D
.a/ D
;
2
1 C x
4
2
so the linearization of f .x/ at a is
1
0
L.x/ D f
.a/.x
a/ C f .a/ D
.x
1/ C
:
2
4
p
x
53. y D e
, a D 1
p
x
Let f .x/ D e
. Then
SOLUTION
p
1
1
0
x
0
; f .a/ D e; and f
f
.x/ D
p
e
.a/ D
e;
2
x
2
so the linearization of f .x/ at a is
1
1
0
L.x/ D f
.a/.x
a/ C f .a/ D
e.x
1/ C e D
e.x C 1/:
2
2
x
54. y D e
ln x, a D 1
x
Let f .x/ D e
ln x. Then
SOLUTION
x
e
0
x
0
f
.x/ D
C e
ln x; f .a/ D 0; and f
.a/ D e;
x
so the linearization of f .x/ at a is
0
L.x/ D f
.a/.x
a/ C f .a/ D e.x
1/:
55. What is f .2/ if the linearization of f .x/ at a D 2 is L.x/ D 2x C 4?
f .2/ D L.2/ D 2.2/ C 4 D 8.
SOLUTION
56. Compute the linearization of f .x/ D 3x
4 at a D 0 and a D 2. Prove more generally that a linear function coincides with
its linearization at x D a for all a.
0
0
Let f .x/ D 3x
4. Then f
.x/ D 3. With a D 0, f .a/ D 4 and f
.a/ D 3, so the linearization of f .x/ at a D 0
SOLUTION
is
L.x/ D 4 C 3.x
0/ D 3x
4 D f .x/:
0
With a D 2, f .a/ D 2 and f
.a/ D 3, so the linearization of f .x/ at a D 2 is
L.x/ D 2 C 3.x
2/ D 2 C 3x
6 D 3x
4 D f .x/:
More generally, let g.x/ D bx C c be any linear function. The linearization L.x/ of g.x/ at x D a is
0
L.x/ D g
.a/.x
a/ C g.a/ D b.x
a/ C ba C c D bx C c D g.x/I
i.e., L.x/ D g.x/.
p
p
57. Estimate
16:2 using the linearization L.x/ of f .x/ D
x at a D 16. Plot f .x/ and L.x/ on the same set of axes and
determine whether the estimate is too large or too small.
1=2
0
1
1=2
0
0
1
Let f .x/ D x
, a D 16, and x D 0:2. Then f
.x/ D
x
and f
.a/ D f
.16/ D
. The linearization to
SOLUTION
2
8
f .x/ is
1
1
0
L.x/ D f
.a/.x
a/ C f .a/ D
.x
16/ C 4 D
x C 2:
8
8
p
Thus, we have
L.16:2/ D 4:025. Graphs of f .x/ and L.x/ are shown below. Because the graph of L.x/ lies above the
16:2
graph of f .x/, we expect that the estimate from the Linear Approximation is too large.
y
5
4
L(x)
3
2
f (x)
1
x
0
5
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15
20
25
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