Math 124 Exam 3 Nov. 3, 2006 SOLUTIONS
Directions
1.
SHOW YOUR WORK and be thorough in your solutions.
Partial credit will only be given for work
shown.
2. Any numerical answers should be left in exact form, i.e, no decimal approximations.
3. You may use calculators.
Good luck!
1)
Find the derivative of
g
(
x
) = cosh
2
(9
x
) with respect to
x
.
Solution:
g
(
x
) = 2 cosh(9
x
) sinh(9
x
)(9) =
18 cosh(9
x
) sinh(9
x
)
2)
Let
h
(
x
) =
√
1 +
x
.
(a) Determine the local linearization of
h
(
x
) at
a
= 3.
(b) Use the local linearization above to approximate
h
(3
.
1).
(c) Is the approximation in part (b) an underestimate or an overestimate? Give a brief reason.
Solutions:
(a) First, note that
h
(
x
) =
1
2
(1 +
x
)

1
2
. Then
h
(3) =
1
4
.
L
(
x
) =
h
(
a
) +
h
(
a
)(
x

a
) =
2 +
1
4
(
x

3)
(b)
h
(3
.
1)
≈
L
(3
.
1) = 2 +
1
4
(3
.
1

3) =
2
.
025
(c) The above approximation is an
overestimate
, as
h
is concave down at
x
= 3. Hence the tangent line
lies above the function
h
.
1
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3)
Consider the family of curves
f
(
x
) =
ax
+
b
x
, where
a
and
b
are parameters.
(a) Determine the critical points of this family in terms of the parameters.
(b) Classify the critical points in part (a) as local minima, local maxima, or neither.
Solutions:
(a)
f
(
x
) =
a

b
x
2
= 0. Hence,
a
=
b
x
2
ax
2
=
b
x
=
±
b
a
(b) We compute a second derivative:
f
(
x
) =
2
b
x
3
. Substituting the above critical points, we have
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 Spring '08
 KENNEDY
 Calculus, Approximation, Optimization, 1 ft, 10ft, 6ft

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