Math 161 – Spring 2009
Exam #2
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Show your work where necessary.
Question
Value
Score
1 2 3 4 5 6 7
24 12 12 10 18 12 12
1. ( 6 pts each) Find the derivative of each the following (do not simplify):
(a) f x =
(b)
x
e3
3x
e
5
g  x=ln x
(c) g  x =ln
[
3
2x5 x −1
x
(d) y=e 2t ln t 1
]
Total
100
2. (12 pts.) Use implicit differentiation to find the slope and the equation of the line tangent to the graph
of
1 2y
x e =5−x 2
2
at the point (2,0).
3. (12 pts) Find the absolute maximum and absolute minimum values for
on the interval [-2, 3].
4. (10 pts) Use the given information to sketch the graph of f (x):
f (0) = -2, f (1) = 0, f (2) = 4;
f '(0) = 0, f '(2) = 0, f '(1) is not defined;
f '(x) > 0 on (0,1) and (1,2);
f '(x) < 0 on −∞ , 0 and 2, ∞ ;
f ''(1) is not defined;
f ''(x ) > 0 −∞ , 1 ; f ''(x ) < 0 1, ∞
f x =
−5
x 21
5. (18 pts.) Given f x = x 3−3x 2 −24x1 find:
(a) intervals where f (x) is increasing and decreasing
(b) local maxima and minima (x – coordinates only)
(c) intervals where f (x) is concave up and concave down.
(d) any inflection points
6. (12 pts) A boat is pulled toward a dock by a winch that is 8 feet above the bow of the boat. If the rope
is being pulled in at a rate of 4 ft./sec., how fast is the boat approaching the dock when 10 feet of rope
is out?
winch

rope
8 ft.

Boat 7. (12 pts) A farmer has $1500 available to build an E-shaped fence along a straight river so as to create
two identical rectangular pastures. The materials for the side parallel to the river cost $6 per foot, and
the materials for the three sections perpendicular to the river cost $5 per foot. Find the dimensions for
which the total area is as large as possible.
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