Calculus I Exam #3
Sections 3.10 – 4.6
Name ______________________________________
Show all work on this paper. Solutions without correct supporting work will not earn credit. Only the best 10 of the 11
problems will count towards your grade. All solutions must be exact unless indicated otherwise in the problem.
1. The graph of
a.
on the interval [-3,2] is shown in the figure.
On what interval(s) is f increasing?
b. Find the critical points of f. Which critical points correspond to local
maxima? Local minima? Neither?
c.
Does f have any inflection points? If so, where?
d.
Sketch the graph of f ‘’ on the same set of axes given here:
e.
Sketch one possible graph of f on the same set of axes given by part (a).
2. Determine the absolute extrema of ( )
3. Find the local maxima and local minima for ( )
on [-8,8].
.
4.
Given that
( )
,
a.
Find the interval(s) where
is increasing: ____________________________
b.
Relatative Extrema: ______________________________
c. Interval(s) where f is concave up: _____________________________
d. Inflection point(s): _________________________________
e.
Sketch a possible graph of f. Make sure you label the x-coordinates of all critical points and inflection
points.
5. Find the points where ( )
why.
(
) has horizontal and vertical tangent lines. If they don’t exist, explain
6. a. Find all asymptotes for ( )
. As always, you must show your work. You will not earn credit for
correct solutions without correct supporting work.
b. Find dy given ( )
(
)
7. Determine whether the Mean Value Theorem applies to ( )
value(s) of c guaranteed by the theorem.
(
) on the interval
. If so, find the
8. A surface ship is moving in a straight line at 10km/hr. At the same time, an enemy submarine maintains a
position directly below the ship while diving at an angle that is 20 below the horizontal. How fast is the
submarine’s altitude decreasing? Approximate your answer to the nearest hundredth.
9. A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window. Find
the dimensions of a Norman window of maximum area if the total perimeter is 16 feet. Give an exact answer.
10. An inverted conical water tank with a height of 12 ft and a radius of 4 ft is drained through a hole in the vertex at
a rate of 2 ft3/sec. What is the rate of change of the water depth when the water is 3 ft? (Recall that the
volume of a right circular cone is given by
.)
11. An island is 3.5 mi from the nearest point on a straight shoreline; that point is 8 mi from a power station. A
utility company plans to lay electrical cable underwater from the island to the shore and then underground
along the shore to the power station. Assume that it costs $2400/mi to lay underwater cable and $1200/mi to
lay underground cable. At what point should the underwater cable meet the shore in order to minimize the cost
of the project?