C Roettger, Spring 15
Math 165 - Practice Exam 2
Problem 1 Suppose a balloon in the shape of a cylinder is inflated, so that
its volume increases at a rate of 20π cubic centimeters per second. At a
time when its radius is 6 cm and its height 4 cm, the height of the cylinder
increases at a rate of 0.2 cm per second. What is the rate of change of the
radius at this moment in time? Round the answer to four digits after the
decimal point.
Problem 2 Suppose an asteroid is observed to have a current speed of
v0 = 1, 200
km
s
Simulations of the asteroid’s projected path lead to this estimate of the distance d of the asteroid to Earth at the time when it will be closest,
d = 180, 000 + 75v0 − 0.001v03
The asteroid’s speed can only be measured with limited accuracy. But the
astronomers are sure that
v0 = 1, 200 ± 40
km
s
Use differentials to estimate the corresponding change in the value of d. Exact
evaluations of the function d will not give credit.
Problem 3 Suppose you know that the derivative of an unknown function
is
(x + 3)2 (x + 1)(x − 2)(x − 5)
f 0 (x) =
x2 + 1
Find all critical points of f (x). Use the First Derivative Test to determine
which are relative maxima and which are minima.
Problem 4 Find all relative extrema of
f (x) = x4 (x2 − 2)
1
Use the Second Derivative Test to determine whether they are relative minima or maxima.
Problem 5 Find all relative extrema of
f (x) =
x2 − 2
x+3
and tell which ones are relative minima and which ones are relative maxima.
Identify the intervals on which f (x) is increasing and the intervals on which
f (x) is decreasing.
Problem 6 Find two positive numbers x, y such that xy = 1 and
1 2
+
x y
is minimal.
Problem 7 Suppose the cost of making x units of FanCytm hairdryers is
C(x) = 1500 + 60x + 0.002x3
(dollars). If you can sell all of them at a price of 75 dollars each, find the
number x which will give you maximum profit. Round the answer to the
nearest whole integer.
2