Practice Problems: Exam 2
1. A chemist prepares a 100-mg sample of 210 Po. Because of radioactive decay, the mass of the sample
after one day is only 99.5 mg.
(a) Find a formula for the mass of the sample after x days.
(b) How long will it take for the mass of the sample to reach 50 mg? Round your answer to the nearest
day.
2. A lighthouse sits on a small island near a rocky shoreline, emitting a rotating beam of light. The
lighthouse is 3 km from the shore, and it emits a beam of light that rotates four times per minute:
shore
B
light
beam
3 km
)
lighthouse
island
(a) Find a formula for x in terms of θ .
dθ
dx
in terms of θ and
.
dt
dt
(c) Based on your formula from part (b), how quickly is the end of the light beam moving along the
(b) Using your answer to part (a), find a formula for
shoreline when θ = 30 degrees?
3. Consider the function f (x) =
36
+ 25x.
x
(a) Find the absolute minimum value of f (x) on the interval [0.5, 2.5].
(b) Find the absolute maximum value of f (x) on the interval [0.5, 2.5].
4. (a) Find
dy
3x − 5
if y = 2
.
dx
x −4
(b) Find f 0 (x) if f (x) = (ln x − ln 2)2 .
(c) Find
d 2 −3x x e
.
dx
(d) Find
dy
if y = x sec(x2 + 1).
dx
(e) Find g 0 (x) if g(x) = sin−1 (3x2 ).
(f) Find
dy
if y = tan3 (x2 + 1).
dx
(g) Find
dθ
if θ = arctan(lnt).
dt
5. Let f (x) = x4 + 8x3 .
(a) Find the critical points of f (x).
(b) Classify each critical point as a local max, a local min, or neither.
(c) On what interval(s) is f (x) increasing? On what interval(s) is f (x) decreasing?
(d) Find the inflection points of f (x).
(e) On what interval(s) is f (x) concave up? On what interval(s) is f (x) concave down?
6. The following formula lets you determine the area of a triangle given the lengths of two sides and the
measure of the angle between them:
)
+
,
1
A = ab sin θ
2
For this problem, the lengths a and b are both constant, with a = 4 inches and b = 5 inches.
(a) Suppose that θ is increasing at a rate of 8.5◦ /min. How quickly is the area of the triangle increasing
at the instant that θ = 23◦ ?
(b) Suppose instead that the area is increasing at a rate of 0.5 in2 /min. How quickly is θ increasing at
the instant that the area is 6 square inches?
7. Given that f 0 (x) = 20x4 +
1
and f (1) = 25, find a formula for f (x).
x
8. Carol plans to create a large open box (a box without a top) from a piece of cardboard. She has a
cardboard rectangle with side lengths 6 feet and 12 feet. She will cut off a square with side length a
from each corner, and then fold the resulting flaps up to create a box.
+
6
+
12
(a) Find a formula for the volume of the box in terms of a.
(b) Find the maximum possible volume of the box.
9. A capacitor holds 20.0 mC of electric charge. The capacitor is attached to a light bulb, and charge
begins to flow out of it at a rate of 0.4 mC/sec.
(a) Assuming the charge left in the capacitor decays exponentially, find a formula for the charge Q
left in the capacitor after t seconds.
(b) How long will it take for half of the charge in the capacitor to be expended?
10. A cylindrical cup with an open top must have a volume of 300 cm3 :
<
2
Find the dimensions of the cup that minimize the amount of material used.
11. Suppose that f (x) is an exponential function. Given that f (0) = 50 and f 0 (0) = 20, find a formula
for f (x).
12. The following table gives some information about the derivative of a function f (x) on the interval [0, 8].
x
0
1
2
3
4
5
6
7
8
f 0 (x) 1.2 0.0 −0.6 −1.0 −0.7 −0.3 0.0 −0.3 −0.6
(a) Based on this data, how many critical points does f (x) have on the interval [0, 8]?
(b) Classify each critical point as a local max, a local min, or neither.
(c) On what interval(s) is f (x) increasing? On what interval(s) is f (x) decreasing?
(d) Based on this data, find the x-coordinate of each inflection point of f (x) on the interval [0, 8].
(e) On what interval(s) is f (x) concave up? On what interval(s) is f (x) concave down?
√
13. Given that f 0 (x) = 12e3x + x + 5 and f (0) = 6, find a formula for f (x).