Practice Problems: Exam 2
1. A chemist prepares a "!!-mg sample of polonium 210. Because of radioactive decay, the mass of
the sample after one day is only **Þ& milligrams.
(a) Find a formula for the mass of the sample after B days.
(b) How long will it take for the mass of the sample to reach 50 milligrams? Round your answer to
the nearest day.
2. The function 0 aBb œ #B%  #&B# is graphed below:
Find the coordinates of the point T .
3. During a certain biology experiment, the population of a certain culture of bacteria after > hours was
given by the following formula:
T œ '!! /!Þ$&>
(a) What was the population of bacteria at the beginning of the experiment?
(b) How quickly was the population of bacteria increasing at the beginning of the experiment?
(c) What was the percentage growth rate for the bacteria colony?
(d) When did the population of bacteria reach #!!!? Your answer must be correct to the nearest
minute.
4. The following picture shows the curve defined by the equation B$  C$ œ *C:
(a) Use implicit differentiation to find a formula for
.C
in terms of B and C.
.B
(b) Use your answer to part (a) to find the slope of the curve at the point a#ß "b.
(c) Use differentials to estimate the C-coordinate of a point on the curve whose B-coordinate
is #Þ!!$.
5. A lighthouse sits on a small island near a rocky shoreline, emitting a rotating beam of light. The
lighthouse is $ km from the shore, and it emits a beam of light that rotates four times per minute:
(a) Find a formula for B in terms of ).
(b) Using your answer to part (a), find a formula for
.B
.)
in terms of ) and
.
.>
.>
(c) Based on your formula from part (b), how quickly is the end of the light beam moving along the
shoreline when ) œ $! degrees?
6. (a) Find
.C
B
if C œ & sinŠ#  ‹Þ
.B
$
(b) Find 0 w aBb if 0 aBb œ aln B  ln #b$ .
(c) Find
. # $B
ˆB / ‰.
.B
(d) Find
.<
if < œ sec# ) tan ).
.)
(e) Find 1w a?b if 1a?b œ
"
.
sin" ?
(f) Find
.C
if C œ tan$ ˆB#  "‰.
.B
(g) Find
.)
if ) œ arctanalog >b.
.>
(h) Find
.C
if C$  sin C œ B  cos B.
.B
7. Recall that the decibel level P of a sound is related to the power T of the sound by the equation:
P œ "! logŒ
T

T!
where T! is a constant equal to "!"# watts.
(a) Find a formula for the derivative
.P
.
.T
(b) Suppose a certain sound has a power of #Þ! watts, with a maximum error in measurement of
!Þ" watts. Use differentials to estimate the maximum possible error in the decibel level of the
sound.
8. Suppose you deposit $2000 in a bank account with a nominal interest rate of 8%Îyear, compounded
continuously.
(a) Find a formula for the balance after > years.
(b) How long until the balance reaches $3000? Your answer should be correct to the nearest day.
9. 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:
Eœ
"
+, sin )
#
For this problem, the lengths + and , are both constant, with + œ % inches and , œ & inches.
(a) Suppose that ) is increasing at a rate of !Þ"& radiansÎminute. How quickly is the area of the
triangle increasing at the instant that ) œ !Þ% radians?
(b) Suppose instead that the area is increasing at a rate of !Þ& in# Îmin. How quickly is the angle )
increasing at the instant that the area is ' square inches?
10. The function 0 aBb œ
$'
 #&B is graphed below.
B
(a) Find the minimum value of 0 aBb on the interval c!Þ&ß #Þ&d.
(b) Find the maximum value of 0 aBb on the interval c!Þ&ß #Þ&d.