MATH 119
TEST 2 (Sample B Key)
NAME:
Class ID #:
1. You open an IRA account with an initial deposit of $10,000 which will accumulate tax-free at 4 % per year,
compounded continuously.
a) How much (to the nearest penny) will you have in your account after 10 years?
$ 14,918.25
b) How long does it take your initial investment to triple?
27.47 years
2. If 500 people have a personal computer in a town of 10,000 employees. If the number of PC was growing at
20% a year and the population at 10% per year. How long will it take to have PC per person? (assume
continuous growth)
29.96 years
3) The population of a certain town is declining exponentially. If the population now is 10% less than it was 5
years ago.
(a) Find the decline rate.
2.107%
(b) When will the population be 50% of the original? (find the half-life)
32.89 years
4) How long does it take amount to double at 8.5% compounded:
a) annually
b) continuously
a) t = 8.496
b) t = 8.154
5) If the quantity of a certain radioactive substance is decreases by 5% in 10 hours, find the half-life.
t = 135.13 hours
6) The population of a certain town is declining exponentially due to immigration. If only 80% of the original
population are still in town after 10 years:
a) Find the decline rate.
2.23%
b) How long will it take for the population to be half what it was?
31.06 years
7) The population of a town in millions is given by: P = 1.2(1.01)t where t is the number of years since the start
of 1998 (i.e. t = 0 corresponds the year 1998). Find:
a) The population in 2000
P = 1.22412 B
b) The average rate of growth between 1998 and 2000:
0.01206 M/Y
c) How fast the population is growing at the start of 1998? (Hint: Estimate the instantaneous rate of change
of P at t = 0 using h = 0.01)
0.01194045 M/Y
8) Use the graph on the right to sketch following:
y
B
a) the line segment corresponding to f (b)  f ( a ) ;
label that line segment as line A;
b) the line whose slope is given by
A
f (b)  f (a )
;
ba
label that line as line B;
C
c) the line whose slope is given by f ' ( c) ;
label that line as line C;
b
c
a
k following table:
9) The distance s traveled by an object as a function of time t is given in the
t (sec)
s (feet)
0
0
1
2
2
5
3
9
4
15
a) Find the average velocity of the object between t = 1 and 4.
Av = 4.3
b) Estimate the velocity of the object at t = 3. (you can use one interval only)
5
27
V=6
x
10) Sketch the graph of the first and second derivatives of the functions given below. Be sure that your
sketches are consistent with the important features of the original functions.
f(x)
f(x)
f ’(x)
f ’(x)
f" (x)
f"(x)
11) Draw a possible graph of y  f ( x ) given the following information about its derivative:

f '( x )  0 for x < 1

f '( x )  0 for 1 < x < 3 and x > 3

f ' ( x )  0 at x = 1 and x = 3
1
2
3
12) Using the following graph, estimate the intervals or points where:
f '( x )  0
b<x<d
f ' ( x)  0
a<x<b
d<x<e
f ' ' ( x)  0
x=c
f ' ' ( x)  0
a<x<c
f ' ' ( x)  0
c<x<e
f ' ' ( x)  0 and
f ' ( x)  0
x=d
f ' ' ( x)  0 and
f ' ( x)  0
x=b
b
a
c
e
d
13) Suppose that f (t ) is a function, that f (10)  7 and that f ' (10)  0.2 . Use this information to estimate
f (12) .
f(12) = 6.6
14) Using the following table:
Quantity: q
Cost: C
Revenue: R
15
140
30
20
250
150
25
340
240
30
400
290
a) Estimate the marginal cost and the marginal revenue produced by the 20th unit.
C'(20)
R'(20)
= (340-250)/(25-20) = 18
b) What happens when q = 20 units?
C ' = R ', then it is a Maximum Profit
= 18
C
R
q
0
20
50
70
15) The graph above shows the Cost and Revenue functions associated with a certain product. Give the
value(s) of q for which
RC
20 < q < 70
CR
q < 20 ; q > 70
CR
q = 20, q = 70
R'  C '
q < 50
C'  R'
q > 50
C '  R'
q = 50
What is the value of q that will maximize profit? q = 50
After producing 30 units, should the manufacturer produce more? Why? yes, R ' > C '
After producing 60 units, should the manufacturer produce more? Why? No, C ' > R '
16) If f(x) = x2 + 3x, find f ' (2) .
(use h = 0.01, and show all steps)
When h = 2.01,
When h = 1.99,
Then
f ' (2) = 7.01
f ' (2) = 6.99
f ' (2) = 7
Bonus (see page 129):
f ( x  h)  f ( x )
17) Use the definition of the derivative f ' ( x)  lim
to show that
h
h 0
If f ( x)  2 x 2  x  1 , then f ' ( x)  4 x  1
4 xh  2h 2  h
 lim 4 x  2h  1  4 x  1
h
h 0
h 0
The last steps are f ' ( x)  lim