Sample Multiple-Choice Questions
For questions 1, 2 and 3 below, let the functions f and g be defined as follows:
f ( x)  x  3
1.
and
g ( x)  x 2  8 x  15 .
 f ( x) 
lim  g ( x) 
Answer: C
x  3
a.
c.
2.
1
8
1
2
b.
0
d.
The limit does not exist.
lim  f ( g ( x) 
Answer: A
x 0
a.
c.
3.
18
12
b.
d.
15
The limit does not exist.
 1 
lim  f ( x) 
Answer: A
x  3
a.
c.
4.


For f ( x ) 
b.
d.
1
,
x2
lim  f ( x) 
0
0
is given by
Answer: C
x 2
a.

b.
c.

d.
0
1
4
Sample Problem 1 [Linear Model]
In the Fahrenheit temperature scale, water freezes at 32° F and boils at 212° F. In the Celsius
temperature scale, water freezes at 0° C and boils at 100° C.
a)
Knowing that the two temperature scales are linearly related, write F as a function of C.
F(C) = 1.8C + 32
b)
What temperature is the same in both scales?
Solving the equation F = C yields a temperature of -40 degrees.
Sample Problem 2 [Linear & Quadratic Model]
You operate a gaming website, www.virtualvegas.com, where users must pay a small fee to log
on. When you charged $2.00 the demand was 280 log-ons per month. When you lowered the
price to $1.50, the demand increased to 560 log-ons per month.
a)
Construct a linear demand function for your website
on fee (in dollars) and is the number of log-ons per month.
q = -560p + 1400
b)
Obtain the monthly revenue , in dollars, as a quadratic function of .
R(p) = -560p2 + 1400p
c)
If your internet provider charges you a monthly fee of $30 to maintain your site,
express your monthly profit as a function of .
Profit = Revenue – Cost
So
d)
, where
is the log-
P(p) = R(p) – C(p) = -560p2 + 1400p - 30
What log-on fee should you charge to maximize your profits? What is your maximum
profit then?
Optimal price = -1400 / (2 x -560) = -1400 / -1120 = $1.25
Maximal profit = $845
Sample Problem 3 [Exponential/Compounding Model]
You invest $20,000 in a mutual fund with an annual yield of 12.4%.
a)
If the earnings are compounded every month, how much money will you have after
three and a half years?
$30,799.71
b)
How much more money would you get if the monthly compounding scheme is changed
to a continuous compounding scheme?
$30,868.38 - $30,799.71 = $68.67
Sample Problem 4
The cost of fighting crime in the U.S. increased linearly in the period 1982-1999.
Total spending (in billions of dollars) on police and courts could be approximated, respectively,
by P(t )  1.745t  29.84 and C (t )  1.097t  10.65 , where t is the time in years since 1980.
Compute
to two decimal places and interpret your result.
The limit is the number 1.745/1.097 = 1.59 (rounded to the nearest hundredth).
So, in the long term (i.e. decades from now), these models predict that for every 1$
spent on courts, $1.59 will be spent on police or, equivalently, that spending on
police will be 59% higher than spending on courts.
Sample Problem 5 [Logistic Model]
There are currently 1,000 cases of a nasty flu going around in a total susceptible population of
10,000. The number of cases, P (t ) , is increasing by 25% each day.
a)
Find a logistic model of the form P (t ) 
N
, where N is the limiting value and
1  Ab  t
A, b are nonzero positive constants.
N = 10,000. This is the limiting value.
Since P(0) = 1,000, we have:
or
10,000 / (1+A) = 1,000
A=9
Since P(1) = 1,250 (b/c the number of cases increased by 25% in one day), we
have:
b)
or
or
or
10,000 / (1 + 9/b) = 1,250
(1 + 9/b) = 8
9/b = 7
b = 9/7
=>
P(t ) 
10, 000
9
1 9  
7
t

10, 000
7
1 9  
9
t
Use your model to predict the number of flu cases three weeks from now.
P(21) = 9,561 cases (rounded to the nearest case).
Sample Problem 6 [Average rate of Change]
The following table shows the amount of money, per year, that companies invested in venture
capital during the period 1995 – 2001 ( t  5 represents 1995):
Year, t
Investment
M (t ) (billions
of USD)
5
0.05
6
0.5
7
1
8
2
9
8
10
16
11
4
ARC[1995-1998] = (2 – 0.05) / 3 = 0.65
ARC[1996-1999] = (8 – 0.5) / 3 = 2.5
ARC[1997-2000] = (16 – 1) / 3 = 5
ARC[1998-2001] = (4 – 2) / 3 = 0.67 (rounded to the nearest hundredth)
During which three-year interval(s) was the average rate of change of M (t ) …
a)
…the greatest?
1997 to 2000
b)
…the least?
1995 to 1998
Sample Problem 7 [Average rate of Change - SARS]
Exercise # 46 on page 237
a)
The ARC of A(t) from t = 18 (April 19) to t = 28 (April 29), rounded to the
nearest one, is given by:
ARC = [A(28) – A(18)] / [28 – 18] = [5,409.7 – 3,654.6] / 10 = 176
So between April 19 and April 28 of 2003, the number of reported SARS
cases was increasing by approximately 176 cases per day.
b)
Graphing this exponential growth (use, for example, a WINDOW where
Xmin = 0, Xmax = 30, Ymin = 0, and Ymax = 5,000), it becomes clear that
the correct answer is A: the number of reported SARS cases is increasing at
a faster and faster rate.