Multiple Choice Identify the
choice that best completes the statement or answers the question.


1.

You are given the graph of a function f. Determine the intervals where
f is increasing, constant, or decreasing.


2.

The graph of the function f shown in the accompanying figure gives the
elevation of that part of the Boston Marathon course that includes the notorious Heartbreak Hill.
Determine the intervals (stretches of the course) where the function f is increasing (the
runner is laboring), where it is constant (the runner is taking a breather), and where it is
decreasing (the runner is coasting).


3.

Find the interval(s) where the function is increasing and the interval(s) where
it is decreasing.


4.

Find the interval(s) where the function is increasing and the interval(s) where
it is decreasing.
A  Increasing on , decreasing on  B  Increasing on , decreasing on  C  Increasing on  D  Decreasing on 


5.

Determine the relative maxima and relative minima, if any.
A  No relative maxima; Relative minimum: f(5) =
0 and f(  5) = 0  B  Relative
maximum: f(0) = 5; No relative minima  C  Relative
maximum: f(0) = 5; Relative minima: f(5) = 0
and f(  5) = 0  D  No relative maxima or
minima 


6.

Find the graph of the derivative of the function.


7.

Find the relative maxima and relative minima of the function.


8.

The Mexican subsidiary of ThermoMaster manufactures an indooroutdoor
thermometer. Management estimates that the profit (in dollars) realizable by the company for the
manufacture and sale of x units of thermometers each week is . Find
the intervals where the profit function P is increasing and the intervals where P is
decreasing.


9.

The height (in feet) attained by a rocket t sec into flight is given by
the function . When is the rocket rising and when is it
descending?
A  The rocket is rising on the interval (0,58) and it is descending on the interval (58,
t) where t is some positive number more than 58.  B  The rocket is rising
on the interval (0,45) and it is descending on the interval (45, t) where t is some positive
number more than 45.  C  The rocket is rising on the interval (0,21) and
it is descending on the interval (21, t) where t is some positive number more than
21.  D  The rocket is rising on the interval (0,17) and it is descending on the interval (17,
t) where t is some positive number more than 17. 


10.

Show that the function has no relative extrema on
.
A  The derivation of the function is
. At every point the derivative exists and does
not equal to 0. So by definition the function has no relative extrema on this
interval.  B  The function has no relative extrema because does not equal
to 0 at any point.  C  The function has no relative extrema by
definition of the derivation. 


11.

You are given the graph of a function f. Determine the intervals where
f is concave upward.


12.

Show that the function is concave upward wherever it
is defined.
A  The second derivative of g(x) is . It is positive
for any value of x, hence the function is concave upward for any x.  B  The first derivative
of g(x) is . It is positive for any value of x, hence
the function is concave upward for any x.  C  The function g(x) is defined for
any value of x and hence the function is concave upward for any
x. 


13.

Determine where the function is concave upward and where it is concave
downward.


14.

Determine where the function is concave downward.


15.

Find the inflection points of the following function.


16.

Find the inflection points, if any, of the function.
A  (0, 0)  B  (0, 9)  C  (1,
1)  D  none 


17.

Find the relative extrema of the following function. Use the second derivative
test, if applicable.
A  Relative maximum:  B  Relative minimum:  C  Relative maximum:  D  Relative minimum:  E  Relative minimum: 


18.

Sketch the graph of the function having the given properties. , , ,
, on , on
, inflection point at


19.

An efficiency study conducted for a company showed that the number of devices
assembled by the average worker t hr after starting work at 8 A.M. is given by At what time during the morning shift is the average worker performing at peak
efficiency?
A  At 8 A. M. the average worker is performing at peak efficiency.  B  At 9 A. M. the
average worker is performing at peak efficiency.  C  At 11 A. M. the average worker is performing at
peak efficiency.  D  At 10 A. M. the average worker is performing at peak
efficiency. 


20.

Find the horizontal and vertical asymptotes of the graph.
A  Vertical asymptote: x = 0 .  B  Horizontal asymptote:
y = 2 ; vertical asymptote x =
0  C  Horizontal asymptote: y = 2  D  Horizontal
asymptotes: y = 2 and y =
3 


21.

Find the horizontal and vertical asymptotes of the graph.
A  Vertical asymptote: x = 0  B  Horizontal
asymptote: y = 0.5; Vertical asymptote: x =
0  C  Horizontal asymptotes: y = 0.5 and
y = 1.5  D  Horizontal asymptote: y =
0.5 


22.

Find the horizontal and vertical asymptotes of the graph.


23.

Find the horizontal and vertical asymptotes of the graph.
A  Vertical asymptotes: x = 2 and x
=  2; Horizontal asymptote: y = 3  B  Horizontal
asymptotes: y = 2 and y =  2  C  Vertical
asymptotes: x = 2 and x = 
2; Horizontal asymptote: y = 1  D  Horizontal
asymptotes: y = 2 and y =  2;
Vertical asymptote: x = 1 


24.

Find the horizontal and vertical asymptotes of the graph of the function.
A  Vertical asymptote: x = 2; Horizontal
asymptote: y = 0  B  Vertical asymptote: x =
2; Horizontal asymptote: y = 3  C  Vertical asymptote:
x = 3; Horizontal asymptote: y =
2  D  Vertical asymptote: x = 2; Horizontal
asymptote: y = 1 


25.

Find the horizontal and vertical asymptotes of the graph of the function.
A  Vertical asymptote: t = 1; Horizontal
asymptote: y = 3  B  Vertical asymptotes: t =
1 and t = 1; Horizontal asymptote:
y = 2  C  Vertical asymptote: t
= 1; Horizontal asymptote: y = 3
and y = 2  D  Vertical asymptote: t
= 3; Horizontal asymptote: y =
1 


26.

Find the horizontal and vertical asymptotes of the graph of the function.
A  Vertical asymptotes: x = 4, x =
1 and x = 0  B  Vertical
asymptote: x = 4; Horizontal asymptotes: y
= 0  C  Vertical asymptotes: x = 4 and x =
1; Horizontal asymptotes: y = 1 and y = 1
 D  Vertical asymptote: x = 4 


27.

One of the functions below is the derivative function of the other. Identify
each of them.
A  Functions are independent of each other  B  g is the
derivative function of the function f  C  f is the derivative function of the
function g 


28.

Use the information summarized in the table to select the graph of f.


29.

The total worldwide boxoffice receipts for a longrunning movie are
approximated by the function where is measured in
millions of dollars and x is the number of years since the movie's release.
Select the graph of the function T.


30.

You are given the graph of some function f defined on the indicated
interval. Find the absolute maximum and the absolute minimum of f, if they exist.
A  Absolute maximum value: 7.4; absolute minimum value: 0  B  Absolute maximum
value: none; absolute minimum value: 0  C  Absolute maximum value: 7.4; absolute minimum
value: none  D  Absolute maximum value: none; absolute minimum value:
none 


31.

You are given the graph of some function f defined on the indicated
interval. Find the absolute maximum and the absolute minimum of f, if they exist.
A  Absolute maximum value: 4; absolute minimum value:  1  B  Absolute maximum
value: 6; absolute minimum value:  1  C  Absolute maximum value: 6; absolute minimum
value: 0  D  Absolute maximum value: 3; absolute minimum value: 
1 


32.

Find the absolute maximum value and the absolute minimum value, if any, of the
given function.
A  Absolute maximum value: ; absolute minimum value:
none  B  Absolute maximum value: ; absolute minimum value:
none  C  Absolute maximum value: none; absolute minimum value:  D  No absolute extrema 


33.

Find the absolute maximum value and the absolute minimum value, if any, of the
function.
A  Absolute maximum value: 9; absolute minimum value: 0  B  Absolute maximum
value: 5; absolute minimum value:  4  C  Absolute maximum value: 4; absolute minimum
value:  5  D  Absolute maximum value: 0; absolute minimum value:  48  E  Absolute maximum
value: 9; absolute minimum value:  48 


34.

Find the absolute maximum value and the absolute minimum value, if any, of the
given function.
A  Absolute maximum value: ; absolute minimum value:  B  Absolute maximum value: ; absolute minimum value:  C  Absolute maximum value: none; absolute minimum value:  D  No absolute extrema 


35.

The quantity demanded each month of the Walter Serkin recording of
Beethoven's Moonlight Sonata, manufactured by Phonola Record Industries, is related to the
price/compact disc. The equation , where p denotes the
unit price in dollars and x is the number of discs demanded, relates the demand to the price.
The total monthly cost (in dollars) for pressing and packaging x copies of this
classical recording is given by . To maximize its profits, how many copies
should Phonola produce each month? Hint: The revenue is , and the profit
is .


36.

Suppose the total cost function for manufacturing a certain product is dollars, where x represents the number of units produced. Find the level of
production that will minimize the average cost. Round the answer to the nearest integer.
A  40 units  B  44 units  C  46
units  D  50 units 


37.

After the economy softened, the skyhigh office space rents of the late 1990s
started to come down to earth. The function R gives the approximate price per square foot in
dollars, R( t), of prime space in Boston's Back Bay and Financial District from
1997 ( ) through 2000, where . What was the highest office space rent during the period in question?
Hint: Use the quadratic formula.
A  $53.07 per sq ft  B  $53.02 per sq ft  C  $52.92 per sq
ft  D  $53.12 per sq ft  E  $52.97 per sq ft 


38.

The owner of the Rancho Los Feliz has 2,600 yd of fencing material with which to
enclose a rectangular piece of grazing land along the straight portion of a river. If fencing is not
required along the river, what are the dimensions of the largest area that he can enclose? What is
this area?


39.

If an open box has a square base and a volume of 500 and is
constructed from a tin sheet, find the dimensions of the box, assuming a minimum amount of material
is used in its construction.


40.

A book designer has decided that the pages of a book should have margins at the top and bottom and margins on the sides. She
further stipulated that each page should have an area of (see the figure). Determine the page dimensions that will result in the maximum printed area on the
page.


41.

If exactly 150 people sign up for a charter flight, Leisure World Travel Agency
charges $250/person. However, if more than 150 people sign up for the flight (assume this is the
case), then each fare is reduced by $1 for each additional person.
Determine how many
passengers will result in a maximum revenue for the travel agency. What is the maximum revenue? What
would be the fare per passenger in this case? Hint: Let x denote the number of
passengers above 150. Show that the revenue function R is given by R(x)
= (150 + x)(250  x).
A  200; $40,000; $200  B  250; $39,000; $250  C  250; $40,000;
$250  D  200; $39,000; $200 


42.

The owner of a luxury motor yacht that sails among the 4,000 Greek islands
charges $600/person/day if exactly 20 people sign up for the cruise. However,if more than 20 people
sign up (up to the maximum capacity of 90) for the cruise, then each fare is reduced by $4 for each
additional passenger.
Assuming at least 20 people sign up for the cruise, determine how many
passengers will result in the maximum revenue for the owner of the yacht. What is the maximum
revenue? What would be the fare/passenger in this case?
A  85; $28,900; $340  B  90; $28,400; $350  C  90; $28,900;
$350  D  85; $28,400; $340 


43.

Neilsen Cookie Company sells its assorted butter cookies in containers that have
a net content of 1 lb. The estimated demand for the cookies is 1,000,000 1lb containers. The setup
cost for each production run is $250, and the manufacturing cost is $.30 for each container of
cookies. The cost of storing each container of cookies over the year is $.20. Assuming
uniformity of demand throughout the year and instantaneous production, how many containers of cookies
should Neilsen produce per production run in order to minimize the production cost?
Hint: Show that the total production cost is given by the function . Then minimize the function on the interval (0,
1,000,000).
A  50,000  B  40,000  C  45,000  D  35,000 

Numeric Response


44.

Suppose the total cost function for manufacturing a certain product is dollars, where x represents the number of units produced. Find the level of
production that will minimize the average cost. Round the answer to the nearest integer.
__________ units

Matching



Match the graph of the function with the graph of its
derivative.
Choose the correct letter for each question.


45.



46.



47.


Short Answer


48.

Find the relative maxima and relative minima, if any, of the function.
Otherwise, answer none. Relative minima: __________
Relative maxima: __________


49.

Find the inflection points, if any, of the following function. Otherwise, answer
no solution.

Essay


50.

The level of ozone, an invisible gas that irritates and impairs breathing,
present in the atmosphere on a certain May day in the city was approximated by , where is measured in pollutant standard index (PSI)
and is measured in hours, with
corresponding to a.m. Use the second derivative test to
show that the function has a relative maximum at approximately . Interpret your results.
