Name: 
 

MAT 121 Section 14                  Test #3 (Chapter 4)



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.

mc001-1.jpg
A
Decreasing on mc001-2.jpg and increasing on mc001-3.jpg
B
Decreasing on mc001-4.jpg and increasing on mc001-5.jpg
C
Decreasing on mc001-6.jpg and increasing on mc001-7.jpg
 

 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).

mc002-1.jpg
A
Decreasing on mc002-2.jpg, increasing on mc002-3.jpg, and constant on mc002-4.jpg
B
Decreasing on mc002-5.jpg, increasing on mc002-6.jpg, and constant on mc002-7.jpg
C
Decreasing on mc002-8.jpg, increasing on mc002-9.jpg, and constant on mc002-10.jpg
 

 3. 

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

mc003-1.jpg
A
Increasing on mc003-2.jpg, decreasing on mc003-3.jpg
B
Increasing on mc003-4.jpg, decreasing on mc003-5.jpg
C
Increasing on mc003-6.jpg, decreasing on mc003-7.jpg
D
Increasing on mc003-8.jpg
 

 4. 

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

mc004-1.jpg
A
Increasing on mc004-2.jpg, decreasing on mc004-3.jpg
B
Increasing on mc004-4.jpg, decreasing on mc004-5.jpg
C
Increasing on mc004-6.jpg
D
Decreasing on mc004-7.jpg
 

 5. 

Determine the relative maxima and relative minima, if any.

mc005-1.jpg
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.

mc006-1.jpg
A
mc006-2.jpg
D
mc006-5.jpg
B
mc006-3.jpg
E
mc006-6.jpg
C
mc006-4.jpg
 

 7. 

Find the relative maxima and relative minima of the function.

mc007-1.jpg
A
Relative minimum: mc007-2.jpg;   Relative maximum: mc007-3.jpg
B
Relative minimum: mc007-4.jpg;   Relative maximum: mc007-5.jpg
C
Relative minimum: mc007-6.jpg;   Relative maximum: mc007-7.jpg
D
Relative minimum: mc007-8.jpg;   Relative maximum: mc007-9.jpg
E
Relative minimum: mc007-10.jpg;   Relative maximum: mc007-11.jpg
 

 8. 

The Mexican subsidiary of ThermoMaster manufactures an indoor-outdoor 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 mc008-1.jpg.

Find the intervals where the profit function P is increasing and the intervals where P is decreasing.
A
increasing on mc008-2.jpg, decreasing on mc008-3.jpg
B
increasing on mc008-4.jpg, decreasing on mc008-5.jpg
C
increasing on mc008-6.jpg, decreasing on mc008-7.jpg
D
increasing on mc008-8.jpg, decreasing on mc008-9.jpg
E
increasing on mc008-10.jpg, decreasing on mc008-11.jpg
 

 9. 

The height (in feet) attained by a rocket t sec into flight is given by the function 

mc009-1.jpg.

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 mc010-1.jpg has no relative extrema on mc010-2.jpg.
A
The derivation of the function  mc010-3.jpg   is   mc010-4.jpg. 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 mc010-5.jpg 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.

mc011-1.jpg
A
mc011-2.jpg
B
mc011-3.jpg
C
mc011-4.jpg
D
mc011-5.jpg
 

 12. 

Show that the function mc012-1.jpg is concave upward wherever it is defined.
A
The second derivative of g(x) is mc012-2.jpg. 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 mc012-3.jpg. 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 mc012-4.jpg hence the function is concave upward for any x.
 

 13. 

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

mc013-1.jpg
A
concave upward on mc013-2.jpg;
concave downward on mc013-3.jpg
B
concave upward on mc013-4.jpg;
concave downward on mc013-5.jpg
C
concave upward on mc013-6.jpg;
concave downward on mc013-7.jpg
D
concave upward on mc013-8.jpg;
concave downward on mc013-9.jpg
E
concave upward on mc013-10.jpg;
concave downward on mc013-11.jpg
 

 14. 

Determine where the function is concave downward.

mc014-1.jpg
A
mc014-2.jpg
B
mc014-3.jpg
C
mc014-4.jpg
D
mc014-5.jpg
 

 15. 

Find the inflection points of the following function.

mc015-1.jpg
A
mc015-2.jpg, mc015-3.jpg, and mc015-4.jpg
B
mc015-5.jpg and mc015-6.jpg
C
mc015-7.jpg and mc015-8.jpg
D
mc015-9.jpg and mc015-10.jpg
E
mc015-11.jpg and mc015-12.jpg
 

 16. 

Find the inflection points, if any, of the function.

mc016-1.jpg
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.

mc017-1.jpg
A
Relative maximum: mc017-2.jpg
B
Relative minimum: mc017-3.jpg
C
Relative maximum: mc017-4.jpg
D
Relative minimum: mc017-5.jpg
E
Relative minimum: mc017-6.jpg
 

 18. 

Sketch the graph of the function having the given properties.

mc018-1.jpg,   mc018-2.jpg,   mc018-3.jpg,   mc018-4.jpg,   mc018-5.jpg on mc018-6.jpg,   mc018-7.jpg on mc018-8.jpg, inflection point at mc018-9.jpg
A
mc018-10.jpg
D
mc018-13.jpg
B
mc018-11.jpg
E
mc018-14.jpg
C
mc018-12.jpg
 

 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

mc019-1.jpg

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.

mc020-1.jpg
A
Vertical asymptote: x = 0 .
B
Horizontal asymptote: y  =  -2vertical asymptote =  0
C
Horizontal asymptote:  y  =  -2
D
Horizontal asymptotes:  y  =  -2  and y  =  -3
 

 21. 

Find the horizontal and vertical asymptotes of the graph.

mc021-1.jpg
A
Vertical asymptote:   x  =  0
B
Horizontal asymptote:  y  =  0.5Vertical 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.

mc022-1.jpg
A
Horizontal asymptote is mc022-2.jpg, vertical asymptote is mc022-3.jpg
B
Horizontal asymptote is mc022-4.jpg, vertical asymptote is mc022-5.jpg
C
Horizontal asymptote is mc022-6.jpg, vertical asymptote is mc022-7.jpg
D
Horizontal asymptote is mc022-8.jpg, vertical asymptote is mc022-9.jpg
E
Horizontal asymptote is mc022-10.jpg, vertical asymptote is mc022-11.jpg
 

 23. 

Find the horizontal and vertical asymptotes of the graph.

mc023-1.jpg
A
Vertical asymptotes:  x  =  2  and  x  =  - 2Horizontal asymptote:  y  = 3
B
Horizontal asymptotes:  y  =  2 and y  = - 2
C
Vertical asymptotes:  x  =  2   and x  =  - 2Horizontal asymptote:  y  =  1
D
Horizontal asymptotes:  y  =  2   and y  = - 2Vertical asymptote: x  =  1
 

 24. 

Find the horizontal and vertical asymptotes of the graph of the function.

mc024-1.jpg
A
Vertical asymptote: x  =  -2Horizontal asymptote:  y  =  0
B
Vertical asymptote: x  =  -2Horizontal asymptote:  y  =  3
C
Vertical asymptote: x  =  3Horizontal asymptote:  y  =  -2
D
Vertical asymptote: x  =  -2Horizontal asymptote:  y  =  1
 

 25. 

Find the horizontal and vertical asymptotes of the graph of the function.

mc025-1.jpg
A
Vertical asymptote:   t  =  -1Horizontal asymptote:   y  =  3
B
Vertical asymptotes:  t  =  -1  and  t  =  1Horizontal asymptote:  y  =  2
C
Vertical asymptote:   t  =  -1Horizontal asymptote:   y  =  3  and  y  =  2
D
Vertical asymptote:   t  =  3Horizontal asymptote:   y  =  1
 

 26. 

Find the horizontal and vertical asymptotes of the graph of the function.

mc026-1.jpg
A
Vertical asymptotes:  x  =  -4,  x  =  -1   and   x  =  0
B
Vertical asymptote:  x  =  -4Horizontal 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.

mc027-1.jpg
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.

mc028-1.jpg
 
mc028-2.jpg
mc028-3.jpg
mc028-4.jpg
mc028-5.jpgmc028-6.jpg
 mc028-7.jpg
mc028-8.jpg
mc028-9.jpg
mc028-10.jpg
 
A
mc028-11.jpg
C
mc028-13.jpg
B
mc028-12.jpg
 

 29. 

The total worldwide box-office receipts for a long-running movie are approximated by the function

mc029-1.jpg

where mc029-2.jpg 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.
A
mc029-3.jpg
C
mc029-5.jpg
B
mc029-4.jpg
D
mc029-6.jpg
 

 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.

mc030-1.jpg

mc030-2.jpg
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.

mc031-1.jpg

mc031-2.jpg
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.

mc032-1.jpg
A
Absolute maximum value: mc032-2.jpg; absolute minimum value: none
B
Absolute maximum value: mc032-3.jpg; absolute minimum value: none
C
Absolute maximum value: none; absolute minimum value: mc032-4.jpg
D
No absolute extrema
 

 33. 

Find the absolute maximum value and the absolute minimum value, if any, of the function.

mc033-1.jpg
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.

mc034-1.jpg
A
Absolute maximum value: mc034-2.jpg; absolute minimum value: mc034-3.jpg
B
Absolute maximum value: mc034-4.jpg; absolute minimum value: mc034-5.jpg
C
Absolute maximum value: none; absolute minimum value: mc034-6.jpg
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  mc035-1.jpg, 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  mc035-2.jpg.

To maximize its profits, how many copies should Phonola produce each month? Hint: The revenue is mc035-3.jpg, and the profit is mc035-4.jpg.
A
mc035-5.jpg
B
mc035-6.jpg
C
mc035-7.jpg
D
mc035-8.jpg
E
mc035-9.jpg
 

 36. 

Suppose the total cost function for manufacturing a certain product is mc036-1.jpg 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 sky-high 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 (mc037-1.jpg) through 2000, where 

mc037-2.jpg.

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?
A
mc038-1.jpg
B
mc038-2.jpg
C
mc038-3.jpg
D
mc038-4.jpg
 

 39. 

If an open box has a square base and a volume of 500mc039-1.jpgand is constructed from a tin sheet, find the dimensions of the box, assuming a minimum amount of material is used in its construction.
A
mc039-2.jpg
B
mc039-3.jpg
C
mc039-4.jpg
D
mc039-5.jpg
 

 40. 

A book designer has decided that the pages of a book should have mc040-1.jpgmargins at the top and bottom and mc040-2.jpgmargins on the sides. She further stipulated that each page should have an area of mc040-3.jpg(see the figure).

mc040-4.jpg

Determine the page dimensions that will result in the maximum printed area on the page.
A
mc040-5.jpg
B
mc040-6.jpg
C
mc040-7.jpg
D
mc040-8.jpg
 

 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 1-lb 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   mc043-1.jpg.

Then minimize the function mc043-2.jpg 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 nr044-1.jpg 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.
A
grp001-1.jpg
C
grp001-3.jpg
B
grp001-2.jpg
 

 45. 

ma045-1.jpg
 

 46. 

ma046-1.jpg
 

 47. 

ma047-1.jpg
 

Short Answer
 

 48. 

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

sa048-1.jpg

Relative minima: __________

Relative maxima: __________
 

 49. 

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

sa049-1.jpg
 

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 

es050-1.jpg, where es050-2.jpg is measured in pollutant standard index (PSI) and es050-3.jpg is measured in hours, with es050-4.jpg corresponding to es050-5.jpg a.m.

Use the second derivative test to show that the function es050-6.jpg has a relative maximum at approximately es050-7.jpg. Interpret your results.
 



 
Check Your Work     Start Over