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.

 A Decreasing on and increasing on B Decreasing on and increasing on C Decreasing on and increasing on

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).

 A Decreasing on , increasing on , and constant on B Decreasing on , increasing on , and constant on C Decreasing on , increasing on , and constant on

3.

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 , decreasing on D Increasing on

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.

 A D B E C

7.

Find the relative maxima and relative minima of the function.

 A Relative minimum: ;   Relative maximum: B Relative minimum: ;   Relative maximum: C Relative minimum: ;   Relative maximum: D Relative minimum: ;   Relative maximum: E Relative minimum: ;   Relative maximum:

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 .

Find the intervals where the profit function P is increasing and the intervals where P is decreasing.
 A increasing on , decreasing on B increasing on , decreasing on C increasing on , decreasing on D increasing on , decreasing on E increasing on , decreasing on

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.

 A B C D

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.

 A concave upward on ;concave downward on B concave upward on ;concave downward on C concave upward on ;concave downward on D concave upward on ;concave downward on E concave upward on ;concave downward on

14.

Determine where the function is concave downward.

 A B C D

15.

Find the inflection points of the following function.

 A , , and B and C and D and E and

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
 A D B E C

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.

 A Horizontal asymptote is , vertical asymptote is B Horizontal asymptote is , vertical asymptote is C Horizontal asymptote is , vertical asymptote is D Horizontal asymptote is , vertical asymptote is E Horizontal asymptote is , vertical asymptote is

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.

 A C B

29.

The total worldwide box-office receipts for a long-running 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.
 A C B D

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 .
 A B C D E

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 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 () 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?
 A B C D

39.

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

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.
 A B C D

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   .

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.
 A C B

45.

46.

47.

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.