MATH 2070
TEST ONE VERSION A
Spring 2015
Multiple Choice: In this section there are 17 multiple choice questions, 13 are worth 3 points each, four are
worth 1 point each, for a total of 43 points. These will not be hand-graded, so it is not necessary to show any
work. But, for your own record, place a small indicating mark (that cannot be seen by others in the room) by
your choices on this test document. You will not receive your Scantron back with your test. As you work, keep
your Scantron covered, so it cannot be read by others in the testing space.
1. A dairy uses process machinery to package milk into individual quart-sized coated paper containers.
The machine fills a continuous coated paper tube with milk, seals and cuts the tube between the quarts,
folds the containers, and pallets the quarts of milk for shipment to stores.
On a particular day, the process machinery began to package the milk at a rate of 400 quarts per
minute. After 4 minutes machine speed was steadily increased, so that after 2 additional minutes the
machine was processing 960 quarts per minute. The machine continued at this rate until the milk was
packaged, which took a total of 23 minutes, as shown in the graph below.
How much milk was packaged in the 23 minutes?
A.
B.
C.
D.
E.
19,280 quarts
17,920 quarts
18,480 quarts
21,760 quarts
22,080 quarts
2. r ( x)  765 x 2  232 x  459 gallons per hour is a rate of change function (check: r(2) = 3055).
If R(x) is an antiderivative of r(x), and it is known that R(4) = 30,068, compute R(1).
A.
B.
C.
D.
E.
14499 gallons
14388 gallons
14455 gallons
14377 gallons
14366 gallons
February 11, 2015
Page 1 of 13
MATH 2070
3.
Spring 2015
r ( x)  765 x 2  232 x  459 gallons per hour is a rate of change function (check: r(2) = 3055).
Find the average value of r(x) on the interval 2  x  2.
A.
B.
C.
D.
E.
4.
TEST ONE VERSION A
1479 gallons per hour
5916 gallons per hour
1386 gallons per hour
4964 gallons per hour
1266 gallons per hour
Given P  x   45x3  21x 2  15x pounds, where x equals the number of hours since 4:00 pm (check:
P(3) = 1449). Find the average rate of change of P(x) on the interval 2  x  6.
A.
B.
C.
D.
E.
5.
4024 pounds per hour
2523 pounds per hour
1386 pounds per hour
4964 pounds per hour
1266 pounds per hour
f ( x)  0.2 x3  4 x 2  6 x  20 is graphed to
the right. Check: f  2   17.6 Use two midpoint rectangles to estimate the
signed area between f(x) and the x-axis on the
the interval 1  x  9 .
A.
B.
C.
D.
E.
15.6
11.2
232
516.8
242.7
February 11, 2015
Page 2 of 13
MATH 2070
TEST ONE VERSION A
Spring 2015
p

d 
6. f(x) is a continuous function.
  f ( x)dx  
dp  4

A.
B.
C.
D.
E.
F(p)
f(x)
F(x)
f(p)
All of the above may be considered correct.
7. The graph shows a rate of change function.
Point a corresponds to what type of feature on a
related accumulation function?
A.
B.
C.
D.
It is a relative minimum
It is a relative maximum
It is an inflection point
It does not correspond to anything
8. The graph shows an accumulation function.
Point a corresponds to what type of feature on a
related rate of change function?
A.
B.
C.
D.
It is a relative minimum
It is a relative maximum
It is an inflection point
It does not correspond to anything
4
9. Find the value of the following expression
  3x
3
 4 x 2  5 x  2 dx
1
A.
B.
C.
D.
E.
144.25
38.75
138.75
32.25
56.50
February 11, 2015
Page 3 of 13
MATH 2070
TEST ONE VERSION A
Spring 2015
10. Given the following information:
10
d
J ( x)  j ( x) ; j(2) = 8; j(10) = 12;  j ( x)dx  22 and J(2) = 10. Find J(10), if possible.
dx
2
A. J(10) = 30
B. J(10) = 34
C. J(10) = - 4
D. J(10) = 32
E. It is not possible to find J(10) with the information given.
11. The height of bamboo plant is P(h)  0.042h3  0.13h 2  7 h  C inches, where h is the number of
hours after 12:00 noon. At 2:00 pm this plant was 16 inches tall. How tall was this plant at 9:00 pm?
A.
B.
C.
D.
E.
49.184 inches
85.272 inches
101.184 inches
32.184 inches
102.184 inches
x
12. d(t) is a rate of change function. D  x    d (t )dt . Which of the following statements is true?
3
A.
B.
C.
D.
E.
D(3) = 0
D(0) = 3
D(x) is always increasing.
D(x) is always decreasing.
None of these statements are always true.
13. The function depicted is y  2 x  0.8 ; find the sum of the signed areas trapped between y and the xaxis on the interval 2  x  2.
A.
B.
C.
D.
E.
-8.5
4
8.32
-5.76
-3.2
February 11, 2015
Page 4 of 13
MATH 2070
TEST ONE VERSION A
Spring 2015
Mrs. Smith owns a hot dog stand.
h(x) hotdogs per hour
Her accountant has supplied to her a function,
h(x), which represents the rate of change of
sales, measured in hotdogs per hour.
Mrs. Smith does not understand Calculus.
Use the following to build a sentence of
practical interpretation to explain to Mrs.
Smith the meaning of the shaded area.
x hours after
10:00 am
QUESTIONS 14 – 17 INVOLVE WRITING A SINGLE SENTENCE. EACH QUESTION IS WORTH 1 POINT.
14.
Begin the best sentence of practical interpretation for the area trapped between h(x) and the
x-axis on the interval of 4  x  7 . The sentence will continue in subsequent questions.
A.
B.
C.
D.
15.
Continue the best sentence of practical interpretation for the area trapped between h(x) and
the x-axis on the interval of 4  x  7 .
A.
B.
C.
D.
16.
…the decrease in the number of hot dogs sold…
…the number of hot dogs sold…
…the change in the number of hot dogs sold…..
…the increase in hot dogs sold…..
Continue the best sentence of practical interpretation for the area trapped between h(x) and
the x-axis on the interval of 4  x  7 .
A.
B.
C.
D.
17.
Between 2:00 pm and 5:00 pm…..
The area under the curve between 4 and 7…..
The integral of h(x) between x = 4 and x = 7…..
From x = 4 to x = 7…..
…was increasing …
…was decreasing…
…increased…
…decreased…
Finish the best sentence of practical interpretation for the area trapped between h(x) and
the x-axis on the interval of 4  x  7 .
A.
B.
C.
D.
…to 210 hot dogs.
…to 210 hot dogs per hour.
…by 210 hot dogs.
…by 210 hot dogs per hour.
This is the end of the multiple choice section of the test.
February 11, 2015
Page 5 of 13
MATH 2070
TEST ONE VERSION A
Spring 2015
Check your Scantron now to make sure it will successfully run. If it does, you will earn one point.
The Scantron will not successfully run if:
 The XID is not filled in correctly: Check that you have the correct number and that you have
bubbled in the number below where you write it in. The initial C is bubbled in as a zero.
 The Test Version (A) is bubbled in incorrectly. Check to make sure that the Test Version you
bubbled in (below the XID) agrees with the Test Version in the header of this page.
 Your responses are not bubbled in boldly enough. If your responses are only lightly shaded, the
machine will not pick up your answers. Check to make sure your marks are heavy enough.
You also will not earn the point if:
 Your name, your instructor’s last name and your section number are missing. Check to make sure
all of these are clearly and legibly indicated. A list of section numbers, class locations and
instructors are included below for your convenience.
When you have finished filling our your Scantron, turn the Scantron over so that it cannot be read by
others in the room and continue to the next section.
Sec
Time
1
2
4
5
08:00
08:00
08:00
08:00
6
7
9
09:30 am-10:45 am
09:30 am-10:45 am
09:30 am-10:45 am
Judith Irene McKnew
Jennifer Marie Hanna
Grady McGowan Thomas
10
11
12
13
14
11:00
11:00
11:00
11:00
11:00
am-12:15
am-12:15
am-12:15
am-12:15
am-12:15
pm
pm
pm
pm
pm
Liem P Nguyen
Tony Thanh Nguyen
Kara Aileen Stasikelis
Mengying Xiao
Luke Martin Giberson
15
16
17
19
12:30
12:30
12:30
12:30
pm-01:45
pm-01:45
pm-01:45
pm-01:45
pm
pm
pm
pm
Judith Irene McKnew
Kevin Wayne Dettman
Fawwaz Tawfiq Batayneh
Margaret Winship Milliken
20
21
22
02:00 pm-03:15 pm
02:00 pm-03:15 pm
02:00 pm-03:15 pm
Hayato Ushijima-Mwesigwa
Jennifer Marie Hanna
Trevor Scott Trees Vilardi
24
25
26
03:30 pm-04:45 pm
03:30 pm-04:45 pm
03:30 pm-04:45 pm
Qijun He
Jennifer Marie Hanna
Yisu Jia
February 11, 2015
Instructor
am-09:15
am-09:15
am-09:15
am-09:15
am
am
am
am
Elaine Marie Sotherden
Garrett Michael Dranichak
Amy Christine Grady
Jennifer Marie Hanna
Page 6 of 13
MATH 2070
TEST ONE VERSION A
Spring 2015
Free Response: The free response portion of the exam is worth 56 points. This portion will be hand graded. Answer all
questions clearly as illegible work will be considered to be incorrect. Correct answers without the appropriate work shown will
receive little or no credit. Where appropriate, for full credit, show mathematical notation, necessary work and units.
t
1. The graph above is of a rate of change function f(x). F (t )   f ( x )dx .
k
All questions below regard the accumulation function F(t) on the interval  2  t  9 (only).
a. Circle the correct answers regarding the critical points:
F(t) has a relative maximum at locations a b c d e f g h i
F(t) has a relative minimum at locations a b c d e f g h i
F(t) has an inflection point at locations a b c d e f g h i
b. Fill in the correct answers regarding concavity on the interval a  t  i
(for example: if your answer is “between b and c” write “ b < t < c”) .
You may not need all the spaces provided.
F(t) is concave up in the following region(s):
and
F(t) is concave down in the following region(s):
and
Points awarded
Possible points
February 11, 2015
9
Page 7 of 13
MATH 2070
TEST ONE VERSION A
Spring 2015
2. Algebraically compute the following integral.
Give the exact answer.
Do not represent values such as e4 or ln 4 with a decimal approximation.
Combine like terms.
Show each step necessary to find the answer (as points are awarded for each step in the process)
Show irrational values (such as 2.33 ) as fractions ( 7 ).
3
6
 (4 x
2
2
 5 )dx 
x
Points awarded
Possible points
3.
6
The rate of change of the weight of a laboratory mouse can be modeled as
13.785
w (t ) 
grams per week , where t equals the number of week after the beginning of an
t
experiment, 1  t  15.
Find the change in the weight of the mouse between the third and ninth weeks. Show your work with
mathematical notation.
Answer: The mouse ______________, __________________ grams.
gained or lost?
Round to 3 places
Points awarded
Possible points
February 11, 2015
3
Page 8 of 13
MATH 2070
TEST ONE VERSION A
Spring 2015
4. You are given the following:
a. G(x) is an antiderivative of g(x).
b. G(a) = -272
c. G(b) = 326
b
Either: find
 g ( x)dx = ________
a
Or: Explain (in less than 25 legible words) why it is not possible to do so with the given information.
Points awarded
Possible points
3
5. You are given the following:
a. During a storm, snow was falling in Boston at a rate of b(x) = 8(0.76x) inches per hour,
where x is the number of hours since 6:00 am, 1  x  6 .
b. At 8:00 am the snow was falling at the rate of 4.62 inches per hour.
c. The function B(x) is an antiderivative of b(x).
Either: find the amount of snow on the ground at 11:00 am.
Or: Explain (in less than 25 legible words) why it is not possible to do so with the given information.
Points awarded
Possible points
February 11, 2015
3
Page 9 of 13
MATH 2070
TEST ONE VERSION A
Spring 2015
6. Find the following. Simplify expressions when possible. Place a box around your final answers.
3
 21x 2  10 x)dx 
a.
 (8x
b.
  dt  6t
c.
d
1
3

 9t 2  6t  14   dt =

3x
  6 x  e
Points awarded
Possible points
6
Points awarded
Possible points
7

 7 x  5 3  dx =

t
d
( 16  32.7 x  15e4 x )dx =
d.
dt 4 x

February 11, 2015
Page 10 of 13
MATH 2070
TEST ONE VERSION A
7. Given two functions, f ( x )  2 x  1
and
Spring 2015
g ( x)  x 2  4 , pictured below.
Find the area trapped between the two curves on the interval  1  x  3.
Show the appropriate mathematical notation to compute this area.
Give the EXACT answer (hint: if your answer is irrational, show your answer as a fraction).
Points awarded
Possible points
February 11, 2015
4
Page 11 of 13
MATH 2070
TEST ONE VERSION A
Spring 2015
8. The rate of change of profit for an on-line business is
given by:
f ( x )  55 x 3  795 x 2  2958 x  2962 thousand dollars ,
year
where x is the number of years since 1999, 1  x  10 .
In the year 2000, the rate of change of profit was 744 thousand dollars per year.
In the year 2004, the profit for the business was 2632 thousand dollars.
a. Compute the average rate of change of profit from 2000 through 2009. Show mathematical notation.
Include units.
b. Recover the function F(x), the profit for the business. That is, find the specific antiderivative F(x).
Include units.
then, find the profit in 2006: ______________________________________include units.
c. What was the change in profit between 2003 and 2007? Show mathematical notation, and include
units.
Points awarded
Part
Possible points
February 11, 2015
a
6
b
6
c
3
Page 12 of 13
MATH 2070
February 11, 2015
TEST ONE VERSION A
Spring 2015
Page 13 of 13