MATH 1020
TEST 1 VERSION A
ANSWER KEY
SPRING 2015
Printed Name: ______________________________ Section #: _______ Instructor:_________________
Please do not ask questions during this exam. If you consider a question to be ambiguous, state your
assumptions in the margin and do the best you can to provide the correct answer. See page two for
formulas, general directions, and calculator troubleshooting tips.
•
•
•
•
Any communication with any person (other than the instructor or the designated proctor) during this
exam in any form, including written, signed, verbal or digital, is understood to be a violation of
academic integrity.
All devices, such as computers, cell phones, cameras and PDAs must be turned off while the student is
in the testing room.
The only calculators to be used are TI-83, TI-83+, TI-84 or TI-84+. You may NOT borrow or share a
calculator with another person taking this test.
Statement of Academic Integrity: I have not and will not give or receive improper aid on this test.
In signing below, I acknowledge that I have read, understand, and agree to these testing conditions.
Student’s Signature: _________________________________________________________________
(This test will not be accepted for grading unless it bears the signature of the student.)
Possible
points
FR#1
FR #2
FR #3
FR #4
FR #5
scantron
Free
Response
Total
Multiple
Choice
Total
Total
Points
Earned
6
10
8
7
6
1
38
62
100
Points
Earned
Useful Formulas:
A(t ) = P(1 + rt )
A(t ) = Pert
 r
A(t ) = P 1 + 
 n
( nt )
Page 1/12
MATH 1020
•
TEST 1 VERSION A
SPRING 2015
General Directions:
Show work where possible. Answers without supporting work (where work is appropriate) may receive
little credit.
ANSWER KEY
•
Do not round intermediate calculations.
•
Answers in context ALWAYS require units.
•
•
•
•
Round your answers to 3 decimal places UNLESS the answer needs to be rounded differently to make
sense in the context of the problem OR the directions specify another type rounding OR the complete
answer has less than 3 decimal places.
When you are asked to write a model, include all components of a model: an equation, a description of
the input including units, a description of the output including units, and the interval when known
When asked to write a sentence of practical interpretation, answer the questions: when?, what?, and
how much? using ordinary, conversational language. DO NOT use math words, terms, or unnecessary
phrases.
Always use a ruler when estimating values off of a graph.
HINTS FOR TROUBLESHOOTING YOUR CALCULATOR:
•
If you lose your L1, L2, etc., you may reinsert them using STAT 5 (set-up editor) enter.
•
The SCATTER PLOT will not show unless Plot 1 has been turned on and there is data in L1 and L2.
•
ZOOM 0 may not work for graphing if Plot 1 is turned on.
•
IF you set the entire x and y window range, use the GRAPH button to see the graph.
•
DIM MISMATCH error usually means that the lists in L1 and L2 are not of equal length.
•
DATA TYPE error usually means that you already have something in Y1 and you need to clear it before you can
paste a new equation.
•
INVALID DIM error usually means that your plot(s) are on, but that you have no data in the lists. Refer to the
second hint above.
•
If your batteries die, raise your hand and hold up your calculator. If your instructor has an extra calculator
available, he/she will loan it to you for a few minutes.
•
SYNTAX ERROR: Try GO TO. This will happen if you use a subtraction minus sign when you should use a
MULTIPLE CHOICE: 62 points
negative sign.
Page 2/12
MATH 1020
Version A:
TEST 1 VERSION A
ANSWER KEY
SPRING 2015
D C D B A| B C C D B| B D A C A| D B A C B| B C A A
Use a #2 pencil and completely fill each bubble on your scantron to answer each multiple choice question.
(For future reference, circle your answers on this test paper.) There is no penalty for guessing on multiple
choice. If you indicate more than one answer, or you leave a blank, the question will be marked as
incorrect. Each question is worth 3 points unless otherwise indicated.
1. Consider the three tables to the right. Which of the relations
displayed in the tables satisfies the definition of a function?
x
-2
-1
0
1
2
y
4
1
0
1
4
Table I
a. Tables I and II only
b. Table II only
c.
d. Tables I and III only
Table III only
x
-2
-2
-1
0
-1
y
3
-1
2
1
0
Table II
x
0
3
4
7
11
y
1
-1
-7
4
6
Table III
Use the following to answer the next two questions.
10667.261
thousand fish gives the population of a certain species of fish in a lake,
1 + 120.735e −1.274 x
x months after the species of fish was introduced into the lake, 1 ≤ x ≤ 7 .
f ( x) =
2. Use the function f ( x ) and the fact that f ( x ) has an inflection point that occurs at x = 2.76 to
complete the following sentence.
The graph of f ( x ) has ___________________ concavity/concavities and the population of the fish
species in the lake was ___________________ 2.76 months after the fish species was introduced.
a.
one; increasing most rapidly
b. one; decreasing most rapidly
c.
two; increasing most rapidly
d. two; decreasing most rapidly
3. What are the equations for the asymptotes of f ( x ) ?
a.
x = 0 and x = 1.274
b. y = 0 and y = 120.735
c.
y = 120.735 and y = 10, 667.261
d. y = 0 and y = 10, 667.261
Page 3/12
MATH 1020
TEST 1 VERSION A
ANSWER KEY
SPRING 2015
Use the following graph to answer the next six questions. [2 pts each]
4. Evaluate f (2) = _______ .
a.
5.
b.
-3
c.
-7
d.
undefined
b.
-3
c.
2
d.
does not exist
b.
1.75
c.
4
d.
does not exist
lim f ( x) = ______ .
x→ 2 −
a.
6.
2
-7
lim f ( x) = ______ .
x→ − 4
a.
−∞
7. On the interval -3 < x < -0.5 , the graph of f ( x ) is __________________________.
8.
a. Concave up and increasing
b. Concave down and decreasing
c.
d. Concave down and increasing
Concave up and decreasing
f ( x ) has __________ input values at which ݂(‫ )ݔ‬is not continuous.
a.
0
b.
1
c.
2
d.
3
d.
1.75, ∞
9. Fill in the blanks: lim f ( x) = ______ and lim f ( x) = ______ .
x → −∞
a.
−∞, ∞
x→ ∞
b.
∞, − ∞
c.
−6, 4.5
Page 4/12
MATH 1020
TEST 1 VERSION A
ANSWER KEY
SPRING 2015
Use the following to answer the next two questions.
T ( x) = 4.767(0.9837 x ) hours gives the maximum dive-time for a no-compression scuba dive at a depth of
x feet, 25 ≤ x ≤ 200 .
10. Write the following statement using function notation. (Hint: pay attention to the units.)
The maximum dive-time for a 165 foot dive is approximately 19 minutes.
a.
c.
b.
T (19) = 165
d.
T (0.317) = 165
T (165) = 0.317
T (165) = 19
11. Find the constant percentage change in the maximum dive-time T ( x) = 4.767(0.9837 ) .
x
a.
98.37 %
b.
-1.630 %
c.
1.9837 %
d.
-98.37 %
12. Matt bought a new car at a cost of $25,000 at the end of 2014. The car depreciates by 15% of its value
each year. Which of the following describes the value of Matt’s car in dollars, t years after 2014?
a.
c.
b.
f (t ) = −0.15t + 25, 000
d.
f (t ) = 25,000(1.15t )
f (t ) = 0.15 t − 25, 000
f (t ) = 25,000(0.85 t )
13. C ( x ) dollars gives the total cost for “Babies-R-Wee” to produce and sell x packages of diapers.
p ( x ) dollars per package gives the sales price for a package of diapers when “Babies-R-Wee” produces
and sells x packages of diapers.
The solution to which one of the following equations yields the break-even point for the production and
sale of diapers by “Babies-R-Wee”?
a.
C ( x) = x ⋅ p ( x)
b.
C ( x) = p ( x)
c.
C ( x) + p( x) = 0
d.
p( x) = x ⋅ C ( x)
Page 5/12
MATH 1020
TEST 1 VERSION A
ANSWER KEY
SPRING 2015
Use the following graph to answer the next two questions. [2 pts each]
14. The slope of the dotted line on the graph of f ( x )
shown to the right represents the ___________
between points A and B.
a. change
b. percent change
c.
d. inverse
average rate of change
15. An inflection point occurs at approximately which point on the graph of f ( x ) ?
a.
Point A
b.
Point B
c.
Point C
d.
Point D
16. t ( x) degrees Fahrenheit gives the temperature at a SC beach, x weeks after June 1, 0 ≤ x ≤ 14 .
v (t ) = −3662 + 909 ln(t ) people gives the number of daily visitors to a SC beach when the
temperature is t degrees Fahrenheit, 70 ≤ t ≤ 100 .
In what way can these two models be combined to create a meaningful new model?
a.
t ( x)
degrees Fahrenheit gives the temperature at a SC beach when there are x visitors daily,
v (t )
70 ≤ x ≤ 100 .
b. t (v ( x )) degrees Fahrenheit gives the temperature at a SC beach when there are x visitors daily,
70 ≤ x ≤ 100 .
c.
t ( x) ⋅ v(t ) people gives the number of daily visitors to a SC beach, t weeks after June 1,
d.
v(t ( x )) people gives the number of daily visitors to a SC beach, x weeks after June 1,
0 ≤ t ≤ 14.
0 ≤ x ≤ 14 .
Page 6/12
MATH 1020
TEST 1 VERSION A
ANSWER KEY
SPRING 2015
17. S (t ) gives the total number of students enrolled at Daniel High School, t years after 2000, 0 ≤ t ≤ 10.
H (t ) students gives the percentage of honors students enrolled at Daniel High School, t years after
2000, 0 ≤ t ≤ 10.
These two functions can be combined using ______________________to make a meaningful new
function that gives the number of honors students at Daniel High school, t years after 2000, 0 ≤ t ≤ 10.
a.
addition
b. multiplication
c.
subtraction
d. composition
18. The table below gives the speed of a boat, based on the speed of its engine.
Engine Speed
(hundred RPM)
Boat Speed
(knots)
9
11
13
15
17
21.5
6.43
7.91
8.98
9.86
10.88
15.24
Find a cubic function to model this data. According to the cubic model, what is the speed of the boat
when the engine speed is 16 hundred RPM?
a.
10.336 knots
b.
10.582 knots
c.
21.997 knots
d.
10.562 knots
19. t ( d ) = 1.34 d + 10.02 billion dollars gives the revenue from state cigarette taxes, where d is the
number of decades since 1987, 0 ≤ d ≤ 1.5 .
Which of the following three statements follow from the given model?
I.
II.
III.
a.
Between 1987 and 2002, revenue from state cigarette taxes increased by 1.34 billion dollars.
The initial value for t ( d ) tells us that in 1987 revenue from state cigarette taxes were 10.02
billion dollars.
t (2) = 12.7 uses interpolation.
I only
b.
I and II only
c.
II only
d.
II and III only
20. [ 2 pts.] A cubic function has _____________ inflection point(s) and ________________ asymptote(s).
a.
one; two
b.
one; zero
c.
zero; two
d. zero; zero
Page 7/12
MATH 1020
ANSWER KEY
TEST 1 VERSION A
21. [ 2 pts.] Which function(s) has (have) two concavities?
I . Cubic
a.
I only
II. Logarithmic
b.
SPRING 2015
III. Logistic
I and III
c.
II and IV
IV. Exponential
d. II and III
Use the following to answer the next two questions.
A $10,000 investment is made for a grandchild’s college education. The investment is compounded
semiannually at an annual percentage rate (APR) of 2.45%. Assume no additional deposits or withdrawals.
22. Find the future value of the investment after 25 months.
a.
$18,381.85
b.
$10,520.40
c.
$10,499.08
d. $10627.69
23. The investment will take ________ to double and has an annual percentage yield (APY) of ________.
a.
28 years 6 months; 2.465 %
b. 28 years; 2.465 %
c.
28 years 6 months; 2.480 %
d. 28 years; 2.480 %
24. Using the data in the table, a linear function b ( g ) can be found to describe the number of bathroom
visits Peter makes during an eight hour workday when he drinks g glasses of water.
g glasses of water
0
2
4
6
8
b bathroom visits
1
2
3
4
5
Which ONE of the following describes the inverse function to the linear function b ( g ) ?
a.
b.
c.
d.
g (b ) = 2b − 2 glasses gives the amount of water Peter consumes during an eight hour workday
when he has b bathroom visits, 1 ≤ b ≤ 5.
g (b ) = 0.5b + 1 glasses gives the amount of water Peter consumes during an eight hour
workday when he has b bathroom visits, 1 ≤ b ≤ 5.
g (b ) = 2b − 2 gives the number of bathroom visits Peter makes when he consumes g glasses of
water during an eight hour workday, 0 ≤ b ≤ 8.
g (b ) = 0.5b + 1 gives the number of bathroom visits Peter makes when he consumes g glasses of
water during an eight hour workday, 0 ≤ b ≤ 8.
Page 8/12
MATH 1020
ANSWER KEY
TEST 1 VERSION A
SPRING 2015
FREE RESPONSE: 38 points
Show work where possible. Read the directions on rounding, inclusion of units, and writing models and
sentences at the front of the test.
1. T ( x ) = 6.099 + 6.108 (ln x ) feet gives the average height of twelve cherry trees when the trees are
x years of age, 1 ≤ x ≤ 11 .
Checkpoint: T (2) = 10.333
A. Find T (4.5) =_____15.286_____ (correct to three decimal places).
Write a sentence of interpretation.
At 4.5 years of age, the average height of twelve cherry trees is 15.286 feet.
Part A:
When: 1 pt
Output description: 1 pt
Output value: 1 pt
Output units: 1 pt
B. According to the model, when will the average height of the twelve cherry
trees reach 21 feet?
Part B: 2 pts
Answer correct to three decimal places.
Solve the equation 0 = Y1-21
11.468 years of age
( _______ / 6 pts ) .
2. The table below shows the average sales price, in thousand dollars, of a house in Suffolk County,
Massachusetts, for various years.
Year
2000
2002
2004
2006
2008
Average sales price, in
thousand dollars
154.5
124.5
115
128
165
A. Align the data to the number of years since 2000.
B. Find a quadratic function to model the data. Write a completely
defined quadratic model.
f ( x ) = 2.795 x 2 − 21.132 x + 154.857 thousand dollars gives the
average sales price of a house in Suffolk County, Massachusetts,
x years after 2000, 0 ≤ x ≤ 8.
Part B:
Aligned quadratic: 3 pts
Output units: 2 pts
Output description: 1 pt
Input units & description: 1 pt
Input data range: 1 pt
Part C: 1 pt
Part D: 1 pt
C. Use the calculator to observe the quadratic function superimposed
on the scatter plot. In how many places does the graph of the quadratic function completely miss (does
not touch) the five points on the scatter plot?
In _zero (or one) (zero/one/two/three/four/five) places.
D. The actual average sales price of a house in Suffolk County, Massachusetts was 165,000 dollars in 2009.
Does the model give an underestimate or an overestimate of the average sales price in 2009?
Since
f (9) = 191.034
Page 9/12
MATH 1020
TEST 1 VERSION A
ANSWER KEY
SPRING 2015
3. For each of the following statements, name ONE of the six functions for which the statement is true:
Linear Exponential Logarithmic Logistic Quadratic Cubic
1 pt per blank
• Write out the name of the function: do not abbreviate.
• There may be more than one correct answer, but you only need to write one
function per blank.
• If a statement is not true for any of the functions, write “none”.
A. This function has either a relative maximum or a relative minimum, but not both. __ Quadratic __
B. This function has two horizontal asymptotes. ___ Logistic____
C. This function has exactly one vertical asymptote. ___ Logarithmic_____
D. This function has exactly one horizontal asymptote. __ Exponential________
E. This function may have both a relative maximum and a relative minimum. ____ Cubic____
F.
This function has no asymptotes. __ Quadratic, or Linear, or Cubic ____
G. This function has constant rate of change. __ Linear_______
H. This function has constant percentage change. ___ Exponential____
( _______ / 8 pts )
4. Use the tables below to determine the behavior of a function g (t ) .
Answers to limit questions are −∞ , or ∞ , or a number.
A.
B.
t → −∞
g (t )
t→∞
g (t )
t → 4−
g (t )
t → 4+
g (t )
-10
100
10
0.1
3.9
-10
4.1
10
-100
1,000
100
0.01
3.99
-100
4.01
100
-1000
100,000
1000
0.001
3.999
-1,000
4.001
1,000
-10000
1,000,000
10000
0.0001
3.9999
-10,000
4.0001
10,000
-100000
10,000,000
100000
0.00001
3.99999
-100,000
4.00001
100,000
lim g (t ) = ___ ∞ ___ lim g (t ) = _____ 0 ____
t →− ∞
;
t→ ∞
lim g (t ) = ___ − ∞ ___ ; lim g (t ) = _____ ∞ ____
t→ 4 −
1 pt per blank
t → 4+
C.
The equation of the horizontal asymptote for the function g (t ) is _____y = 0____________.
D.
The equation of the vertical asymptote for the function g (t ) is ______x = 4_____________.
E.
Is the function g (t ) continuous at t = 4? ____no____(yes/no)
( _______ / 7 pts )
Page 10/12
MATH 1020
ANSWER KEY
TEST 1 VERSION A
SPRING 2015
5. The following table shows the amount of crude oil, in thousand carloads, transported on US railroads in
various years. (Do not model this data.)
Year
Carloads of crude
oil, in thousands
2008
9.5
2009
10.84
2010
29.605
2011
65.751
2012
233.698
2013
407.761
A. Calculate the change (correct to three decimal places) between 2010 and 2013 to
complete the following sentence. Include units.
1 pt per blank
407.761 − 29.605 = 378.156
Between 2010 and 2013, the amount of crude oil transported on US railroads increased by
___378.156____
__thousand carloads_______________________________________.
B. Calculate the percentage change (correct to three decimal places) between 2010 and 2013 to complete
the following sentence. Include units.
407.761 − 29.605
⋅100% = 1, 277.338%
29.605
Between 2010 and 2013, the amount of crude oil transported on US railroads increased
1, 277.338 % .
C.
Calculate the average rate of change (correct to three decimal places) between 2010 and 2013 to
complete the following sentence. Include units.
407.761 − 29.605
= 126.052
3
Between 2010 and 2013, the amount of crude oil transported on US railroads increased on average by
____ 126.052 _______________
_____thousand carloads per year._________.
( _______ / 6 pts )
6.
If your scantron is correctly bubbled with a #2 pencil, with your correct XID, your correct test version,
AND the front of your test is completed with your signature on the academic integrity statement, you
earn 1 point.
END OF TEST
Page 11/12