MthSc 102
Test 1 – Version 1
Spring 2006
Name: ___________________________________________ CU ID#: ____________________________
Section #: ____________ Instructor’s Name: ________________________________________________
Please do not pose questions to your instructor or proctor during this exam. It is the policy of the
Department of Mathematical Sciences that, as answering questions during an exam is disruptive to surrounding
students, NO questions will be answered during examinations. 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.
General Directions:
•
Show work where possible. Answers without supporting work (where work is appropriate) may receive
little credit. Illegible answers will receive no credit.
•
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.
•
Use a ruler when drawing lines on a graph and when asked to estimate values from a graph.
•
Do not write in red pen.
Statement of Academic Integrity:
I have not and will not give or receive improper aid on this test.
Formulas:
 r
A = P 1 + 
 n
Signature: _______________________
nt
A = Pe rt
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. Turn
off/on scatter plots by using the arrows to highlight the item you want off/on and using the enter key to
change the status.
•
ZOOM 0 may not work for graphing when setting the window for the x-min and x-max and allowing the
calculator to find the appropriate y-min and y-max 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.
DO NOT WRITE IN THIS TABLE
Question
1 – 13
14
Possible
46
10
15
4
16
18
17
4
18
14
19
4
TOTAL
Earned
1
MthSc 102
Test 1 – Version 1
Spring 2006
The first part of this test consists of multiple choice questions. Each question has one correct answer.
Using a #2 pencil, bubble in the letter of the response that best answers each question on your
scantron. For your own record, circle your responses. Only your scantron answers will be graded. This
section will account for 46% of the test grade.
1.
The value of an antique armoire is given by V(x) = 975(1.24 x ) dollars, where x is the number of
years past 1999. Which of the following statements about V(x) is TRUE?
(4 pts)
a. The value of the armoire increases by 124 dollars each year past 1999.
b. The value of the armoire increases by 8.75% each year past 1999.
c. The value of the armoire increases by 975 dollars each year past 1999.
d. The value of the armoire increases by 24% each year past 1999.
e. The value of the armoire increases by 124% each year past 1999.
2. The following table describes the value of a manufacturer’s shipments of CDs between 1993 and
2001.
Year
Shipment Value
(millions of $)
1993
6511
1995
8465
1997
9377
1999
9935
2001
9915
If you were given an equation modeling the above data, which of the following would be an example
of extrapolation?
(3 pts)
a.
Estimating the value of CD shipments in 1998.
b. Estimating the value of CD shipments in 2003.
c.
Estimating the value of CD shipments in 1997.
d. Determining when the value of CD shipments reached $5000 million.
e.
Determining when the value of CD shipments reached $9000 million.
3. To set aside money for emergencies, you place $2000 in an emergency savings account earning 1.5%
interest compounded quarterly. After 3 years and 8 months, your car breaks down and you need to
withdraw all of the money from this account. How much will the account be worth at this time?
(4 pts)
a. $7443.95
b. $2117.09
c.
$7105.59
d. $2107.60
e.
$2112.86
2
MthSc 102
Test 1 – Version 1
Spring 2006
The graphs shown below are the weekly cost and revenue functions (denoted C(x) and R(x) respectively)
for an independently owned oil change business where x is the number of cars serviced. Use the
graphs to answer the following questions:
R(x)
C(x)
4. What are the weekly fixed costs for the business?
a. $0
b. $3700
c. $500
(3 pts)
d. $6000
5. How many cars must be serviced each week for the business to break-even?
a. 8
b. 100
c. 500
d. 600
6. What is the profit when 100 cars are serviced in a week?
e. $1000
(3 pts)
e. 0
(3 pts)
a. $5400.00
b. $2100.00
c. $7500.00
d. $3400.00
e. $3800.00
7. What is the average cost per car when 50 cars are serviced in a week?
a.
(3 pts)
$6 per car
b. $45 per car
c.
$0.025 per car
d. $25 per car
e.
$70 per car
3
MthSc 102
Test 1 – Version 1
Spring 2006
W(x) = 1.321 + 4.743 * Ln(x) centimeters is the length of an average earthworm where x is its age in
weeks, x > 0.
8. Evaluate the model at x=2 and interpret your result.
a.
(4 pts)
The average 2-week old earthworm will be about 1.154 cm long.
b. The average 4.6-week old earthworm will be about 2 cm long.
c.
The average 1.154-week old earthworm will be about 2 cm long.
d. The average 2-week old earthworm will be about 4.609 cm long.
e.
The average 2-week old earthworm will be about 1.321 cm long.
9. Choose the correct mathematical notation for the following sentence:
“When an average earthworm is 5.5 weeks old, it will be about 9.407 cm long.”
a.
(3 pts)
W(5.5) = 9.407
b. W(5.5 weeks) = 9.407
c.
W(9.407 cm) = 5.5
d. W(x) = 9.407
e.
W(9.407) = 5.5
10. Which of the following statements about a logarithmic function, f(x) = a + b lnx, is TRUE? (4 pts)
a.
f(x) is defined on the interval (-∞, ∞)
b. f(x) is the inverse of a logistic function
c.
The graph of f(x) shows an inflection point
d. The graph of f(x) has a horizontal asymptote
e.
lim f(x) = ∞ or lim f(x) = - ∞
x →∞
x →∞
4
MthSc 102
Test 1 – Version 1
11. The number of country clubs in a small country is given by C(x) =
where x is the number of years since 1980, 0 ≤ x ≤ 25.
Spring 2006
3
hundred clubs
1 +100e -0.427x
(4 pts)
Which of the following statements about C(x) is true?
a.
C(x) is an increasing function and the number of clubs will approach 100 as the years increase.
b. C(x) is a decreasing function and the number of clubs will approach 300 as the years increase.
c.
C(x) is an increasing function and the number of clubs will approach 300 as the years increase.
d. C(x) is a decreasing function and the number of clubs will approach zero as the years increase.
e.
C(x) is a decreasing function and the number of clubs will approach 100 as the years increase.
12. B(x) gives the number of bushels of tomatoes harvested by a farmer on the xth day of May.
P(x) dollars is the market price of one bushel of tomatoes on the xth day of May.
B(x) and P(x) can be combined to form a meaningful new function. Which statement most
accurately describes the new function?
a.
(4 pts)
P(x)
dollars per bushel gives the profit from the sale of tomatoes on the xth day.
B(x)
b. B(x)⋅P(x) dollars per bushel gives the revenue from the sale of tomatoes on the xth day.
c.
B(x)
bushels per dollar gives the revenue from the sale of tomatoes on the xth day.
P(x)
d. B(x)⋅P(x) dollars gives the total revenue from the sale of tomatoes on the xth day.
e.
B(x)⋅P(x) dollars per bushel gives the profit from the sale of tomatoes on the xth day.
5
MthSc 102
Test 1 – Version 1
Spring 2006
13. V(x) is the number of volcanoes, in hundreds, at x degrees of latitude.
H(x) is the number of volcanoes higher than 10,000 feet at x degrees of latitude.
V(x) and H(x) can be combined to form a meaningful new function. Which statement most
accurately describes the new function?
a.
(4 pts)
100V(x) – H(x) volcanoes gives the number of volcanoes 10,000 feet or less at x degrees of
latitude.
b. 100V(x) – H(x) hundred volcanoes gives the number of volcanoes 10,000 feet or less at x
degrees of latitude.
c.
V(x) – H(x) hundred volcanoes gives the number of volcanoes 10,000 feet or less at x degrees
of latitude.
d. V(x) – H(x) volcanoes gives the number of volcanoes 10,000 feet or less at x degrees of
latitude.
e.
V(x) – 100H(x) volcanoes gives the number of volcanoes 10,000 feet or less at x degrees of
latitude.
END OF THE MULTIPLE CHOICE
6
MthSc 102
Test 1 – Version 1
Spring 2006
Read each question carefully. Provide only one clearly indicated answer to each question. If your
answer is illegible, it will be graded as incorrect.
____________________________________________________________________________
14. Using the graph below, answer the following questions:
(10 pts)
f(x)
x
a.
lim f(x) = _____
x →1 +
d. limf(x) = _____
x→ 4
b.
lim f(x) = _____
x →1 -
e. limf(x) = _____
x→ 6
c. limf(x) = _____
x →1
f. lim f(x) = _____
x →∞
g. f(1) = ____
h. List the x-value(s) where f(x) is NOT continuous.
7
MthSc 102
Test 1 – Version 1
Spring 2006
7x2 - x
numerically. Show your work by filling in the table. Report your answer on the line
x →∞ 5 + x3
provided.
(4 pts)
15. Find lim
x →∞
7x2 - x
5 + x3
7x2 - x
= ____________
x →∞ 5 + x3
lim
16. The table below shows the end-of-the-year population of the funnel-web spider on a particular
island.
a.
Year
1981
1983
1984
1985
1987
1988
1991
Population
(in thousands)
68.5
58.6
55.3
52.8
49.5
48.4
46.5
Look at a scatter plot of the data. List one characteristic of the scatter plot that indicates that
an exponential model may fit the data.
(2 pts)
b. Align the input data to 1980. Shift the output data down by 45 units and then fit an exponential
model to the data. Completely define your model, showing the vertical shift as part of the
equation. Name the model P(x).
(10 pts)
c.
Write the equation(s) of any asymptotes of your model.
d. Find P(6) and report your answer in a sentence of practical interpretation.
(3 pts)
(3 pts)
8
MthSc 102
Test 1 – Version 1
Spring 2006
17. When diabetics take insulin shots, the insulin breaks down rather quickly. Typically, 10 milligrams
of insulin are injected into the bloodstream and the amount of insulin in the bloodstream will
decrease at a rate of 3.85 percent per minute.
What is the equation that best describes the situation given above? Report your equation by
completing the model below.
(4 pts)
The amount of insulin in the bloodstream can be described by the equation
I(x) = _____________________________________________________________
milligrams where x is the number of minutes after the injection, x ≥ 0.
18. Under the Servicemen’s Dependents Allowance Act of 1942, the U.S. Government helped men and
women serving their country during WWII take care of dependents by means of family allowances.
The table lists the amounts that were paid to certain Class A Dependents.
Number of dependents
Monthly Allowance ($)
a.
Find the best-fitting linear model for this data.
1
50
2
80
3
100
4
120
(8 pts)
b. Consider the interpretation of the slope of the model you found in (a). Complete the sentence of
interpretation by filling in the blanks below.
(3 pts)
The amount of money paid to Class A Dependents was __________________ (increasing or
decreasing) by ____________(slope)_________________________________ (units).
c.
Should the model you found in (a) be interpreted without restriction or with discrete interpretation?
(circle one)
Explain your choice in a sentence or two.
(3 pts)
turn the page 9
MthSc 102
Test 1 – Version 1
Spring 2006
19. An initial investment of $12,500 is placed in an account earning 2.75% interest compounded
continuously. How long will it take for the initial investment to double? Show your work.
(4 pts)
10