ANSWER KEY TEST 3 VERSION A
MATH 1020
FALL 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
•
•
•
•
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.)
•
General Directions and Calculator Hints are located on the last page.
Possible
points
FR#1
FR #2
FR #3
FR #4
FR #5
scantron
Free
Response
Total
Multiple
Choice
Total
Total
7
11
8
6
4
1
37
63
100
Points
Earned
Formula:
f L ( x) = f (c) + f ′(c)( x − c)
Page 1 of 12
ANSWER KEY TEST 3 VERSION A
MATH 1020
FALL 2015
MULTIPLE CHOICE: 63 points
(Each question is worth 3 points unless otherwise noted.)
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 questions. If you indicate more than one answer, or you leave a blank, the question will be marked
as incorrect.
A
C
D A B|
D
B
D
A
B|
D
D
B
E
C|
A
D
C
B
C|
D
B
C
A
1. Identify an outside function g ( h ) and an inside function h( x ) for f ( x) = 3ln(5x − 4) .
2
a. g (h ) = 3ln( h ) ; h( x) = 5x − 4
2
c. g (h) =
3
2
; h( x) = 5x − 4
h
2
b. g ( h ) = 3 ; h( x) = ln(5x − 4)
d. g (h) = 5h − 4 ; h ( x ) = 3 ln( x )
2
2
2. Find the derivative of f ( x) = 3ln(5x − 4) .
a. f ′( x) =
3
5x − 4
b. f ′( x) =
c. f ′( x) =
30 x
5 x2 − 4
d. ln(5 x − 4) + 3 ⋅
2
3
10 x
2
10 x
5x2 − 4
3. Given f ( x ) = 7(3ln x ) , find f ′( x ) = _________________ .
b. 7(ln 3)(3ln x )
1
a. 7(3 x )
ln x
c. (ln 21)(21
1
) 
x
ln x
d. 7(ln 3)(3
1
) 
x
2
4. Given f ( x ) = g ( x ) ⋅ h( x ) , where g ( x ) = x and h( x) = ( x + 1) 3 , find f ′( x ) = _____________ .
a.
( x + 1)
c. x ( x + 1)
2
2
−1
2x
( x + 1) 3
3
3
+
3
1
2
+ ( x + 1) 3
3
b.
−1
2
( x + 1) 3
3
2( x + 1)
d. 1 +
3
−1
3
Page 2 of 12
ANSWER KEY TEST 3 VERSION A
MATH 1020
FALL 2015
5. Given f ( x ) = ( x 2 + 3 x + 1) 5 ln( x ) , find f ′( x ) = _____________ .
 2x + 3 

 x 
a. 5( x 2 + 3 x + 1) 4 
1
x
b. 5( x 2 + 3 x + 1) 4 (2 x + 3) ln( x ) + ( x 2 + 3x + 1)5  
1
 x
c. 5(2 x + 3) 4 ln( x ) + ( x 2 + 3x + 1)5  
1
x
d. 5( x 2 + 3 x + 1) 4 (2 x + 3)   + ( x 2 + 3 x + 1)5 ln( x )
6. Given u ( x ) =
du
4 x3
, find
= _____________ .
x
1 + 5e
dx
12 x 2
5e x
−2
b. − 12 x 2 (1 + 5e x ) (5e x )
a.
(
)
(1 + 5e )
c. 12 x 2 1 + 5e x
d. 12 x 2
−1
x −1
+ 4 x 3 ( −1) (1 + 5e x )
−2
+ 4 x 3 ( −1) (1 + 5e x ) (5e x )
−2
Use the following to answer the next two questions.
The demand for smart thermostats in North America is given by D( x) million thermostats, when the price
is x dollars/thermostat. Assume demand is equal to the number of units sold. Then, revenue R ( x )
generated from the sale of smart thermostats in North America is given by R ( x ) = x ⋅ D ( x ) million dollars.
7. When the price of a smart thermostat is 222.16 dollars, demand for smart thermostats in North
America is 2.5 million thermostats. At that price, demand for a smart thermostat is decreasing by 0.02
million thermostats per dollar/thermostat in price. How quickly is the revenue from the sale of smart
thermostats changing when the price is 222.16 dollars/thermostat?
a. -4.4432
b. -1.9432
c. -0.05
d. -0.02
8. What are the units for the previous question?
a. thermostats per dollar
b. million thermostats per dollar/thermostat
c. dollars per dollar/thermostat
d. million dollars per dollar/thermostat
Page 3 of 12
ANSWER KEY TEST 3 VERSION A
MATH 1020
FALL 2015
9. To find the critical points of a function f ( x ) , solve the equation _______________ and to find possible
inflection points of f ( x ) , solve the equation _________.
a. f ′( x ) = 0 ;
f ′′( x ) = 0
c. f ′( x ) = f ′′(0) ;
f ′′( x ) = 0
b. f ( x ) = 0 ;
f ′( x ) = 0
d. f ′′( x ) = 0 ; f ′( x ) = 0
10. On the interval −5 ≤ x ≤ 5 , the function g ( x ) has an absolute minimum at x = −5 and
a relative maximum at x = 3 . Which of the following could be a graph of g ( x ) ?
a.
b.
c.
d.
Page 4 of 12
ANSWER KEY TEST 3 VERSION A
MATH 1020
FALL 2015
The graph of f ( x ) is shown twice below. Use the graphs to answer the next six questions. Use the graph
on the left for the first four questions and the graph on the right for the last two.
[2 points each]
11. On the given interval, the function f ( x ) has a relative maximum that occurs at point ____________.
a.
A
b.
B
c.
C
d.
D
e. E
12. On the given interval, the function f ( x ) has an absolute maximum that occurs at point ___________.
a.
A
b.
B
c.
C
d.
D
e. E
13. On the given interval, the function f ( x ) has a relative minimum that occurs at point _____________.
a.
A
b.
B
c.
C
d.
D
e. E
14. On the given interval, the function f ( x ) has an absolute minimum that occurs at point ____________.
a.
A
b.
B
c.
C
d.
D
e. E
15. The graph of the second derivative f ′′( x ) will have an x-intercept at point(s) x = ________ .
a. -0.5 and 0.6 and 2.1
b. 0 and 1.5
c. 0.8
d. cannot be determined
16. The graph of the second derivative f ′′( x ) will lie above the x-axis on the interval(s) ________.
a. −0.7 < x < 0.8
b. −0.7 < x < −0.5 and 0.6 < x < 2.1
c. 0 < x < 1.5
d. 0.8 < x < 2.3
Page 5 of 12
MATH 1020
ANSWER KEY TEST 3 VERSION A
FALL 2015
Use the following to answer the next two questions.
f ( x) =
77.917
gives the percentage of online 30 – 49 year olds who use social networking
1 + 32.414 e−0.854 x
Checkpoint: f (2) ≈ 11.3345
sites, ‫ ݔ‬years since 2004, 0 ≤ ‫ ≤ ݔ‬9.
17. Find the point on the graph of f at which the percentage of online 30 – 49 year olds who use social
networking sites is increasing most rapidly.
a. (4.091, 16.634)
b. (4.091, 39.252)
c. (4.073, 16.635)
d. (4.073, 38.958)
18. At the point at which the function f is increasing most rapidly, which one of the following statements
about the second derivative is TRUE?
a. f ′′ has a relative maximum
b. f ′′ has a relative minimum
c. f ′′ has a zero
d. f ′′ is positive
19. Given the following three graphs, determine which is the graph of f , of f ′ , and of f ′′ .
GRAPH I
GRAPH II
GRAPH III
a. f is graph I, f ′ is graph II, f ′′ is graph III
b. f is graph III, f ′ is graph II, f ′′ is graph I
c. f is graph II, f ′ is graph I, f ′′ is graph III
d. f is graph III, f ′ is graph I, f ′′ is graph II
Page 6 of 12
MATH 1020
ANSWER KEY TEST 3 VERSION A
Use the following to answer the next three questions.
FALL 2015
[2 points each]
The plot below gives the rate of change graph f ′( x ) for the function f ( x ) .
f ( x ) gives the number of hours of daylight Clemson receives throughout the year, where x = 1 is the end
of January, x = 2 is the end of February, etc.
Rate of Change Graph
20. Use the graph to approximate when the number of daylight hours in Clemson is decreasing most
rapidly.
a. At the beginning of March ( x = 2.1 )
b. Near the end of June ( x = 5.7 )
c. In early October ( x = 9.2 )
d. Near the end of December ( x = 11.8 )
21. The number of hours of daylight that Clemson receives is at a relative minimum_____________.
a. At the beginning of March ( x = 2.1 )
b. Near the end of June ( x = 5.7 )
c. In early October ( x = 9.2 )
d. Near the end of December ( x = 11.8 )
22. Use the above graph of f ′( x ) to determine where the function f ( x ) is concave down.
a.
0 < x < 9.2
b. 2.1 < x < 9.2
c.
0 < x < 5.7
d. 5.7 < x < 11.8
Page 7 of 12
MATH 1020
ANSWER KEY TEST 3 VERSION A
FALL 2015
Use the following information to answer the next two questions.
P ( x ) trillion dollars gives the nominal GDP (gross domestic product) in the US, when the US population is
x million people, data from 240 ≤ x ≤ 315 . GDP is 13.860 trillion dollars when the US population is 300
million people and is increasing by 0.145 trillion dollars per million people.
23. Using linearization, the nominal GDP in the US is estimated to be __________ trillion dollars when the
US population is 300.5 million people.
a. 7.075
b. 14.005
c. 13.9325
d. 14.360
24. Write a linearization for the function P ( x ) with respect to c = 300 .
a. PL ( x) = 13.860 + 0.145( x − 300)
b. PL ( x) = 0.145 + 13.860( x − 300)
c. PL ( x) = 13.860 + 300( x − 0.145)
d. PL ( x) = 0.145 + 300( x − 13.860)
Test continues on next page…….
Page 8 of 12
ANSWER KEY TEST 3 VERSION A
MATH 1020
FALL 2015
FREE RESPONSE: 37 points
Show work where possible. Read the directions on rounding, inclusion of units, and writing models and
sentences at the back of the test.
1. Find the derivative of the following functions. It is only necessary to simplify exponents. Use exact
numbers. For full credit, clearly show work and use proper mathematical notation in your work as well
as your answer.
3
x
Part A: 3.5 points
a. f ( x) = 5 ⋅ x + 2
Part B: 3.5 points
f ′( x ) =
1
−
5 3
( x + 2 x ) 2 [3 x 2 + (ln 2)(2 x )]
2
b. f ( x ) = 3 e 5 x x
−1
2
1 −3
+ 3e 5 x ( − )( x 2 )
2
1
3
−
−
3
= 15e 5 x x 2 − e 5 x x 2
2
f '( x ) = 3 ⋅ 5e 5 x x
−
1
2
( ____ / 7 pts )
2.
G( x) = −2.283x3 + 46.543x2 −141.668x + 367.501 dollars per ounce gives the price of gold at the end of
the year, x years after 2000, 0 ≤ x ≤ 14 .
G (2) = 252.073
Identify the following, correct to THREE decimal places. Answers that are not written correct to three
decimal places will be marked wrong.
Abs. Max: 2 pts first blank; 1 pt second blank
Abs. Min: 2 pts first blank; 1 pt second blank
Inflection Pont: 3 pts first blank; 1 pt second blank; 1 pt third blank
On the interval 0 ≤ x ≤ 14 , the price of gold was at its highest_ 11.845_____ years after 2000. At that
time, the price of gold was _________1425.493___ dollars per ounce.
On the interval 0 ≤ x ≤ 14 , the price of gold was at its lowest__ 1.746______ years after 2000. At that
time, the price of gold was _______249.884_____ dollars per ounce.
On the interval 0 ≤ x ≤ 14 , the price of gold was increasing most rapidly____ 6.796___ years after 2000. At
that point in time, the price of gold was increasing by ____174.619_ dollars per ounce per year and the
price of gold was _837.760*_ dollars per ounce.
*answers vary by calculator.
( ____ /11pts )
Page 9 of 12
MATH 1020
3.
ANSWER KEY TEST 3 VERSION A
FALL 2015
n( x ) students gives the number of out-of-state students attending the University of South Carolina (USC),
x years since 2000, 0 ≤ x ≤ 14 .
t(x) thousand dollars per student gives the annual out-of-state-tuition for a student attending the
University of South Carolina, x years since 2000, 0 ≤ x ≤ 14 .
In 2008, the number of out-of-state students attending USC was 8,320 students and that number
was increasing by 457.6 students per year.
Also in 2008, annual out-of-state tuition at USC was 22.000 thousand dollars per student and
was increasing by 1.500 thousand dollars per student per year.
Part A: 2 points
Part B: 2 points
Part C: 4 points
a. The functions n( x ) and t ( x) can be combined to create a new function f ( x) using
(circle one) multiplication or composition.
b. Fill in the blanks to complete the model for the new function f ( x) , using the given functions to
describe f ( x) .
f ( x) =___ n( x ) ⋅ t ( x ) ______ _____________ thousand dollars _____________ gives the
(notation for new function)
(units)
total tuition for all out-of-state students attending USC, x years since 2000, 0 ≤ x ≤ 14 .
c. How quickly is the total tuition for out-of-state students attending USC changing in 2008?
Show work and include units with the answer (correct to 3 decimal places).
f ′( x) = n′( x)t ( x) + n( x)t ′( x)
f ′(8) = n′(8)t (8) + n(8)t ′(8)
= (457.6)(22) + (8320)(1.500)
= 22,547.200 thousand dollars per year
( ____ / 8 pts )
One more page…….
Page 10 of 12
MATH 1020
ANSWER KEY TEST 3 VERSION A
FALL 2015
4. The figure to the right shows the annual cost for a one-milliondollar term life insurance policy for a male as a function of the
age of the insured person. The slope of the tangent line to the
function f ( x ) at x = 60 is 0.127.
a. Write a linearization for the
function f ( x ) with respect to
c = 60 .
Part A: 3 points
Part B: 2 points
Part C: 1 point
f L ( x ) = f (c ) + f '(c )( x − c ) = f (60) + f '(60)( x − 60)
= 4.827 + 0.127( x − 60)
b. Use the slope of the tangent line to estimate the annual cost for a one-million-dollar term life insurance
policy for a 58-year-old male. Show work. Write the answer correct to three decimal places and include
units with the answer.
f L (58) = 4.827 + 0.127(58 − 60)
= 4.827 + 0.127(−2)
= 4.573 thousand dollars
c. Use the graph to predict whether the estimate in part b) is an overestimate or an underestimate (circle
one).
( ____ / 6 pts )
5. Find the first and second derivative formulas for the function f ( x ) = 3 x 4 − 2.5 x 3 + 1.5 x 2 − 6.05 .
f ′( x ) = 12 x 3 − 7.5 x 2 + 3 x
f ′′( x ) = 36 x 2 − 1 5 x + 3
( ____ / 4 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.
Page 11 of 12
MATH 1020
ANSWER KEY TEST 3 VERSION A
FALL 2015
General Directions:
•
Show work where possible. Answers without supporting work (where work is appropriate) may
receive little 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.
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.
•
DIM MISMATCH error usually means that the lists in L1 and L2 are not of equal length.
•
1. Let f ( x ) = g ( x ) h ( x ) , g (10) = −4 , h(10) = 12 , g '(10) = 0 , h '(10) = 35 . Find f '(10).
DATA a.
TYPE
error usually means that youb.already
−140
−48 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.
MULTIPLE CHOICE: 69 points
(Each question is worth 3 points unless otherwise noted.)
Use a #2 pencil and completely fill each bubble on your scantron to answer each multiple choice question.
future reference,
circle
this if
test
There is no
penalty
guessing
on multiple
• (For
SYNTAX
ERROR: Try
GOyour
TO. answers
This will on
happen
youpaper.)
use a subtraction
minus
sign for
when
you should
use a negative
choice
questions.
If
you
indicate
more
than
one
answer,
or
you
leave
a
blank,
the
question
will
be marked
sign.
as incorrect.
Page 12 of 12