Math 131
Exam 3
F04M131.3.1
Name ________________________________
ID Number ____________________
Part I consists of 14 multiple choice questions (worth 5 points each) and 5 true/false
question (worth 1 point each), for a total of 75 points. Mark the correct answer on
the answer card. For Part I, only the answer on the card will be graded.
Ð$BÑ  ,B  "
1. If lim cos(B
œ ), what is , ?
 sin ,B
BÄ!
A) !
F)  #
B) 1
G)  $
C) 4
H)  '
D) 7
I)  8
E) 9
J)  "!
2. A spherical snowball melts in such a way its radius is decreasing at a rate of 3 cm/min
at the instant when its radius is 20 cm. At that moment, how fast is its volume changing?
(Round your answer to 1 decimal place.)
A)
12478.6 cm$ /min
D)  23156.7 cm$ /min
G)  14682.3 cm$ /min
J)  24978.7 cm$ /min
B) 14682.3 cm$ /min
E) 16783.8 cm$ /min
H) 14292.9 cm$ /min
C)  15079.6 cm$ /min
F)  15983.7 cm$ /min
I)  12478.6 cm$ /min
F04M131.3.2
3. A certain cone is growing in such a way that its height is always twice its radius. Use
differentials to estimate how much the volume changes as the radius grows from 10 m to
10.05 m. (Round your answer to 2 decimal places.)
A) 31.12 m$
F) 31.42 m$
B) 31.27 m$
G) 31.55 m$
C) 31.57 m$
H) 31.59 m$
D) 31.66 m$
I) 31.99 m$
E) 31.75 m$
J) 31.03 m$
4. Suppose 0 ÐBÑ œ #B#  B  *. The Mean Value Theorem states that there is a
number - between 0 and 3 with a certain property. What is - ?
A) !
F) "#
B) 1
G) &%
C) 2
H) (%
D) &#
I) *%
E) $#
J) ""
%
F04M131.3.3
5. What is the slope of the tangent line to the curve œ
B œ sin >  cos >
C œ sin >  cos 2>
at the point corresponding to > œ 1 ? (See the figure.)
A)  "
F) $%
B) 
G) 1
"
#
C) !
H) &%
D) "%
I) &#
6. There two times > in Ò!ß #1Ó for which the curve
Ð! Ÿ > Ÿ #1Ñ
E) "#
J) 2
B œ sin >  cos >
œ C œ sin >  cos >
has a vertical
tangent line. What are those times? (Note: the equations are slightly different from the
equations in problem #5.)
A) !ß 1
F) #$1 ß &$1
B) 1% ß &%1
G) $%1 ß (%1
C) 1# ß $#1
H) %$1 ß ($1
D) 1' ß ('1
I) 1, #1
E) 1$ ß %$1
J) 1$ ß #$1
F04M131.3.4
7. The point T œ Ð"ß "Ñ is on the graph of #B ln C  C ln B œ !. What is the slope of the
tangent line to the graph at T ?
A)  #
F) "#
B) 
G) "
$
#
C)  "
H) $#
D) 
I) /
"
#
E) !
J) #/
8. A rectangular box has a base in which the length is always 3 times the width. What is
the largest volume for such a box if its surface area (4 sides  top  bottom) must be
1152 in# ?
A) 1728 in$
F) 1480 in$
B) 512 in$
G) 1624 in$
C) 695 in$
H) 1848 in$
D) 1048 in$
I) 2142 in$
E) 1246 in$
J) 2304 in$
F04M131.3.5
9. If 0 ÐBÑ œ lnÐ
A) %$
"!
F) "%
"(
%
È
#B  & †Ð$B  &Ñ)
Ñ,
È'B &
B)
G)
#"
$
#&
$'
what is 0 w Ð!Ñ?
C) &#
H) !
D) %"
"&
I) %$
E) #$
#&
J) #$
10. 0 is a function defined on the interval Ò!ß &Ó, and 0 Ð!Ñ œ 0 Ð&Ñ œ "ß 0 Ð$Ñ œ  "Þ
Suppose 0 w ÐBÑ œ ÐB  "ÑÐB  #Ñ# ÐB  $Ñ$ ÐB  %Ñ% Þ
Then exactly three of the following statements are true. Which three are true?
i) 0 is increasing on the interval "  B  #
ii) 0 has a local min at B œ "
iii) 0 has neither a local max nor a local min at B œ #
iv) 0 has a local min at B œ $
v) 0 has its absolute min at B œ $
A) i, ii, iii
F) i, iv, v
B) i, ii, iv
G) ii, iii, iv
C) i, ii, v
H) ii, iii, v
D) i, iii, iv
I) ii, iv, v
E) i, iii, v
J) iii, iv, v
F04M131.3.6
11. Suppose 0 ÐBÑ œ lnÐÐarctan BÑ3 Ñ for B  !Þ What is 0 w Ð"Ñ ? (Note: arctan B is the
“inverse tangent function” which the text sometimes also writes as tan" B.)
A) !
F) "#
B)
G)
'
1
$
#1
C) 1"
H) 21
D) 1%
I) $%1
E)
J)
$
1
%1
$
+#
12. On the interval Ò"ß $Óß the absolute minimum of the function 0 ÐBÑ œ B+  #B
# occurs
at B œ #. What is the absolute maximum value of 0 ÐBÑ on Ò"ß $Ó ?
A) &#
F) 3
B) 0
G) (#
C)  #
H)  $#
D) "#
I) 
"
#
E) 2
J) 1
F04M131.3.7
#
"
13. For 0 ÐBÑ œ $BÐB  %Ñ $ , the derivative 0 w ÐBÑ œ ÐB  %Ñ $ Ð&B  "#ÑÞ What is the
largest interval listed on which 0 ÐBÑ is concave down? (For your convenience: the right
endpoints in the intervals listed below are increasing as you advance through the list.)
B) Ð  _ß !Ñ
C) Ð  _ß '& Ñ
D) Ð  _ß 4"Î$ Ñ
E) Ð  _ß (% Ñ
F) Ð  _ß "#
& Ñ
G) Ð  _ß 3Ñ
H) Ð  _ß 4Ñ
I) Ð  _ß #%
& Ñ
J) Ð  _ß $'
& Ñ
A) Ð  _ß 
&
"# Ñ
F04M131.3.8
"4Þ If lim Ð#B  $Ñ
BÄ_
Ð + ln" B Ñ
œ "!, what is + ?
A) 1
F)
"
ln "!
B) ln #
G)
"
ln $
C)
"
ln #
D) ln 10
E) ln $
H)
"
ln '
I) /
J) / ln "!
F04M131.3.9
Questions 15-19 are “true/false” questions
15. lim Ð"  B" Ñ'B œ _
BÄ_
A) True
B) False
"6. Mary drives the 280 miles from St. Louis to Kansas City in 5 hours. At some time
during the trip she was traveling 56 miles/hr.
A) True
B) False
17. If - is a critical point of 0 and 0 ww Ð-Ñ  !, then 0 ÐBÑ has an absolute minimum at
B œ -Þ
A) True
B) False
18. There exists a differentiable function 0 such that 0 Ð&Ñ œ #!!ß 0 Ð"Ñ œ ! and
0 w ÐBÑ  '! for all B.
A) True
19Þ
.
.B
lnÐ)Ñ œ
A) True
B) False
"
)
B) False
F04M131.3.10
Name ______________________________________ ID Number__________________
Please put your name and ID number above and on each following sheet of Part II,
in case the sheets get separated during the grading process.
Part II: (25 points) In each problem, clearly show your solution in the space
provided. “Show your solution” does not simply mean “show your scratch
work”  you should cross out any scratch work that turned out to be wrong or
irrelevant and, where appropriate, present a readable, orderly sequence of steps
showing how you got the answer. Generally, a correct answer without supporting
work may not receive full credit.
20. a) Find all the critical numbers for the function 0 ÐBÑ œ /B ÐB#  $ÑÞ
b) What are the absolute maximum and minimum values for 0 ÐBÑ œ /B ÐB#  $Ñ on the
interval Ò!ß #Ó? (Be sure to give the exact max and min values  although you can also
include a decimal approximation for these values if you like.)
F04M131.3.11
Name ______________________________________ ID Number__________________
.C
21. a) Find .B
if C œ log# Ð B##B " Ñ (No simplification is necessary after you get to a
.C
correct formula .B
œ ÞÞÞ Ñ
b) Find
formula
.C
.B
.C
.B
if C œ Barctan B
œ ÞÞÞ Ñ
(No simplification is necessary after you get to a correct
c) Find lim Ð B"  csc BÑ (You must show the steps leading to an answer: a guess based
BÄ!
on using a calculator is not adequate  although it is a way to check your work.)