1. No category

Math 1M03, Spring/Summer 2009, Final Exam Sample Answers, page 1 of "&
McMaster University
MATH 1M03,Spring-Summer 2009
Final Exam-Sample-Answers
Time: August 6, 2009
Duration: 3 hours
Instructor: Z. Kovariik
Name:____________________________ ID #:______________
Instructions:
This test paper contains 20 multiple choice questions printed on both sides of 18 pages.
Letter answers to all questions must be recorded on the lower part of this page.
Part A contains 14 multiple choice questions worth 2 marks each; no part marks
will be given.
Ä A missing answer or a multiple answer receives zero marks.
Part B contains 6 multiple choice questions worth 4 marks each; however,
incorrect answers will be reviewed for possible part marks.
Maximum total marks: 52
The questions are on pages 2 to 14. Pages 15 through 17 are available for scrap work.
Helpful formulas are provided on page 18.
Only the McMaster standard calculator, Casio fx 991, is permitted.
YOU ARE RESPONSIBLE FOR ENSURING THAT YOUR COPY OF THE
PAPER IS COMPLETEÞ BRING ANY DISCREPANCY TO THE ATTENTION
OF THE INVIGILATOR.
Question
Answer
Question
Answer
Question
Answer
Marks
A-1
A-2
A-8
A-9
B-15
B-16
A-3
A-10
B-17
A-4
A-5
A-6
A-11
A-12
A-13
B-19
B-20
B-18
A-7
A-14
Total
á Continued on next page
Math 1M03, Spring/Summer 2009, Final Exam Sample Answers, page 2 of "&
PART A: Enter all answers into the appropriate boxes on the front page.
THERE ARE NO PART MARKS IN THIS SECTION.
A-1
Suppose the graph below gives several level curves of the function D œ 0 aBß Cb.
Then which of the following functions (a), (b), (c), (d) can be 0 aBß Cb?
(a) B  C#
(b) B#  C
(c)
C
B#
(d)
B
C#
Solution (not required)
The typical equation is B œ GC# , corresponding to CB# œ G , as in (d).
_________________________________________________________________
A-#
'"" $B .B œ
(a) # ln $
(b) $ ln) $
Solution (not required)
' $B .B œ ' /B ln $ .B œ /lnB ln$$  G
'"" $B .B œ
"
/B ln $
ln $ ¹"
œ
" ˆ
ln $ $
(c)
 $" ‰ œ
#
ln $
(d)
) ln $
$
)
$ ln $
á Continued on next page
Math 1M03, Spring/Summer 2009, Final Exam Sample Answers, page 3 of "&
Enter your choice on the front page!
ÈBC
B# C# b
A-3 Consider the functions J aBß Cb œ lnaBC"
ß KaBß Cb œ BC#
Which of the following points aBß Cb are in the domain of J  K À
(a) a!ß "b
(b) a"ß !b (c) a  #ß #b
(d) a#ß "b
Solution (not required)
J a"ß !b  Ka"ß !b œ
lna"# !# b
a"ba!b"

È"!
a"ba!b#
is well-defined (equal to  "# )
The other three points always make one of J ß K undefined. Try out on your own.
_________________________________________________________________
A-4 A tree is currently 2 meters high, and in B years from now it will grow at a rate
5
meters per year. Its height at the end of year 4 will be
a1Bb#
(a) 6 meters
Solution (not required)
L w aBb œ
&
a"Bb#
L aBb œ 
&
"B
L aBb œ ( 
(b) 3 meters
 Gß L a!b œ # œ 
&
"B
L a%b œ #  '!
or shorter:
%
&
.B
a"Bb#
&
"
 Gß
L a% b œ ( 
œ#ˆ
(c) 7 meters (d) 5 meters
&
&
& ‰%
"B k!
G œ(
œ'
œ#
&
&

&
"
œ'
á Continued on next page
Math 1M03, Spring/Summer 2009, Final Exam Sample Answers, page 4 of "&
Enter your choice on the front page!
A-5 If 1w aBb œ &B%  " and 1a"b œ  1 then 1a  "b œ
(a) !
(b) '
(c)  '
(d)
Solution (not required)
"
1a  "b œ  "  ' ˆ&B%  "‰.B
&
"
œ  "  aB&  Bbk" œ  "  a  "  "b  a"  "b
"
œ &
_________________________________________________________________
A-6 If /BC œ # and /#BC œ % then /B œ
(a) 4
(b) 2
(c) È#
(d) "#
Solution (not required)
Shortcut: Multiply a/BC ba/#BC b œ /$B œ a#ba%b œ )
Long way:
/B œ a/$B b
"Î$
œ )"Î$ œ #
B  C œ ln #
#B  C œ ln %
solve for B by your favourite method to find
Bœ
ln #ln %
$
œ
ln )
$
œ
$ ln #
$
œ ln #
/B œ /ln # œ #
á Continued on next page
Math 1M03, Spring/Summer 2009, Final Exam Sample Answers, page 5 of "&
Enter your choice on the front page!
A-7 Suppose $1000 is invested at 5% interest compounded continuously. Then it will
double in value after how many years?
(a) 20
(b) 2 ln #0
(c) ln#0#
(d) #0 ln #
Solution (not required)
Rate < œ !Þ!&ß Ea>b œ Ea!b/<>
Data: Ea!b œ "!!!ß
Ea>b œ #!!!
#!!! œ "!!! /!Þ!& >
# œ /!Þ!& >
!Þ!& > œ ln #
> œ #! ln #
_________________________________________________________________
#
'#_ B"
A-8
.B œ
(a) ln $
(b) ln %
(c) #$
(d) a divergent integral
Solution (not required)
#
'#R B"
œ # lnaR  "b  # ln $ß
diverges to _ as R Ä _
á Continued on next page
Math 1M03, Spring/Summer 2009, Final Exam Sample Answers, page 6 of "&
Enter your choice on the front page!
' lnBB .B œ
A-9
(a) B"#  lnB#B  G
(b) #" aln Bb#  G
(c) ln ln B  G
(d) B# ln B  G
Solution (not required)
Change variable: ln B œ ?ß
' lnBB .B œ ' ? .? œ
"
B .B
" #
#?
œ .?ß
 G œ #" aln Bb#  G
_________________________________________________________________
A-10 Suppose the function 0 aBß Cb has partial derivatives
0B œ $B#  'Cß
0C œ  'B  #C
Then the point aBß Cb œ a'ß ")b is
(a) a point of relative minimum
(b) a point of relative maximum
(c) not a critical point
(d) a saddle point
Solution (not required)
Check critical point:
0B a'ß ")b œ a$ba'b#  a'ba")b œ "!)  "!) œ !
0C a'ß ")b œ a  'ba'b  a#ba")b œ  $'  $' œ !
Second derivatives:
0BB œ 'B
0BC œ  '
0CC œ #
#
H œ a0BB ba0CC b  a0BC b œ "#B  $'
Ha'ß ")b œ a"#ba'b  $' œ $'  !
not a saddle point
0BB a'ß ")b œ a'ba'b œ $'  !
Ä local minimum
á Continued on next page
Math 1M03, Spring/Summer 2009, Final Exam Sample Answers, page 7 of "&
Enter your choice on the front page!
A-11 Consider 0 aBß Cb œ B/BC Þ Then the partial derivative 0BC aBß Cb is
(a) BC# /BC
(b) #B/BC  B# C/BC
(c) B/#BC
(d) #C/BC  BC# /BC
Solution (not required)
0B œ a"b/BC  BC/BC
0BC
œ B/BC  aB/BC  B# C/BC b
œ #B/BC  B# C/BC
_________________________________________________________________
A-12 For a random variable \ with the probability density function
Ú!
B!
"
0 aBb œ Û ) B ! Ÿ B Ÿ %
Ü!
B%
the probability of \ 2 is
(a) "#
(b) $%
(c) "%
(d) ()
Solution (not required)
T a\ #b
œ '# 0 aBb.B œ '# ") B .B  !
_
œ
"
#
"' a%
%
 ## b œ
"#
"'
œ
$
%
á Continued on next page
Math 1M03, Spring/Summer 2009, Final Exam Sample Answers, page 8 of "&
Enter your choice on the front page!
A-13 Suppose a random variable \ has normal distribution with mean ' and standard
deviation 3. Then the probability T a& Ÿ \ Ÿ )b is
(a) greater than T a% Ÿ \ Ÿ (b
(b) less than T a% Ÿ \ Ÿ (b
(c) equal to T a% Ÿ \ Ÿ (b
(d) twice as much as T a% Ÿ \ Ÿ (b
Solution (not required)
Standard: ^ œ \'
$ :
Tˆ 
"
$
Tˆ 
#
$
5Ÿ\Ÿ8
Ä
&'
$
Ÿ^ Ÿ
)'
$
Ä

"
$
Ÿ^ Ÿ
#
$
%Ÿ\Ÿ(
Ä
%'
$
Ÿ^ Ÿ
('
$
Ä

#
$
Ÿ^ Ÿ
"
$
Ÿ ^ Ÿ #$ ‰ œ T ˆ 
"
$
Ÿ ^ Ÿ !‰  T ˆ! Ÿ ^ Ÿ #$ ‰
Ÿ ^ Ÿ "$ ‰ œ T ˆ 
#
$
Ÿ ^ Ÿ !‰  T ˆ! Ÿ ^ Ÿ "$ ‰
œ T ˆ! Ÿ ^ Ÿ "$ ‰  T ˆ! Ÿ ^ Ÿ #$ ‰
œ T ˆ! Ÿ ^ Ÿ #$ ‰  T ˆ! Ÿ ^ Ÿ "$ ‰
á they are equal
Of course the symmetry can be observed already with \ .
_________________________________________________________________
.C
A-14 The differential equation .B
œ BC has a general solution
(a) C œ GB
(b) C œ GB
(c) B#  C# œ G
(d) B#  C# œ G
Solution (not required)
Separated:
.C
C
œ
.B
B
ln kCk œ ln kBk  G"
C œ „ /G" B œ GB
á Continued on next page
Math 1M03, Spring/Summer 2009, Final Exam Sample Answers, page 9 of "&
PART B: For all the remaining questions, continue to enter your answers into the
appropriate box on the front page. If you record an incorrect answer, your work on the
question will be reviewed for possible part marks. Please show your work.
B-15 The temperature X a>b at time > in hours changes from midnignt to 6am by the
formula X a>b œ ")  "' a'  >b# (in degrees Celsius), ! Ÿ > Ÿ 'Þ The average
temperature over this part of the day is
(a) 13°C
(b) 14°C
(c) 15°C
(d) 16°C
Solution
'
Average œ "' ' ˆ")  "' a'  >b# ‰.>
!
œ "' ˆ")> 
"
") a'
 >b$ ‰¸! œ '" ˆ"!) 
'
" ‰
") !
 "' ˆ! 
"
‰
") #"'
œ ")  # œ "'
Another arrangement:
X a>b œ ")  "' a'  >b# œ "#  #>  "' >#
Average œ
"
'
'!' ˆ"#  #>  "' ># ‰.> œ "' ˆ"#>  >#  ")" >$ ‰¸'!
œ "' ˆ(#  $' 
#"' ‰
")
œ "' a"!)  "#b œ "'
á Continued on next page
Math 1M03, Spring/Summer 2009, Final Exam Sample Answers, page 10 of "&
B-16
Enter your choice on the front page and show your work!
The lifetime \ of a light bulb is a random variable with probability density
"
/BÎ)!!! B !
0 aBb œ œ )!!!
0
B!
where B is measured in hours. The probability that the light bulb will work for
more than 8000 hours is
(a) "/
(b) "  "/
(c) "#
(d) È"/
Solution
_
_
"
T a\ )!!!b œ ')!!! 0 aBb.B œ ')!!! )!!!
/BÎ)!!! .B
Improper integral:
R
"
BÎ)!!!
')!!!
.B œ
)!!! /
 /BÎ)!!! ¸)!!!
R
œ /R Î)!!!  /"
lim ˆ/R Î)!!!  /" ‰ œ
R Ä_
"
/
á Continued on next page
Math 1M03, Spring/Summer 2009, Final Exam Sample Answers, page 11 of "&
Enter your choice on the front page and show your work!
B-17 The random variable \ is uniformly distributed over the interval # Ÿ B Ÿ &Þ Then
its variance is
È
(a) %&
(b) $%
(c) #$
(d) #$
Solution
Probability density: 0 aBb œ "$ for # Ÿ B Ÿ &ß 0 aBb œ ! otherwise.
I a\ b œ '_ B0 aBb .B œ '# "$ B .B œ "' a&#  ## b œ
_
&
_
&
'_
B# 0 aBb .B œ '# "$ B# .B œ
œ
""(
*
" $
* a&
(
#
 #$ b
œ "$
Vara\ b œ "$  ˆ (# ‰ œ "$ 
#
%*
%
œ
$
%
á Continued on next page
Math 1M03, Spring/Summer 2009, Final Exam Sample Answers, page 12 of "&
Enter your choice on the front page and show your work!
.C
B-18 Consider the differential equation B .B
 #C œ B with initial condition Ca"b œ #Þ
The solution is
(a) $B#  B
(b)  $B#  &
(c) $B#  "
(d) $B#  B
Solution
.C
#
Standard:
.B  B C œ "
M aBb œ /'  B .B œ
#
" ˆ .C
B# .B
"
B# C
 B# C‰ œ
œ 
"
B
"
B#
. ˆ " ‰
.B B# C
œ
"
B#
to integrate:
G
C œ  B  GB#
Initial condition:
# œ  "  G"#
G œ$
C œ  B  $B# like in (d)
á Continued on next page
Math 1M03, Spring/Summer 2009, Final Exam Sample Answers, page 13 of "&
Enter your choice on the front page and show your work!
B-19 The minimal value of
0 aBß Cb œ B  C# subject to the constraint 1aBß Cb œ BC#  "' œ ! is
(a) "'
(b) %
(c) )
(d) #
Solution
Using Lagrange multipliers:
P œ B  C#  -aBC#  "'b
PB œ "  -C# œ !
PC œ #C  #-BC œ !
P- œ  aBC#  "'b œ !
C Á ! because BC# œ "'
From PC œ #C  #-BC œ ! À
By whatever correct method - mine would be
C
- œ C"# œ BC
ß
BC œ C$ ß
B œ C#
Constraint:
BC# œ C# C# œ C% œ "'ß
C œ #ß
OR
C œ  #ß
#
0 a%ß #b œ 0 a%ß  #b œ %  # œ )
B œ ## œ %
Bœ%
Eliminating a variable:
BC# œ "'
0 Š "'
C# ß C ‹ œ
Bœ
"'
C#
"'
C#
 C # œ 2 aC b
Critical pont: 2w aCb œ 
Again, C œ # or  #ß
$#
C$
 #C œ
$##C%
C$
B œ "'
%
œ!
œ%
We are also in a position to classify the extremum:
2ww aCb œ *'
C%  #  !
relative minimum in both places.
á Continued on next page
Math 1M03, Spring/Summer 2009, Final Exam Sample Answers, page 14 of "&
Enter your choice on the front page and show your work!
'"/ %B ln B .B œ
B-20
(a) /#  "
(b) /#  "
(c) #/#
(d) %/
Solution
By parts,
?w œ %B
? œ #B#
@ œ ln B
@w œ B"
' %B ln B .B œ #B# ln B  ' #B .B œ #B# ln B  B#  G
'"/ %B ln B .B œ a#B# ln B  B# bk/"
œ a#/#  /# b  a!  "b œ /#  "
á Continued on next page
Math 1M03, Spring/Summer 2009, Final Exam Sample Answers, page 15 of "&
Helpful formulas
a,B bC œ ,BC
a+,bB œ +B ,B
,! œ "
,B ,C œ ,BC
," œ ,
log, aBCb œ alog, Bb  alog, Cb
,B œ ,"B
8
,"Î8 œ È
,
log, Š BC ‹ œ alog, Bb  alog, C b
log, aB< b œ <alog, Bb
log, " œ !
log, , œ "
log, a,B b œ B
,log, B œ B
-B
log, B œ log
ln B œ log/ B
lna/B b œ B
/ln B œ B
log- ,
Exponential growth/decay: T a>b œ T! /<>
.
" "
. B
B
.B log, B œ ln , B
.B , œ aln , b,
' 0 a?aBbb ?w aBb .B œ ' 0 a?b .?
'+, ? .@ œ a?@bk,+  '+, @ .?
'+, 0 aBb.B œ  ',+ 0 aBb.B
'++ 0 aBb.B œ !
'+, 0 aBb.B œ '+- 0 aBb.B  '-, 0 aBb.B
R
,
lim ! 0 aB3 b?B œ '+ 0 aBb.B
where ?B œ ,+
R
R Ä_ 3œ"
'+_ 0 aBb.B œ
average:
"
,+
'+, 0 aBb.B
R
lim '+ 0 aBb.B
lim R : /5R œ ! if 5  !ß :  !
R Ä_
.D
.>
œ
`D .B
`B .>
R Ä_

`D .C
`C .>
HaBß Cb œ 0BB aBß Cb0CC aBß Cb  a0BC aBß Cbb#
0B aBß Cb œ -1B aBß Cb
0C aBß Cb œ -1C aBß Cb
Cw  T C œ U À
C œ M" Š' MU .B  G ‹
I a\ b œ '_ B0 aBb.B
_
1aBß Cb œ 5
where M œ /' T .B
5 a\ b œ ÈZ +<a\ b
Z +<a\ b œ '_ aB  I a\ bb# 0 aBb.B œ '_ B# 0 aBb.B  aI a\ bb#
_
_
END OF TEST PAPER