Code: 1642
DEPARTMENT OF MATHEMATICS
University of Toronto
MAT 136H1S
Term Test
Wednesday, February 27, 2013
Time allowed: 1 hour, 45 minutes
NAME OF STUDENT:
(Please PRINT full name
and UNDERLINE surname):
STUDENT NUMBER:
SIGNATURE OF STUDENT
(in INK or BALL-POINT PEN):
TUTORIAL CODE (e.g. M4A, R5D, etc.):
TUTORIAL TIME (e.g. T4, R5, F3, etc.):
NAME OF YOUR T.A.:
NOTE:
1. Before you start, check that this test
has 14 pages. There are NO blank pages.
2. This test has two parts:
PART A [48 marks]: 12 multiple choice questions
PART B [52 marks]: 7 written questions
Answers to both PART A and PART B are
to be given in this booklet. No computer
cards will be used.
3. No aids allowed.
NO CALCULATORS!
FOR MARKERS ONLY
QUESTION
MARK
PART A
/ 48
B1
/7
B2
/7
B3
/7
B4
/7
B5
/8
B6
/8
B7
/8
DO NOT TEAR OUT ANY PAGES.
TOTAL
Page 1 of 14
/ 100
Code: 1642
PART A [48 marks]
Please read carefully:
PART A consists of 12 multiple-choice questions, each of which has exactly one correct answer.
Indicate your answer to each question by completely filling in the appropriate circle with
a dark pencil .
MARKING SCHEME: 4 marks for a correct answer, 0 for no answer or a wrong answer. You are
not required to justify your answers in PART A. Note that for PART A, only your final answers (as indicated by the circles you darken) count; your computations and answers
indicated elsewhere will NOT count.
DO NOT TEAR OUT ANY PAGES.
1. If f 0 (x) = 6x2 − 3 and f (1) = 0 , then f (2) =
A
15
B
14
C
6
D
11
E
8
2. Find the Riemann sum for f (x) = x2 + 3 on [0,6] by partitioning [0,6] into 3 subintervals of
equal length and choosing each sample point to be the left endpoint of the subinterval.
A
58
B
50
C
68
D
64
E
52
Page 2 of 14
5
Z
Z
{2f (x) + g(x)} dx = 20 ,
3. If
0
A
12
B
6
C
13
D
7
E
9
5
Z
f (x) dx = 3 and
0
1
A
B
C
D
E
0
√
2
√
6 2
√
3 2
√
2−1
Page 3 of 14
Z
g(x) dx = 5 , then
3
Z x6 q
√
4. Let g(x) =
1 + t dt . Then g 0 (1) =
5
3
Code: 1642
g(x) dx =
0
Code: 1642
2
5. Find the area of the region bounded by the curves y = 2x and y = x − 2x .
A
25/2
B
32/3
C
28/3
D
23/2
E
21/2
√
6. Let R be the region bounded by the curves y = x and y = x3 . Find the volume of the
solid generated by revolving R about the y -axis.
A
5π/6
B
π/2
C
3π/10
D
2π/5
E
3π/5
Page 4 of 14
Code: 1642
3
7. Find the average value of f (x) = 4x − 6x + 2 on the interval [1,3].
A
30
B
40
C
50
D
20
E
35
Z
8.
1
2
dx
=
x(x2 + 1)
A
1
2 (2 ln 3
− 3 ln 2)
B
1
2 (2 ln 2
− ln 3)
C
1
2 (3 ln 2
− ln 5)
D
1
2 (2 ln 3
− ln 5)
E
1
2 (3 ln 3
− 2 ln 5)
Page 5 of 14
Z
Code: 1642
1
9.
|3x2 − x| dx =
−1
A
50/27
B
49/27
C
58/27
D
53/27
E
55/27
Page 6 of 14
Code: 1642
10. f is a differentiable function whose graph is symmetrical about the line x = 5 . If f (5) = 7
Z 5
Z 10
0
and
xf (x) dx = 8 , find the value of
f (x) dx .
0
A
50
B
54
C
60
D
44
E
64
0
Page 7 of 14
Z
11.
Code: 1642
π/4
sin2
0
A
B
C
D
E
x
2
cos2
x
2
dx =
π−2
16
π−4
16
π−4
32
π−3
16
π−2
32
Page 8 of 14
Code: 1642
12. A hemispherical bowl has radius 10 inches. A heavy spherical metal ball with diameter 8
inches sits at the bottom of the bowl. Water is then poured into the hemispherical bowl to
a depth of 5 inches. Find the volume of the water in the bowl.
A
150 π cu. in.
B
160 π cu. in.
C
156 π cu. in.
D
144 π cu. in.
E
158 π cu. in.
Page 9 of 14
Code: 1642
PART B [52 marks]
Please read carefully:
Present your complete solutions to the following questions in the spaces provided, in a neat and
logical fashion, showing all your computations and justifications. Any answer in PART B without
proper justification may receive very little or no credit. Use the back of each page for rough work
only. If you must continue your formal solution on the back of a page, you should indicate clearly,
in LARGE letters, “SOLUTION CONTINUED ON THE BACK OF PAGE
”. In this case,
you may get credit for what you write on the back of that page, but you may also be penalized
for mistakes on the back of that page.
MARKS FOR EACH QUESTION ARE INDICATED BY [ ] .
DO NOT TEAR OUT ANY PAGES.
Z
1. Find
xe3x dx .
[7]
Z
2. Find
(4 + 2 cos x)(2x + sin x)259 dx.
[7]
Page 10 of 14
Code: 1642
Z
3. Find
100
tan
4
x sec x dx.
[7]
Z
4. Find
√
ex
dx.
25 − e2x
[7]
Page 11 of 14
Code: 1642
Z
5. Find
dx
√
.
2
x 4 − x2
[8]
Page 12 of 14
Code: 1642
Z
6. Find
dx
√
√ .
x+1+ x
[8]
Page 13 of 14
Code: 1642
7.
NOTE: This question will be marked very strictly. Very little or no credit will
be given unless your solution is completely correct.
Z
dx
.
Find
x
2 + 22−x − 2
[8]
Page 14 of 14
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