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DEPARTMENT OF MATHEMATICSUniversity of Toronto
MAT136H1STerm Test
Winter 2012Time allowed: 1 hour, 45 minutes
NAME OF STUDENT:(Please PRINT full nameand UNDERLINE surname):
STUDENT NUMBER:
SIGNATURE OF STUDENT:
TUTORIAL CODE(e.g.,M4A, R5D, etc.):
TUTORIAL TIME(e.g.,T4,R5,F3, etc.):
NAME OF YOUR TA:
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 tobe given in this booklet.No computer cards will be used.
3. No aids allowed.NO CALCULATORS!
4. DO NOT TEAR OUT ANY PAGES
FOR MARKERS ONLYPART A /48
B1 /7
B2 /7
B3 /7
B4 /7
B5 /8
B6 /8
B7 /8
TOTAL /100
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PART A [48 marks]
Please read carefully:PART A consists of 12 multiple-choice questions, each of which has exactly one correct answer. Indica
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 requirto justify your answers in PART A. Note that for PART A, only your final answers (as indicated bthe circles you darken) count; your computations and answers indicated elsewhere will NOcount.DO NOT TEAR OUT ANY PAGES
1. If f(x) = 8x3 4x + 5 and f(1) = 6, then f(1) =.A
O
6
BO 0C
O1
DO
4E
O 5
2. If 32 {5f(x) + 4g(x)}dx = 5 and
32 g(x)dx = 5, then
32 f(x)dx =.
AO
6B
O 0C
O5
DO 3E
O8
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5. Find the area of the region enclosed between the graphs ofy = 3x and y = 3x26A
O
272
BO
24
5C
O
252
DO
263
EO
253
6. Let R be the region bounded by the curves y = 4x and y = 2x2. Find the voluof the solid generated by revolving R about the y-axis.
AO
143
BO
133
CO
163
DO
173
EO
132
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7.20|x |dx =
AO
22 2B
O
22
CO
2 + 1D
O 2
EO 22
8. Water flows from the bottom of a storage tank at a rate r(t) = (200 4t) literminute, where 0 t 50. Find the amount of water that flows from the tankduring the first 10 minutes.
AO
1,650 litersB
O1,700 liters
CO
1,750 litersD
O1,600 liters
EO 1,800 liters
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9. Find the value of
limn
(1
n + 2n2+
2
2n + 2n2+
3
3n + 2n2+ . . . +
n
n2 + 2n2).
AO
1 2 ln 3 + 2 ln 2B
O 1 + 3 ln 3 3 l n 2C
O 1 + 2 ln 3 2 l n 2D
O 1 + 2 ln 3 ln 2E
O1 2 ln 3 + ln 2
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10. Let f be a continuous function on [3, 6] such thatthe average value off(x) on [3,1] is 5,the average value off(x) on [
1, 2] is 4,
the average value off(x) on [2, 6] is 2.Let g(x) = 2 + f(x). Find the average value of g(x) on [3, 6].
AO
174
BO
173
CO
133
DO
132
EO
163
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11. Evaluate87
x
x6dx.
Hint: the formula
dxx2
a2
= ln
|x +
x2
a2
|+ C may be useful.
AO
4 7 + 6 ln 3 3 ln(4 + 7)B
O8 7 + 6 ln 3 3 ln(4 + 7)
CO 8 7 6 l n 3 3 ln(4 + 7)D
O 4 +
7 6 l n 3 3 ln(4 + 7)E
O 8 +
7 6 l n 3 3 ln(4 + 7)
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12.21 ln(x +
x2 1)dx =
AO
2 ln(1 +
2) 1B
O 2 ln(1 + 2) 1C
O
2 ln(1 +
2) 2
DO ln(1 +
2) 1
EO
2 ln(1 +
2)
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PART B [52 marks]
Please read carefully:Present your complete solutions to the following questions in the spaces provided, in a neat and logical fas
ion, showing all your computations and justifications. Any answer in PART B without proper justificatimay receive very little or no credit. Use the back of each page for rough work only. If you must continyour formal solution on the back of a page, you should indicate clearly, in LARGE letters, SOLUTIOCONTINUED ON THE BACK OF PAGE . In this case, you may get credit for what you write 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.
1. Findxe5xdx.
[7]
2. Find
(2x3 + cosx)(x4 + 2 sin x)45dx
Note: This is a very easy problem!
[7]
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3. Find
tan74 x sec4 xdx.
[7]
4. Find
dx
(25x2)3/2 .
[7]
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5. Find
5x2+3x+64x(x2+16)
dx.
[8]
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6. Find
dxx+x
x
.
[8]
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7. Note: This is not an easy question and will be marked very strictly. Very littleno credit will be given unless your solution is completely correct.
Given that10 (arcsinx)6dx = k, find the value of
10 (arcsinx)8dx in terms ofk
Simplify your final answer as much as possible.
[8]
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l't/I^ * 5( 2 *3 + "fr:O( *++2a;^-)."d r. = !t ,^t"d,t^ =_(r,f+2--a..i^>c)'- tC. .g2-t t6 , Lez(+ 6) \ \'--
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86. /.Xq=,JT. T4-*'.,\'l.{c-r^.-.- f {x :-f) fid'hrr^JT l
: 2( a-rdr 2 r.^ o{ ,.,,* = oLZ
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6
- y*) t* = 2o o * *2_-t'1f'=fgo o-+-=9tlt#' - .{^ in* Ln- n-?co t1 ?, it2"rn n+"o ^,h, h*,r_) i= |Alt ^E(#a,.=Jiffi:l:#65 (*t3)dw, A ntt-- - ^\e tr !a^(x lr;o:,,)g ="r*t:F r,e- ).;* t *fJG fu 94lr \.r tJ-
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