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Unformatted text preview: Code: 2782 UNIVERSITY OF TORONTO
Faculty Of Arts and Science APRIL 2010 EXAMINATIONS , 4
MAT135Y1Y «11) Q/
Calculus I . ﬂ}; .
Duration — 3 hours . iiix
.P
3213 NAME
(Please PRINT full name,
and UNDERLINE surname): STUDENT NO: SIGNATURE OF STUDENT
(in INK or BALL—POINT PEN): This exam has two Parts:
PART A: 7 questions (50 marks).
PART B: 20 multiple choice questions (50 marks). Indicate your answer to each multiple—choice question in PART B by completely
ﬁlling in the appropriate circle in the ANSWER BOX on this front page. (Use a dark pencil!) .
ANSWERBOX
FORPARTB
1 ® © ® ®
2.
NOTE: FOR MARKERS ONLY 3. g 8 8 g
4 Q9 © ® CE)
1. Before you start, check that 5. @ © @ ®
this test has 21 pages. 6. (A) © ® ®
. 7 @ © (D) (E)
2. No aidsallowed. 8. @ © ® ®
No CALCULATORS! 9. ® @ ® ®
3. DO NOT TEAR OUT A5 /7 10' ® © ® ®
ANY PAGES' 11. @ © ® ®
 A6 /8 12. @ © ® ®
4. COMPUTER CARDS A7 )8 13 (19 © ® (E
AND ANSWER BOOKS —~]—— 14~ @ © @ ®
WILLMBE USED. PARTB /50 15 g g g g
16. e
No SCRAP PAPER. TOTAL /100 17. @ © ® ®
18. (A) © CD) ®
19. ® @ © ®
20. @ © (D) CED Page 1 of 21 PART A [50 marks] Answer all questions in PART A in spaces provided. Show all your work for PART
Without proper justiﬁcation may receive little or no credit. Use the back of each p for each question in PART A are indicated by [ /.
DO NOT TEAR OUT ANY PAGES. 1. Find /:ce‘r’$ dsc. 2. Find / cot38 a: csc4 m d:c. Page 2 of 21 Code: 2782 A. Any answer in PART A
age for rough work. Marks pm'arwrlwe‘aduirw‘ WarsawA, o: er» » svcyx‘i‘m‘EWLMgwmm 1mm 3W. " , wwwmuwwn . 1
3. Flnd /m dac. Page 3 of 21 Code: 2782 Code: 2782 4. Let R be the region bounded by the curves y = 256 + 1 and y = 3:2 + 1. Find the volume of the solid
generated by revolving R about the y—axis. Page 4 of 21 Code: 2782 d
5. Find the solution of the differential equation i = (9:102 — 41‘)(1 +312) that satisﬁes the condition y(1) 2 0. Express the solution explicitly as a function of ac. Page 5 of 21 6. Find the interval of convergence of the series 2 ( m ) n=1 ﬁ 377. Page 6 of 21 Code: 2782 . Remember to fully justify your answer. Code: 2782 7. NOTE: This is a hard question and will be marked very strictly. Very little or no credit
will be given unless your solution is completely correct. Find /Cos__($______ +azsin)x dar.
(53+ cosac) Page 7 of 21 INDICATE YOUR ANSWERS ON THE FRONT PAGE
Penalty for not doing so is MINUS 4 marks! Code: 2782 PART B [50 marks] PLEASE READ CAREFULLY: Each of the following multiple—choice questions has exactly one correct
answer. Indicate your answer to each question by completely ﬁlling in the appropriate circle in
the ANSWER BOX on the front page. Use a dark pencil. MARKING SCHEME: 21/2 marks for a correct answer,
0 for no answer, a wrong answer or giving more than one answer. You are not required to justify your answers in PART B.
NOTE: If there is any discrepancy between the circles your darken on these inside pages and those you
darken on the front page, the circles you darken on the front page will be regarded as your final answers.
Note that only the circles you darken will count. For PART B, your computations and answers (other than
the Circles your darken) will NOT count.
WARNING: If you darken the circles on these inside pages but do not darken the circles
on the front page, you will still get credit for your correct answers, but there will be a
PENALTY of minus 4 marks.
YOU MUST NOT TEAR OUT ANY PAGES OF THIS EXAM. 3:3 + 42:2 + r — 6 1. lim—————————=
:c—rl 553*7r+6 ® 5
l
3
© 3
® 0
® undeﬁned m3+$2—2093+12 2. The graph of y 2 has a slant (i.e., oblique) asymptote, which is the line a32+6rc
@y=$
.yzr—S
©y=x+1
®y=$+2
1
®y=$+6 Page 8 of 21 «mix... “A m mvivywuawwa :x‘aripl—mn . INDICATE YOUR ANSWERS ON THE FRONT PAGE
Penalty for not doing so is MINUS 4 marks! Code: 2782 3. The product of two positive numbers is 144. What is the smallest possible value of their sum?
@ 26
18
© 24
® 20
® 22 4. The length of a rectangle is decreasing at 2cm / sec, while the width is increasing at 3 cm / sec. At the
moment when the length is 10cm and the width is 20 cm, the ares of the rectangle will be @ increasing at 10 sq cm / sec
increasing at 8 sq cm / sec
© decreasing at 10 sq cm / sec
® decreasing at 8 sq cm / sec ® neither increasing nor decreasing Page 9 of 21 INDICATE YOUR ANSWERS ON THE FRONT PAGE
Penalty for not doing so is MINUS 4 marks! Code: 2782 ~WWv~mmW§k¢$ﬁﬂﬁ wars; 5. The function f(x) 2 :53 + 3:52 — 45x + 8 has a local maximum at a: :
® —5
2
© —4
® 0
® 4 it 6. If Mac) 2/ (t2 + t + 1N5]: — 6 dt, then h’(2) = 3/2 @ 14
0 ,
© undeﬁned ; 23
© 3
® 10 Page 10 of 21 INDICATE YOUR ANSWERS ON THE FRONT PAGE
Penalty for not doing so is MINUS 4 marks! Code: 2782 8. Find the area of the region enclosed between the curves y = x2 and y = 2:): + 3. 2
@E
@g Page 11 of 21 INDICATE YOUR ANSWERS ON THE FRONT PAGE
Penalty for not doing so is MINUS 4 marks! Code: 2782 2333—1 fCC<17
9. Let f($):{3$2~233 if$> 1. Then /: f(a:) dasz
@ 4
5 00 :c
10/
*009+CE6 Page 12 of 21 INDICATE YOUR ANSWERS ON THE FRONT PAGE Penalty for not doing so is MINUS 4 marks! Code: 2782
11. Consider the predator—prey system (ii—1:. 2 6R — 2RW, ﬂag: 2 —4W + SRW.
When the system is in equilibrium with W 75 0, R 75 0, then RW =
10
® 3—
5
12
© 57
® 1
® 3 12. For any integer n 2 1, /(1n 3:)” dx =
@ $(ln :10)” — 71/(lncc)“_1 dz
(111m)" — n /(1n ac)“_1 da:
@ (In :13)"‘1 _ n / (In x)”'1 dag
@ 3:2(ln :6)" — 71/011507"1 d2:
@ :c(1n 3:)”_1 — n / (my1 dzc Page 13 of 21 INDICATE YOUR ANSWERS ON THE FRONT PAGE
Penalty for not doing so is MINUS 4 marks! Code: 2782 13. Consider the following three series: 2% (1)”)
arctan (f+1) 11:4; arctan( (\/— + 5) Decide which of the series converge
® 11 only V
11 and 111 only
© 111 only @ none
® I and 11 only 3?.
V
g:
3p
3
is €
5‘
a
”9
¥
» Page 14 of 21 INDICATE YOUR ANSWERS ON THE FRONT PAGE
Penalty for not doing so is MINUS 4 marks! Code: 2782 14. Consider the following two series: 00 1
I. — ”—
75:25 1) mm + 1)
0° n 3n+1 !
11. 52(4) —( (7203) Which of the following statements is correct?
@ I and II both converge conditionally.
I converges conditionally and II diverges. © 1 converges conditionally and II converges absolutely.
® I and II both diverge.
® I converges absolutely and II diverges. Page 15 of 21 INDICATE YOUR ANSWERS ON THE FRONT PAGE
Penalty for not doing so is MINUS 4 marks! Code: 2782 15. Find the coefﬁcient of x4 of the Maclaurin series for f(:c) = 6226 sin 3x.
Q9 8 ,
21 T
_~2_
5
© 5 13
@3 @—5 .f wnquwwﬁww—gm11.11%...» w Page 16 of 21 INDICATE YOUR ANSWERS ON THE FRONT PAGE
Penalty for not doing so is MINUS 4 marks! Code: 2782 . sin a: — arctan a:
16. 11m ————————
$~>O 332111<1 + 33) Page 17 of 21 INDICATE YOUR ANSWERS ON THE FRONT PAGE
Penalty for not doing so is MINUS 4 marks! Code: 2782 < "r}MWIi“i“&\ssﬁX§ésWW¢: hum 67 7 .
17. Find the arc length of the curve 12mg — 4y4 2 3 from the point (15,1) to the pomt (511, 2). 59
® 5; T
29 ;
1—2
37
14
37
@ 1—6
43
18 Page 18 of 21 INDICATE YOUR ANSWERS ON THE FRONT PAGE
Penalty for not doing so is MINUS 4 marks! Code: 2782 7r/2 2
18. / emsmamoswdccz
o (e7r + 1) @@ (27r G)

—
v A
(D
2:

;_a
v v HberIOOIHOOIr—lyblr—I
A
a:
:1
i
1.4 @©@ 00 l
/‘\
m
:1
+
b—l
v Page 19 of 21 INDICATE YOUR ANSWERS ON THE FRONT PAGE
Penalty for not doing so is MINUS 4 marks! Page 20 of 21 Code: 2782 3
2
§
1‘
z
I , pathwwﬁ‘ INDICATE YOUR ANSWERS ON THE FRONT PAGE
Penalty for not doing so is MINUS 4 marks! Code: 2782 20. 11m n—+oo <1+n+2+n+3+n+4+n+ +n+n>
12+n2 221—712 32+n2 42+n2 n2+n2
7T 1
— —1 2
@ 4+2n
71' 1121112 7? 1
—— —12
© 2+2“
@g+1n2 7r
~12
®4+n Page 21 of 21 ...
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