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Unformatted text preview: FACULTY OF ARTS AND SCIENCE We: 0888 University of Toronto FINAL EXAMINATIONS, APRIL/MAY 2009
MAT 135Y1Y Calculus I Iﬂ‘j
Duration — 3 hours W
NAME ’ 91M ( Please PRINT full name,
_ and UNDERLINE surname): STUDENT NO: SIGNATURE OF STUDENT
(in INK 01' BALLPOINT PEN): This Exam has 2 Parts: PART A: 8 questions (55 marks). PART B: 18 multiple choice questions (45 marks).
Indicate your answer to each multiplechoice question in PART B
by completely ﬁlling in the appropriate circle in the ANSWER BOX
on this front page. (Use a dark pencil!) FOR MARKERS
ONLY «  ANSWER BOX
FOR PART B NOTE: 1. Before you start, check that
this Exam has 20 pages. 2. No aids allowed.
NO CALCULATORS!
3. DO NOT TEAR OUT
THIS PAGE OR ANY
OTHER PAGE. M®l COMPUTER CARDS
AND ANSWER BOOKS
WILL NOT BE USED. NO SCRAP PAPER! 9°51???pr Page 1 of 20 Code: 0888 PART A [55 marks] Answer all questions in PART A in Spaces provided. Show all your work for PART A. Any
answer in PART A Without proper justiﬁcation may receive very little or no credit. Use the
back of each page for rough work. Marks for each question in PART A are indicated by [ ].
DO NOT TEAR OUT ANY PAGES. 1. Find jmcosxdm. [5/ 2. Find / tan82 a: Sec4 xdsc. Page 2 of 20 Code: 0888 1
3. F' d de.
1n ijVm2~4 [7/ Page 3 of 20 Code: 0888 4. Find the arc length of the curve 3/ = 1n(sec cc) , 0 S a: < 1’.
4 . [7/ Page 4 of 20 Code: 0888 d 5. Find the solution of the differential equation 8: = 2:239:32 + 1)e'3y that satisﬁes the condi— tion 31(1) =1n2 . [7] Page 5 of 20 [8] Page 6 of 20 COde: 0888 Code: 0888
nxn m . Remember to fully Justlfy your 00
7. Find the interval of convergence of the series 2 n=4
answer. [8] Page 7 of 20 e Code: 0888 8. NOTE: This is a hard problem and will be marked extremely strictly. Very little
or no credit will be given unless your solution is completely correct. Find [\lx— \/$225 dzc. Hint: Investigate V3; + 5 —— Va: — 5. Page 8 of 20 PART B [45 marks] Code: 0888
18 multiple choice questions 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: 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 13. NOTE: If there is any discrepancy between the circles you 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 you 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. 1. Find the value of lirn w
37—)0 a:
1
® ”:2:
undeﬁned
© 0
1
© 2;
1
® “‘3‘ 2. Let f(:v) = 3:4  2x3 — 3691:2 +533 4 , for all x. Then the graph of f has a point of inflection at ® cc=~2 and m;3 only.
so a 3 only. © :1: z: 2 and :1: = ~3 only.
® a: 2 2 only. ® w=2andm=3only Page 9 of 20 Code: 0888 INDICATE YOUR ANSWERS ON THE FRONT PAGE.
Penalty for not doing so is MINUS 4 marks. 3. The length of a rectangle is increasing at 5 ft/sec while its width is decreasing at 4 ft: / sec.
At what rate will the area of the rectangle be changing when the length of the rectangle is
500 ft and the Width is 300 ft? increasing at 400 sq ft/sec.
decreasing at 500 sq ft/ sec.
increasing at 450 sq ft / sec. increasing at 500 sq ft / sec. @©@@® decreasing at 400 sq ft / sec. 4:133 + 33:2 — 103: + 1 4 The graph 0f :9 = has two vertical aysmptotes and one slant (Le. oblique) asymptote. The slgﬁt+a§§mptote is the line
® 3/ = 4x + g:
y = 4x — 2 © y = 43: @ y = 4:3 + 3 ® 3/ 2 4:1: — 5 Page 10 of 20 Code: 0888 INDICATE YOUR ANSWERS ON THE FRONT PAGE.
Penalty for not doing so is MINUS 4 marks. 5. The sum of two positive numbers is 16. What is the smallest possible value of the sum of
their squares? @ 132
126
© 134
® 128
® 130 6. Find the area of the region enclosed between the curves y 2 1 + 2:1; and y = 1+ :1: + V593 69%
:
©i
@2
@2 Page 11 of 20 Code: 0888 INDICATE YOUR ANSWERS ON THE FRONT PAGE.
Penalty for not doing so is MINUS 4 marks. 7. Let R be the region enclosed between the curves y = a: and y =2 x2 . Find the volume of
the solid generated by revolving R about the a:  axis .
57r
@ 1‘5
71' .3 8. Find the average value of the function f (as) =2 3562 + 83:3 on [1, 3] . ® 190
80
© 93
® 65
® 127
. dR dW
9. ConSIder the predator—prey system ”cl? = 5R — 4RW, ~22;— = 3W + 2RW. When the
system is in equilibrium with W # 0 , R 73 0, then RW =
15
@ 5‘
5
5
7
© ‘2:
7
© 2
5
® 5 Page 12 of 20 Code: 0888 INDICATE YOUR ANSWERS ON THE FRONT PAGE.
Penalty for not doing so is MINUS 4 marks. 10. Suppose that a population grows according to a logistic model, i.e. the growth is modeled dP
by the differential equation ~22},— = kP(1 — E?) . Suppose that the carrying capacity K is 10, 000 . Suppose further that the initial population is 2, 000 and that it grows to 4, 000 after
one year. What will be the population after another year (Le. 2 years from the beginning)? @ 6,800
6, 400
@ 7,000
@ 6, 600
 ® 6, 500 Page 13 of 20 Code: 0888 INDICATE YOUR ANSWERS ON THE FRONT PAGE.
Penalty for not doing so is MINUS 4 marks. 11. Consider the following three series: CO 1
I. 7; (n + 2)\/ln(n + 2) Decide which of the Series converge (or converages).
@ II only ' II and III only
© I, and II and III
® III only (15) I and III only Page 14 of 20 Code: 0888 INDICATE YOUR ANSWERS ON THE FRONT PAGE.
Penalty for not doing so is MINUS 4 marks. 12. Consider the following series: nO n4+n2+25 °° n (1+n)“
;(_1)+1 n! Which one of the following statements is correct? I and II both converge conditionally. I converges absolutely and II converges conditionally.
I converges conditionally and II diverges. I converges conditionally and II converges absolutely. @©©®@ I and II both converge absolutely. Page 15 of 20 Code: 0888 INDICATE YOUR ANSWERS ON THE FRONT PAGE.
Penalty for not doing so is MINUS 4 marks. 13. Find the coefﬁcient of :34 of the Maclaurin series for f(x) = sin2 a: . 1
@723 Q
l
4:: @©@
1
on! 22.2. 14. Let an 2 (1 +
’n n
) . Then the sequence {can} converges to 1.
converges to 2 .
diverges.
converges to 1112. converges to e2 . @©@®® Page 16 of 20 Code: 0888 INDICATE YOUR ANSWERS ON THE FRONT PAGE.
Penalty for not doing so is MINUS 4 marks. my
15. If y is a differentiable function of m such that (1 + :63)y2 + 4 f V 53:2 + t2 dt = 112 , ﬁnd
2:1: the value of ill: at the point Where y = 2 (1.1.93 :3 = 2).
27 @"5‘5’ not determinable due to insufﬁcient information 18
’35
23
"Z4"
28
”3'9? ®©© Page 17 of 20 INDICATE YOUR ANSWERS ON THE FRONT PAGE. Penalty for not doing so is MINUS 4 marks. 16. Find the value of Adex
@ 1%
3;
© *3
© 32:
® '2: Page 18 of 20 Code: 0888 Code: 0888 INDICATE YOUR ANSWERS ON THE FRONT PAGE.
Penalty for not doing so is MINUS 4 marks. °° 1 + 8x — 2:02
1 , Th ‘ ' t 1 ————— d
7 e 1mproper 1n egra /1 8m4 + 21:3 + 43:2 + :1: ac 6 ® converges to In (E) . 5
converges to In (5) . 0 diverges. converges to In (—3) . @©@ 5
converges to 1n (1) . Page 19 of 20 we». Code: 0888
INDICATE YOUR ANSWERS ON THE FRONT PAGE.
Penalty for not doing so is MINUS 4 marks.
18. Let f(”)(a) denote the value of the nth derivative of f at a. If ﬁns) 2 23—+8:2 , then #95) (0) == @ (950294
(950295
© (950297
@ (950293
® (950296 Page 20 of 20 ...
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