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Unformatted text preview: FACULTY OF ARTS AND SCIENCE Code‘ 0622 University of Toronto FINAL EXAMINATIONS, APRIL / MAY 2008
MAT 135Y1Y Calculus I Duration — 3 hours NAME
{Please PRINT full name,
' and UNDERLINE surname): STUDENT NO: SIGNATURE OF STUDENT
(in INK or 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 . eoreyousar,cec a
this Exam has 22 a es. ® © ® ®
 No CALCULATORS! 3 ® @ ® ®
3. DO NOT TEAR OUT 4. @
THIS PAGE OR ANY / 5 3
OTHER PAGE. A4 7 '
/ 6®e©®®
AND ANSWER BOOKS 8 ® © ® ®
WILLNOTBEUSED'
No SCRAP PAPER! n Page 1 of 22 Code: 0622 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 [erZdn [5/ 2. Find / cosh7O :c sinh3 a: da: . Hint: You may use the identity cosh2 a: — sinh2 ac = 1 . [5/ Page 2 of 22 Code: 0622 _ . dz:
3. F1nd [7/ Page 3 of 22 Code: 0622 4. Let R be the region bounded by the graphs of y = :1: and y = :02 — 2x. Find the volume
of the solid generated by revolving R about the y—axis. [7/ Page 4 of 22 Code: 0622 5. Find the orthogonal trajectories of the family of curves y2 = km3 . [7/ Page 5 of 22 Code: 0622 6. [8] Populations of birds and insects are modeled by the equations d
l = 0.411: — 0.0022631 gig dt dt —0.2y + 0.000008my (a) Which of the variables, x or y, represents the bird population and which represents the
insect population? Explain. (You can assume that the birds eat the insects.) (b) Find the equilibrium solutions and explain their significance. (0) Find an expression for dy/dw. Page 6 of 22 , Code: 0622 6. (Cont.) ‘
(d) The direction ﬁeld for the differential equation in part (C) is shown. Use it to sketch the
phase trajectory corresponding to initial populations of a: = 40,000 and y = 100. Then use
the phase trajectory to describe how both populations change. Y
400 I//’—~\\\\\\
l, 1//’—~\\\\\\
l///—~\\\\\\
. lI//—~\\\\\\
30011//—\\\\\'\\
lI//—\\\\\\\
ll//\\\\\\\
a';II/.\\\\\\\
200 l\\/IIIII'I
l\\\:—’//////
\\'\\—,’/////
\ \__.__.—’//’/
100 \:~ _ _ _ a _,,,,
\\ N — — — — — — — —.——
\ _ _ _ _ _ _ _ _ _ __ 40000 (e) Use part (d) to make rough sketches of the bird and insect popoluations as functions of time,
using the given axes. How are these graphs related to each other? Page 7 of 22 00
7. Find the interval of convergence of the series A 2 n=3
your answer. Page 8 of 22 (a? + 2)” n1n(n — 1). Code: 0622 Remember to fully justify [8/ Code: 0622 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.
71 ' i ' ' exists. If it does, ﬁnd its value. If the limit does not exist, explain Decide Whether lim n—voo n Why. Page 9 of 22 PART B [45 marks] Code: 0622
18 multiple choice questions PLEASE READ CAREFULLY: Each of the following multiplechoice 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 B. 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. _ m — .—
1. lim 5—2—1 = :z:—>0 3:
® ~—oo
+00
© 0 1
® '2‘ 1 ® _.._ 2. Let ﬂaw{ng3D ifac23 If f is continuous everywhere, what must be the value of the constant k ? 1
@n
«3"
1
©§
@3
@O Page 10 of 22 Code: 0622 INDICATE YOUR ANSWERS ON THE FRONT PAGE.
Penalty for not doing so is MINUS 4 marks. 3. The graph of y = meg” is concave up on the interval 69 (—5, 3)
(100)
© (3, 5)
® (4, 4)
® (—00, —2) 4. A light is at the top of a pole whose height is 15 feet. A boy 4 feet tall is walking away from
the pole at 3 feet per» second. At what rate will the length of the boy’s shadow be increasing when the boy is 20 feet from the pole? 12 '
® H ft/sec. 10 —9 ft/sec. : ft/sec. 13
E ft/sec. @©@ 7
—8 ft/sec. Page 11 of 22 Code: 0622 INDICATE YOUR ANSWERS ON THE FRONT PAGE. '
Penalty for not doing ‘so is MINUS 4 marks. ' 4x2+4x+5 2:2: — 1
other asymptote is the line 5. The graph of y = has one vertical asymptote and one other asymptote. This @ y=2az—2
y=2x+3
© y=2m @ y=2az—4
® y=2x+2 Page 12 of 22 Code: 0622 INDICATE YOUR ANSWERS ON THE FRONT PAGE.
Penalty for not doing so is MINUS 4 marks. 7. The product of two positive numbers is 16. What is the smallest possible value of their sum? ® 10
9
© 6
® 8
® 7 8. Find the area of the region bounded by the curves y = at + 2 and y : 3:2 . @2
©;
cw; Page 13 of 22 Code: 0622 INDICATE YOUR ANSWERS ON THE FRONT PAGE.
Penalty for not doing so is MINUS 4 marks. dy m+1 9. The function y = f(x) satisﬁes the differential equation E 2 6y . If f (O) = 0, then
f (2) =
® x/ln—2
2ln5
© 1n2
® 21n2
® 1n5 2
10. Find the arc length of the curve y = gag/2, 0 S a: S 3. 16
’3'
14
_3"
10
3
13
3
17
? ® @©@ Page 14 of 22 Code: 0622 INDICATE YOUR ANSWERS ON THE FRONT PAGE.
Penalty for not doing so is MINUS 4 marks. 11. Consider the following three series: I f: (5—4 + cos”(e4))
V 11:1 00 ,
arctan(n + 1)
n 2 ————
n=1 ﬂ Decide which of the series converge (or converges). ® II only I and III only
© I only ® I, II and III
® none Page 15 of 22 12. INDICATE YOUR ANSWERS ON THE FRONT PAGE. Penalty for not doing so is MINUS 4 marks. Consider the following series: °° n!
I (n + 1);; I oo 2 '
n n +3n+1
H Z(_1) (n+2)3
n=1 Which one of the following statements is correct?
I and II both diverge. I and II both converge conditionally. I diverges and II converges absolutely. I diverges and II converges conditionally. @©@@® I converges conditionally and II converges absolutely. Page 16 of 22 Code: 0622 Code: 0622 INDICATE YOUR ANSWERS ON THE FRONT PAGE.
Penalty for not doing so is MINUS 4 marks. I3. Find the coefﬁcient of x3 in the Maclaurin series for f = tan (Z + 4 ©§ Page 17 of 22 Code: 0622 INDICATE YOUR ANSWERS ON THE FRONT PAGE.
Penalty for not doing so is MINUS 4 marks. 3 2
52: +30
. ———————— d =
14 /0:c4+13x2+36 3:
® 7r 3
Z + arctan 7T 2
Z + arctan 7r 3
© —2 — arctan arctané) — arctan ®
77 2
® 5 — arctan Pagé 18 of 22 Code: 0622 INDICATE YOUR ANSWERS ON THE FRONT PAGE.
Penalty for not doing so is MINUS 4 marks. 1:18 :1;
we
15. / _— dart:
' 1n3 V1+61 —4+2an2—81n3
—4+161n2—41n3
—4+161n2—'81n3
—4+ 181n2—41n3
—4'+201n2 —61n3 ®©@@@ Page 19 of 22 Code: 0622 INDICATE YOUR ANSWERS ON THE FRONT PAGE.
Penalty for not doing so is MINUS 4 marks. 00 00
16. Given that / 6’32 dx = 3g? , ﬁnd the value of / 9226—9”2 den.
0 0 @244
i;
©§
<9?
$3? Page 20 of 22 Code: 0622 INDICATE YOUR ANSWERS ON THE FRONT PAGE.
Penalty for not doing so is MINUS 4 marks. 3 ——
17. / ill—dw— _1a;—21+le ‘
® 2
2
© 3
® 2
® Page 21 of 22 Code: 0622 INDICATE YOUR ANSWERS ON THE FRONT PAGE.
Penalty for not doing so is MINUS 4 marks. 71’ I 4
18. Find the value of / arccos(tan 2:) dx. 7T 4
HINT: Investigate the relationship between arcsinQ and arccos 9 . 2 ®%
©O ©2
69% Page 22 of 22 ...
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