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Unformatted text preview: DEPARTMENT OF MATHEMATICS
University of Toronto MAT 135Y AUGUST 2008 EXAMINATION \S’kb?) Time allowed: 3 hours Please PRINT in INK or BALLPOINT PEN: NAME OF STUDENT:
(Please PRINT full name
and UN DERLINE surname): STUDENT N 0.: SIGNATURE OF STUDENT
(in INK or BALLPOINT PEN): SECTION: DAY EVENING (Please circle) FOR MARKERS ONLY
QUESTION MARK NOTE: 1. Before you start, Check that this test
has 22 pages. There are NO blank pages.
2. This test has two parts: PART A [45 marks]: 18 multiple choice questions
PART B [55 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. DO NOT TEAR OUT ANY PAGES. Page 1 of 22 PART A [45 marks]
Please read carefully: PART A consists of 18 multiple—choice questions, in each of which you are to select exactly one
answer. Indicate your answer to each question by completely ﬁlling in the appropriate
circle with a dark pencil or pen. MARKING SCHEME: 2.5 marks for a correct answer, 0 for no answer or a wrong answer,
or multiple answers. 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. 1. What is the absolute maximum value of the function f (:17) : x3 —3:r+ 1 on the interval [—3, 3]? @3
.0 ©23
@19
@34 2. The area bounded by the curve y = \/1 — a: and the as—axis, between :5 = —2 and :c = 1,
revolves about the x—axis to create a solid of revolution. The volume of this solid is: 7f 995
.7r 57r
©3— 97r ® 2
® None of the above Page 2 of 22 3. Calculate the arc length of the curve 3/ = ln(cos x), for 0 S as S %.
® 1110/5) ® ln(2 + x/E’i) © ln(2 + ﬂ) ® —ln(2+ ﬁl—lnﬁ) ® —ln(2— V3) (1
4. A population develops according to logistic equation 3% = y(0.5—0.01y). Determine tliig3 y(t). ® The limit does not exist because it grows to +00 The limit does not exists because it is possible for y(t) to be 0 or to tend to inﬁnity @ 0.5
@ 100
® 50 Page 3 of 22 cos 4:1: —— cos 6a: 5. What is the value of lirn —————— ?
x—+O 11:2 @ It does not exist as the numerator changes sign frequently
® It does not exist as the denominator goes to O fastest © 0 ® 10 ® '1 6. Consider the two series given by 5n+1
I ‘93 n, h = 7 n z n
( ) 2140 W ere a1 2 a +1 4n+3a
and 2+
(II) 2:157“ Where b1 2 17 an : ﬂbn \/ﬁ Select the correct assertion about the convergence of these two series: Both series converge absolutely
(I) converges and (II) diverges
(I) diverges and (II) converges Both series diverge @©@@® (I) converges and (II) only conditionally converges Page 4 of 22 7. The Monotone Sequence Theorem states that: ® Any convergent sequence must be monotone (increasing or decreasing).
® Any monotone sequence converges or diverges. © Any monotone sequence is bounded. ® Any monotone sequence converges if it is bounded. ® Any convergent sequence Which is bounded must be monotone. 7T 8 If “37) = 117 + sinx, then What is the value of (f‘1)’(2 ® ®
@ 1 —7r @7 ® It is not deﬁned +1)? 71' E
1
2 Page 5 of 22 d
9. Given that g = x3y and that y(0) = —1, it follows that y(2) is: 2 10. If f(a:) = [:5 g(t)dt and g(:z:) = f: e_‘2dt then f”(:r) is: @ 6”
® fe—t2dt
1
I —t2 —z:2
© 2 e dt+2xe
1
(D) ~2ze‘3‘2
22
® 2 e—t2dt+4$2€—I Page 6 of 22 @@@@ 12 @ @@@@ we 8 H “W 00 . What is the value of the series 2
0 n 271
(—1) 7r ,, n_ 621:9”)! ' Page 7 of 22 13. @©@@@ 14 @©@@® 0 4(ﬁ— 1) E MRItug 7r
—1 "—1 Page 8 of 22 7T 7r
. Find the average value of the function f (at) = cot2 a: on the interval [1, 5]. 15. Find the ﬁrst three terms of the power series representation of
/ ln(1 — t) dt, where 0 < a: < 1.
o Hint: Integrate the appropriate power series. :52 3:3 ® $+§“+E‘ x+as2+333 @©@@

+

+
l l
16. Calculate the surface area generated by revolving the curve y 2 3T? about the y—axis, from
at = 0 to .’L‘ : 2. ® 3371(5x/5—1) ® gwsn
© [Riff
© 3\/§7r
® %’—’(5¢5—1) Page 9 of 22 2
3)). 17. Evaluate cos(2 sin_1( @0 e © @
‘°'“‘°°"*53w 18. Find n such that the sum 5 of the alternating series can be replaced by 3" with an error of
at most 0.001; where pk; Le;
2\/§ 3V3 xxx/74‘
@ 10
® 100
© 5
® 1000
® 50 Page 10 of 22 [2] [1/ PART B [55 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 will 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 [ ]. 1.
1/ 2 2 _
a) Evaluate liIn w
x—wo x b) State the deﬁnition of the derivative. Page 11 of 22 [2/ [2/ c) If f(a:) = 0052(ema), ﬁnd f’(m). d) Find —((ln 505“”). Page 12 of 22 [2/ 2. Evaluate the following indeﬁnite integrals: a) / x2 sin(a:3) d9: b) / sin3 9: cos4 a: dac Page 13 of 22 [3/ [3/ c) [41.6fm
9—2551:2 x2+x—1
d /———————d
) x(x2+1) x Page 14 0f 22 [3] a) Find the area between the curve 3; = 3:2 —~ 350 + 2 and the :c—axis, from a: = —1 to a: = 2. Page 15 of 22 [4/ b) Calculate the volume of the solid of revolution obtained when the area bounded by the
graphs of y = 8 — x2 and y = x2 is revolved about the line at = —4. Page 16 of 22 [3/ [1/ a) Consider the differential equation d
(x2 +1)ﬁ + 333(y —— 1): 0. i) Find the general solution for the above differential equation. ii) Solve the initial value problem with y(0) = 2. Page 17 of 22 [4/ b) A population grows according to the logistic growth model d_P_
dt— KPOekt P . 
[9130— E) w1th solutlon PU) — W' If K = 200, P0 = 100 and P(10) = 2—30, then what is P(20)?
Hint: You do not need to ﬁnd k7 just ﬁnd 610’“. Page 18 of 22 [3/ [3/ 5. Decide whether or not the following series converge or diverge. Justify your answers.
a)
f: x/n + 2 2n2+n+1 n=1 Page 19 0f 22 [3/ [3/ Page 20 of 22 [2/ [2/ 1
3 — x
Hint: you do not want to use differentiation. a) Expand the function f (x) = 11:2 — 22: + 5 + as a Taylor series about as = 2. b) Determine the radius of convergence and the interval of convergence for this Taylor expan
sion. 0) What is f (135)(2)? (That is7 what is the 135th derivative of f 7 evaluated at a: = 2)? Page 21 of 22 [2/ a) State the comparison theorem for improper integrals. °° 2x+3 1' ~51: dac converges. Page 22 of 22 ...
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 Spring '08
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