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Unformatted text preview: UNIVERSITY OF TORONTO MISSESSAUGA
APRIL 2009 FINAL EXAMINATION (Version A)
MAT135Y5Y
Caiculus for Sciences
John Alexander
Duration  3 hours
No aids allowed including calculator. You may be charged with an academic offence for possessing the following items
during the writing of an exam unless otherwise specified: any unauthorized aids,
including but not limited to calculators, cell phones, pagers, wristwatch calculators,
personal digital assistants (PDAs), iPods, MP3 players, or any other device. if any of
these items are in your possession in the area of your desk, please turn them off and
put them with your belongings at the front of the room before the examination begins.
A penalty may be imposed if any of these items are kept with you during the writing
of your exam, Please note, students are NOT allowed to petition to REWRITE a final examination. NAME (PRINT): Last/Surname First lGiven Name
STUDENT N0: SIGNATURE:
INSTRUCTIONS
 Answer all 10 questions. To get part marks show your work in the space provided. You may
use the back of the sheets for your rough work.
 Budget your time. __. m—
o This exam has 12 different pages including this page. w—l—vf Continued on page 2... Page 3 of 12
FOR THE MULTIPLE CHOICE QUESTONS 1, 2, 3, 4 AND 5 CIRCLE THE LETTER BESIDE
THE CORRECT ANSWER. Question 1: (2 marks each, total 10 marks) 1.1 Find the smailest value in the domain of the function f(x) = J23: — 5 . (A) 2/5
(B) 5/2
(C) 2
(D) 5
(E) —2 1.2 Let f(x) = 2x and (f o g)(x) = x2. Find the value of g(4). (A) 6
(B) 2
(C) 16
(D) 8
(E) 0 1 i 1.3 Find the value of the limit, ling Y ‘75::
I“; x  (A) 1112
(B) 1/6
(C) —1/6
(D) m
(E) 1/9 2
1.4 Find the vaiue ofthe limit, iimw 11—») (A) 2
(B) 1
(C) 2
(D) 4
(E) 4 (x 1)2 xz—l fail to have a iimit? 1.5 For what value of x does the ﬁmction (A) 2
(B) —1f2
(C) 1 Continued on page 4. . . .. Page 4 of 12
Question 2: ( 2 marks each, total 10 marks) 02—):2 ifx<2 . 2.1 Find the constant(s) c for which the function f(x) = IS continuous on 91‘ .
2(c+x) if x Z 2 (A) 4,—2
(B) 2,4
(C) 2
(D) 4
(E) —2,4 2.2 Suppose that F(x) : f(g(x)) and g(3) = 5, f'(3) ; 1, f’(5) = 4, F '(3) =12. Find the value ofg '(3). (A) 2
(B) 4
(C) 3
(D) 9
(E) 12 2.3 Find the equation of the line tangent to f(x) = x2 — 4x at the point (3,—3). (A) 2X—y=9
(B) xm2y=9
(C) y~2xn9
(D) 2y—x=9
(B) x~—y=9 2.4 Let y = {(x), ifxf' + xy= 6 and f(3) = 1, ﬁnd f'(3). (A) 0
(B) 1
(C) 2
(D) 1/3
(E) 4/6 2.5 Find the interval on which f(x) = x — 25inx, 0 s x s 271; is increasing. (A) to, E}
(13) ($5?)
(C) tigmni
(D) (0,122] (B) [gig—i] Continued on page 5. Page 5 of 12
Question 3: (2 marks each, total 10_ marks) 3.1 If f(x) 2 eeszx, ﬁnd the value of W0) (A) 1
(B) 1
(C) 128 (D) 256
(E) —256 3.2 f(x) = x2 on the interval [0, 2]. Let the interval be divided into two equal subintervals. Find the value of the Riemann sum 2 f (xi )Axi If each x: 18 the midpomt of its submterval.
i=1 (A) A
US!
v E (E) A O v
#Immlmhlﬂmlﬂmlm 3.3 The radius of a circle is given 10 cm, with a possible error of measurement equal to .1 cm. Use
differentials to estimate the maximum error in the area, in cmz. (A) 1071:
(B) 21:
(C) 3n
(D) TE
(E) 81: b
3.4 If I3x2dx = 37, ﬁnd the value ofb.
3 (A) 4
(B) 5
(C) 6
(D) 7
(E) 3 3.5 Find the area of the region bounded by the curves y = X2 and f = x. (A) 1/6
(B) 1/4
(C) 1/3
(D) 1/15
(E) 1/9 Continued on page 6.... Page 6 of 12
Question 4: (2 marks each, total 10 marks) 4.1 Find the vaiue of the limit lim \fo + 2x —— x. x—)—03 (A) 00 (B) 00 (C) 2 (D) 9 (E) J5
4.2 Find the value of the limit lim(1+—2] . x—)ao x (A) 3 (B) 1 (C) e2 (D) e3 (E) None of the above. . . . . 6’52"
4.3 Find the value of the 11m1t, ling . 1—) x (A) 0 (B) 1 (C) In? (D) 3 (E) 1n3. 4.4 Determine on so that the function f(X) = x2 + 2': has an inﬂection point at x = i.
x (A) 3
(B) 2
(C) 1
(D) 0
(E) 1 . . . . l .
4.5 Determlne the largest interval on which the function f(x) = x + — IS concaVe upward. x (A) (—1, 0_)
(B) (00, 1)
(C) (00, 0)
(D) (0,00)
(E) (1,00) Continued on page 7.  Page 7 of 12 Question 5: ( 2 marks each, total 10 marks)
00 211 5.1 Find the sum of the series 2: 11:1 3 11—1 (A) 4/3
(B) 5/3
(C) 4
(D) 5
(E) 6 x11 03
5.2 Find the interval of convergence of 2
n=0 3n + 1 (A) [3, 3]
(B) (1, 1)
(C) (3, 3)
(D) [‘1’ 1]
(E) [~1,1) 00
5.3 The series 2 not converges if and only if
n=1 (A) 0K1
(B) 1<oc<1
(C) a 2 1
(D) 0c >1
(E) oc<~1 . . °° (x — 2)11
5.4 Fmd the radlus of convergence of 2 11:1 1‘13n (A) 0
(B) 00
(C) 2
(D) 3
(E) 4 5.5 Find the coefﬁcient of x5 in the Maciaurin series for f(x) 2 Icos(x2)dx. (A) 1110
(B) 1/15
(C) 1/5
(D) 2/5
(E) 2/5 Continued on page 8.... Page 8 of 12 6 a). [5 marks] Find the dimensions of a rectangle of largest area that has its base on the xaxis and its other two vertiees above the x—axis and lying on the parabola y = 8 — x2. 6. b). [5 marks} Find the volume of the solid obtained by rotating the region bounded by the curves y = x
and y = x2 about the line y = 2. Continued on page 9.... Page 9 of 12
7 Evaluate the following integrals: _3
5?de
x 7 a) [3 marks] I 7 b) [3 marks] I l
7 c) [4 marks] Iarctan xdx
0 Continued on page 10... Page 10 of 12
x2 —2x—1 8 a) [ 5 marks] Evaluate: x
x" x + 1
8 b) [ 5 marks] Evaluate j lnxdx if possible. 0 Continued on page 11 Page 11 of 12 9 a) E 5 marks] Solve the separable differential equation gig: I — t + x  xt.
9.1)) [5 marks} Find the solution of the initialvalue problem x2 :—y + xy = 1, x > 0 and y(1) = 2.
x Continued on page 12. . .. Page 12 of 12
10.a) [5 marks] Find the Iength ofthe curve y r» —1 dt, 1 S x 316.
l 2 10 b) [5 marks] Find the sum of the series 2 {11(1—m—L).
11 11:2 The end. ...
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