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Unformatted text preview: UNIVERSITY OF TORONTO MISSISSAUGA August 2008 Examination
MATH 135Y5
John Alexander
Duration  3 hours
No aids aliowed (including calculators) 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. it 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. Student Name: ,
Family First Student No:
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.
 Do not tear any pages out from this exam. . .. . . . . .. . _
0 This exam has 12 different pages including this page. Qn ll Value Score
0 Budget your time. ' ' I ' ' 2 1 0
Signature . . Continued on page 2... Page 2 of 12
SOME USEFUL FORMULAS. (DO NOT tear out this page!) n+1
1) jsinxdxzcosx+c 12) Ix“ dx=X +c,n¢l
n+1
2) [cosx dx= sinx+c 13) IX'1 dx=fl dx= 1nx[ +c
x
2 ax
3) [sec x dx:tanx+c 14) fax dx=w+cna
4) jcsczxdx=cotx+c 15)] 1 dx=arcsinx+c
1);2
5) J'secx tanx dx= secx +0 16) j‘ 1 2 (ix: arctanx+ C
1+);
1
6) j'cscx cotx dx=—cscx+c 17) I— dxzarcsecx+c
X‘dxz —1
7) J'cotx dx=1nlsin x] + c 18) sin(x + y) = sinx cosy + cosx siny.
8) Itanx (ix =lnlsecx + c 19) cos(x + y) = cosx cosy — sinx siny.
9) jseex dx=1nisecx + tan X] + c 20) sin2x = 28inx cosx
10)joscx dx=lnlcscx—cotx +0. 21) cos2x = coszx—sinzx
ll) Indv=uv— jvdu =2cos2X~—1=1—28in2X
Seguences: If lim [an] =0, then lim 311:0.
11—)00 11—)00 The sequence {In} is convergent if —1 < r S l and diverges for all other values of r, and lim In 3 0ifir<l
n—>oo 1 if 1‘ = i The Integgal Test: Suppose f is a continuous, positive, decreasing function on [1, co) and let 2111 = f(n). Then the
00 00
series 2 an is convergent if and only if the improper integral I f (X)dx is convergent.
n=1 1 . . °° l .
9—Series: The serles 2 —p IS convergent only for p > 1.
n=1 1’1
. . . °° °° . . . . . an
The Limit comparison Test: Suppose 2 an and 2 1311 are serles With posmve termslf 11m —— = L > 0,
n=1 11:1 new bn then either both series converge or both them diverge. CO
The Alternating Series Test: If all an 2 0, an” S an and Iim an 2 0 then 2 (—1)n an converges. n—>°° n=1
The Ratio test:
. a . °° .
(a) If lnn “1 = L < 1, then the series 2 an 18 absolutely convergent (and therefore convergent).
n—)oo an n=1
 an+l __  an+1 .~ 00  
(b) If hm — L > 1 or hm = 00, then the senes 2 an is dlvergent.
n—)oo an new an n=1 Continue to page 3 for question #1... Page3 of 12 THE FOLLOWING NIULTIPLE CHOICE QUESTONS #1, #2 #3 AND #4 CIRCLE THE LETTER
BESIDE THE CORRECT ANSWER. Question 1: ( 2 marks each, totai 10 marks 1.1 If iogﬁ (9) =x, thenx: 1.2 Find the value of the limit: Iim 51? 5x .
H0 smx (A) 0.
(B) s.
(C) 5.
(D) 1/5.
(E) 1 —x
1.3 Find the value of the limit: lim (e cosx). X—)00 (A) 0.
(B) e.
(C) — 00.
(D) 1.
(E) 00 1
1.4 Find the value of the limit: lim+  111x) .
X—>0 (A) This limit does not exist.
(B) —l. (C)  00 (D) 00. (B) 0 3
1.5 Find the value of the limit: lim(1+ 3;)" . X—)CD (A) I.
(B) e.
(C) e3.
(D) 00
(E) 0 Continued on page 4. . Page 4 of 12
Question 2: (2 marks each, total 10 marks) l
— 'f 2.1
2.1 Find the constant(s) c for which the function f(x) = X 1 X is continuous at x = 1.
2x + c if x <1 '
(A) “2
(B) —1
(C) 0
(D) 1/2
(E) —1, 2
iX3l .
2.2 meg): x3 I” i 3 then f(x) is
1 if x = 3 (A) Undeﬁned at X = 3. (B) Continuous from the left at x = 3 (C) Continuous at x = 3 (from both sides).
(D) Continuous from the right at x = 3.
(E) Differentiable at X 2 3. 2.3 If f(x) =1a(e3x + 2), then 9(0). (A) 3/2
(B) 0
(C) 1
(D) 1/3
(E) 213 2.4 What is the equation of the tangent line to {(X) = 5 + 3x ﬂ 2x2 at the point (1, 6). (A) y = 5x+l
(B)y=7—x
(C) y =~x+6
(D) y =3—4X
(EweX 2.5 A spherical balloon is being inﬂated in such a fashion that the radius is increasing at a rate of l cmfs. In
cm3/s, how fast is the volume increasing 3 seconds after inﬂation starts? (A) 361:
(B) 36 (C) 127:
(D) 187:
(E) 2411 Continued on page 5 . . Page 5 of 12
Question 3: ( 2 marks each, total 10 marks) 3.] Find the absolute minimum of the function f(x) = x3 — 3x2 + l on the interval ——1/2 S x s 4.
(A) 17
(B) —3
(C) 1
(D) 1/8
(E) 1/8 1
3.2 Consider the function f(x) = x2 on the interval [0, E ]. According to Mean Value Theorem, there must be a  1
number 0 in (0, 5) such that f ’(c) is equal to a particular value (1. What is d? (A) 2/3
(B) 1
(C) 1/2
(D) 2
(E) 3/2 2 X
3.3IfF(x)= 1 41+ 8t3dt,ﬁnd thevalueofF'(1). o (A) 2
(B) 1
(C) 4
(D) 6
(E) 3 i ‘ 3 3
3.4 If Jf(x)dx = 7 and if(x)dx = 6, ﬁnd the value of i f(x)dx. 0 O 1 (A) 1 (B) "1 (C) 0 (D) "3 (E) Cannot be determined 0
3.5 Find the value of the integral 1 IX + II dx. —2 (A) 1 (B) 1/2 (C) 1/4 (D) 3/4 (E) 0 Continued on page 6.... Pageﬁ of 12
Question 4: ( 2 marks each, total 10 marks) b
4.1 If I 3x2dx = 37, ﬁnd the value ofb.
3 (A) 6
(B) 4
(C) 5
(D) 7
(E) 8 ﬂ
4 4.2 Find the value of the integral I cos2x dx.
0 (A) 3—4 (B) 76 00 Xn I}
4.3 Find the intewal of convergence of 2 (»1) n 511 .
n=1 (A) [5353
(B) [5, 5)
(C) (—5, 5)
(D) 05,51
(E) (433,03) 4.4 Determine the limit of the sequence 3n = [ 1n(n+l) w111(11) ]
1
(A) e
(B) 2
(C) 0
(D) 1
(E) Divergent
m 2111 4.5 Find the sum of the series 2 3n .
n=1 (A) 4/3
(B) 1
(C) 6
(D) 2
(E) 3 Continued on page '7. . .. Page? of 12 . x2 ' x(2—x2) n (2+x2) I
[10 marks] 5.Given;f(x) = Fiﬁ) :— andf (x) = WAnswer the followmg. Put your
"X ' answer to each part in the box provided:
SHOW YOUR WORK! a) Domain of f. i b) Is f odd, even, or neither? i c) Find the horizontal and vertical asynnptote(s) (if any) d) Find the interval(s where the function is increasing (if any) e) Find the intewaKs) Where the function is decreasing (if any) i) Find the intervaKs) where the function is concave up (if any) g) Find the interval(s) where the function is concave down (if any) i h) Find the local extrema (max./min.), (if any) 1) Find the point(s) of inﬂection (if any) 3') Sketch the graph of y = f(x) y Continued on page 8. PageS of 12 X2 6 a) [5 marks} If X sin(1tx) = I f(t)dt , where f is continuous function, ﬁnd f(4).
O 6. b ) .{5 marks] Tom measures the circumference C of a spherical ball at 40 cm and computes the hall’s
volume V. Estimate the maximum possible error in V if the error in C is at most 2 cm Continued on page 9. .. Page 10 of 12
8. Evaluate the following integrals: si11(ln x ) a) [3 marks] I ab: x b)[3 marks} Ixz 111de c) [4 marks} Isin(lnx)dx Continued on page 1 1... Page 11 of 12 9.21) [3 marks] Evaluate [tan2 x 8604 xdx X29 X 9.b) [4 marks] Evaluate I dx x2 9.0) [3 marks] Evaluate I dx 1+ng6 Continued on page 12.... Page 12 of 12
10.a) [5 marks] Solve the differential equation: xzy' + 2):)! 2 coszx. 10. b) [5 marks] Find the equation of the line through the point (3, 5) that cuts off the least area from the ﬁrst
quadrant. The end. ...
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 Spring '08
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