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Unformatted text preview: UNIVERSITY OF TORONTO MISSISSAUGA
APRIL 2010 FINAL EXAMINATION (Version B)
MAT’i 35Y5Y
Calculus for Sciences
John Alexander
Duration  3 hours
No aids allowed including calculator. The University of Toronto Mississauga and you, as a student, share a commitment to
academic integrity. You are reminded that you may be charged with an academic
offence for possessing any unauthorized aids during the writing of an exam, including
but not limited to any electronic devices with storage, such as cell phones, pagers,
personal digital assistants (PDAs), iPods, and MP3 players. Unauthorized calculators
and notes are also not permitted. Do not have any of these items in your possession in
the area of your desk. Please turn the electronics off and put all unauthorized aids with
your belongings at the front of the room before the examination begins. if any of these
items are kept with you during the writing of your exam, you may be charged with an
academic offence. A typical penalty may cause you to fail the course. Please note, you CANNOT petition to REWRITE an examination once you have begun
writing. NAME (PRINT): Last/Surname First lGiven Name STUDENT N0: SIGNATURE: INSTRUCTIONS m'Vame' __ score .
0 Answer all 10 questions. To get part marks show your ‘n’ork in  . . . _ _ _  1 10
the space provided. You may use the back of the sheets for your 
 rough work.  Budget your time. 
° This exam has 12 different pages including this page. 
  Continued on page 2... 1) jsinx dx=— cosx+e 2) [cosx dx= sinx + c 3) [sec2 x dx=tanx+c 4) lose2 it dx=  cotx + c 5) [sec x tanx dx : secx + c 6) Icscx cotx dx=  cscx+ c 7) jcotx dx=]nIsinxl +c 8) jtanx dx21n[secxi +6 9) [secx dx=1nsecx+tanxl + c 10)jcscx dx=1nlcscx~cotx +c. 11) Iudv=uv~ fvdu SOME USEFUL FORMULAS. xn+1 n+1
I 12) IX” dx= +c,n¢l 13) [x'1 dx=;w dx= 111le +c
X dx = arcsinx + 0 dx = arctanx + c ax
14) fax dx=—+0
1na
15) I 1
1~x2
1
16)
I1+):2
17) 1—1“ dx= xszml EI‘CSCCX ”i‘ C 18) sin(x + y) = sinx cosy + cosx siny. 19) cos(x + y) m cosx cosy— sinx siny. = 2 cos2 Seguences: If 1in1 (an! =0, then lim an= 0. Il—>CO 11—)00 20) sin2x = 23inx cosx
21) cost = coszx __ sinzx x—1=1—23in2x The sequence {1'“} is convergent if 1 < r S 1 and diverges for all other values of r, and n—>oo lim r“ = 0 if]r<l
lif r =1 Page2 0f12 The Integral Test: Suppose f is a continuous, positive, decreasing i. ;, don on [1, co) and let an = f(n). Then 00
the series 2 an is convergent if and only if the improper integral n=1 °° 1 9—Series: The series 2 — is convergent only for p > 1. 11:1 11p CO CO
The Limit comparison Test: Suppose 2 an and 2 bn are series with positive tennslf Iim n=1 n=1 then either both series converge or both them diverge. 00
I f (X)dX is convergent.
1 an_ n—)oo bn (D
The Alternating Series Test: If all an 2. 0, an“ S. an and lim a1.l = 0 then Z (—1)11 an converges. The Ratio test: (a) If lim an“
11—900 an (b) If lirn an“ =L>lor lim
n——>oo an ““900 11—)00 nzl an+1
an n31 n=1 CO
= 00, then the series 2 a11 is divergent. L>0, DO
5 L < 1, then the series 2 an is absolutely convergent (and therefore convergent). Continue to page 3 for question #1... Page 3 of 12
1) Write your answer in the space provided (1 mark each, total 10 marks) (Express your answer in terms of a speciﬁc real number,  00, + 00 or “does not exist”.) Question Answer lim «IX 5 lim (1+ 3x)1/x x—>0+ Continue to page 4 Page 4 of 12
FOR THE MULTIPLE CHOICE QUESTONS 2, 3, 4 AND 5 CIRCLE THE LETTER BESIDE THE CORRECT ANSWER.
Question 1: ( 2 marks each, total 10 marks) 2.1 If f(x) 2: «1x2 m 4 and g(x) = 2x. Find the domain of {g}: . (A) (00, 2) (B) R (C) (2,00) (D) ("003"2)U (2:00)
(B) (4,2) 2.2 Find the smallest value in the range of the function f(x) = [2x[+12x + 3. (A) 2
(B) 5
(C) 3
(D) 1/2
(E) 0 2.3 Let f(x) = 2x + 3 and g(x) 2 — 3x. Find (fog)(x) when x m 1. (A) 0
(B) 1
(C) 5
(D) 5
(E) Undeﬁned. 2.4 Find the constant(s) c for which the function f(x) = { ‘3X If)? 51 is continuous on SR .
( xc)(x+c) 1fx >1
(B) —1
(C) 0
(D) 2
(E) —2 2.5At how many different values of x does the curve y = x3 + 2x have a tangent line parallel to the line y=x? (A) 1
(B) 0
(C) 2
(D) 3
(E) 4 Continued on page 5. . ?age 5 of 12
Question 3: ( 2 marks each, total 10 marks) 3.11fy = f(x). x373 + xy 2 6 and f(3)= 1, ﬁnd f'(3).
(A) 0
(B) 1
(C) 2
(D) (1/6)
(E) (1/3)
3.2 Let F(x) = sin(g(x)), where g(x) is differentiable. Find F’(x).
(A) g'(X)COS(g(X))
(B) g(x)sinx
(C) ~g'(X)Sin(g(X))
(D) ~g'(x)cosx
(E) g(X)COS(g(X))
3.3 Find all value(s) of c (if any) that satisfy the conclusion of the Mean Value Theorem for the function
f(x) = (x  12)3 on the interval [0, 2].
(A) 4
(B) 2 + .2— 3
(C) 0
2
(D) 2 J3
3” i.
Ji J3
3.4 Let f(x) = xmﬂ—x) for x 2 0. Find the absolute maximum of f(x) on the interval [0, 4]. (A) 0 (E)2+ and2~_ 2
(B) 7; (C) 3% (D)  6
(E) 2J5
3.5 Let {(x) = x on the interval [0, 2]. Let the intervai be divided into two equal subintewals. Find the value 11
of the Riemann sum 2 f (x: )Axi if each x: is the right endpoint of its subinterval.
izl 1
(A) 5
(B) 2 5
(C) *2“
7
(D) 3:
(E) 3 _ Continued on page 6. . Page 6 of 12 Question 4: (2 marks each, total 10 marks) 10
4.1 Let F(x) = I l+8t2 dt . Find the value ofF ’(1). (A) 0
(B) 3
(C) 3
(D) 1
(E) Cannot be determined 5 5 5
4.2 ) If Ifmdx = 25 and ]g(x)dx m 9 then ﬁnd I{2f(x)—g(x)}dx
0 0 0 (A) 34
(B) 7
(C) 0
(D) 41
(E) 7 0
4.3 Find the value of the integral le—Zldx .
w3 (A) 0 (B) 21
(C) 32
(D) 21/2 (B) 2 1 2 )6? dx
0 (x341)2 4.4 Find the value of the integral eee @ (E)
4.5 Find the area of the region bounded by the curves y L“ 2x u x2 and y = X2. GNleIqqlwwhlw (A)
(B) O ( A D) 3
2.
3
2
)5"
0
1
E_
()2 Continued on page 7. 5. Question 5: ( 2 marks each, total 10 marks)
21} 00
5.1 Find the sum of the series 2 (—1)“1 n
3 n=1 (A) 1/3
(B) 1/2
(C) 2/5
(D) 1
(E) 3/5 00 00
5.2 Which ofthe three series converges? 1) z —“— 2) z (A) None
(B) 1 (C) 2
(D) 1, 3
(E) 3 °° 1 5.3 The series X, e converges if and only if 0‘.
112111 (A) o.<1
(B) —1<0t<1
(C) oc S 1
(D) 0L 2 1
(E) 0t>1 5.4 Find the interval of convergence of 2
11:1 (A) (00100)
(B) (113)
(C) [113)
(D) [115)
(B) ("115) 5.5 Find the radius of convergence of the Maclaurin series for f(x) i (A) 1
(B) 1/8
(C) 2
(D) 00
(E) 1/2 n$111+] °° (x — 2)n n3n 11
1!; 11:1 3n d3)§ 1
an nleh/E
1 4+x2' Page 7 of 12 Continued on page 8.... Page 8 of 12 _ 10x3 10x2(X2——3) 20x(3 + x2 ) .
6. [10 marks] Givenzﬂx) E 7—,f’(x) a W and f "(x) 2* —(——2—I)3— . Answer the followrng.
x — x — x —
Put your answer to each part in the box provided:
SHOW YOUR WORK! 3) Domain of f. H b) Is f odd, even, or neither? 0) Find the horizontal and verticai asiitote(s) (if any)
d) Find the inteiya1(si where the function is increasing (if any) e) Find the interval(si where the function is decreasing (if any) i) Find the interval(s) where the function is concave up (if any) g) Find the intervai(s) where the function is concave down (if any) h) Find the local extrema max./min.), (if any)
E) Find the iieinrs) ofinﬁection (if any) j) Sketch the graph of y = f(x) y HI Continued on page 9.... Page 9 (1le 8602 (In x) 7. a). [2 marks] Evaiuate I dx x 2Vx21 7.b). [4 marks] Evaluate I
x dx .2 _
7.0). [4 marks] Evaluate Igi—x—de x3 +4); Continued on page 16.... Page 10 0f12 8 a) [Smarks]. Evaluate ifpossiblc: Ixz lnxdx
0 3
8 b) [5 marks] Find the length of the curve y =§6— + —1— , where i5 x51.
2x 2 Continued on page 11... Page ll (3le
l 2
x + 3x + 2
y = 0 from x = 0 to x 2 1 rotate about the y— axis. Sketch the region, the solid, and atypical disk or shell. 9 a) [ 5 marks] Find the volume of the resulting solid if the region bounded by the curves y z 9 b) [ 5 marks] Solve the differential equation ~32:  2tan(x) y $ 3sinx, —% < x < 325.
x Continued on page 12.... Page I2 of 12
10.3) {5 marks] Find the highest and the lowest points on the curve x2 + xy + y2 = 12. 10. b) [5 marks] if a rectangle has its base on the Xaxis and the two vertices on the curve y = e?"2 , show
that the rectangle has the largest possible area when the two vertices are at the point of inﬂection of the
curve. The end. ...
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 Spring '08
 LAM

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