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Dec 2006 Term #2 - ~ Sat/2X Department of Mathematics...

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Unformatted text preview: ~ Sat/2X Department of Mathematics University of Toronto WEDNESDAY, December 6, 2006 6: 10-8: 00 PM . MAT 133Y TERM TEST #2 Calculus and Linear Algebra for Commerce ' Duration: 1 hour 50 minutes Aids Allowed: A non-graphing calculator, with empty memory, to be supplied by student. Instructions: Fill in the information on this page, and make sure your test booklet contains 11 pages. In addition, you should have a multiple-choice answer sheet, on which you should fill in your name, number, tutorial time, tutorial room, and tutor’s name. This test consists of 10 multiple choice questions, and 4 written-answer questions. For the multiple choice questions you can do your rough work in the test booklet, but you must record your answer by circling the appropriate letter on the answer sheet with your pencil. Each correct answer is worth 4 marks; a question left blank, or an incorrect answer, or two answers for the same question is worth 0. For the written-answer questions, present your solutions in the space provided. The value of each written-answer question is indicated beside it. ENCLOSE YOUR FINAL ANSWER IN A BOX AND WRITE IT IN INK. TOTAL MARKS: 100 FAMILY NAME: GIVEN NAME: STUDENT NO: SIGNATURE: TUTORIAL‘TIME and, ROOM: REGCODE and TIMECODE: T.A.’S NAME: —fi-—_m T0101A TOBOID BF 323 T0101B T0601A MP 137 TOlOlC TOGOIB RW 143 TOZOIA T0701A MP 137 T0201B T07OIB $81087 TO201C T07OIC LM 162 TOZOID T0801A 881083 FOR MARKER ONLY Multiple Choice -- - T0301A TOSOIB 881073 T0301B T5101A 881069 T0401A T5101B 882108 T0401B T5201A 882110 T0501A T0501B T0501C Page 1 of 11 NAME: 1. [4 marks] mZ—l 50—1 lim m—>1 (4a: 2 A. equals ——2 B. equals 1 C. equals 2 D. equals 0 E. does not exist 2. [4 marks] lim 2:02 ——6x _ md—Ooxz—lliv-l—3 _. A. +00 B. 1 C. ——2 D. 2 E. --00 STUDENT NO: PART A. Multiple Choice g”; Luv-26m)- é“ — qu 3"! Ya, = fl... 2.. = 2/ X‘a' x+l 1‘“ Page 2 of 11 NAME: ______________________________ STUDENT NO: 3. [4 marks] If $500 grows to $1,000 in 8 years then interest must be compounded continuously at an annual rate that is closest to ' A. 7.72% 3(- B. 6.25% 1000 a: {006 C. 8.66% 44,2. -: 6F _ D. 9.13% . 3 ,L Zn?— ,1: .0395 E. 9.05% 6' 4. [4 marks] The function given by f (:16) = la; _ 2| on the interval 0 S x < 4 has A. no absolute extrema B. one absolute=minimum but no absolute maximum C. one absolute maximum but no absolute minimum D. one absolute maximum and one absolute minimum E. two absol te maxima and one absolute minimum Page 3 of 11 NAME: ______________________________ STUDENT NO: 5. [4 marks] - —-———-———————-1n(2 + Z) __ 1x12 when a: > 0 _ f(as) = a when :13: O 6“”b when an < 0 If f is continuous at a: = 0 then A. a=1n2andb=0 é,” (36¢): 2%It ,1th 1 “li B. a=1 and b=1 x-—)O+ 1 - H C. a=~andb=2 ‘ 2 b Coat D. a=2 and (72% \I £0) E.a=%and b21112 H at: $5360 ‘ 6 x—W’ 6. [4 marks] . If y = (Ex/4:3 + 3, then the tangent line to the curve y = f (as) is horizontal . V‘tx’r3 ‘H" B. a =0, :0 W 3y 7': 2m I D. ata:=1,y=\/7 “.1 6)”? 1 Y 4x+3 l & '5 ”I L o wW‘ X Z a | the“ 7': ’JiW”"i A. never Page 4 of 11 NAME: STUDENT NO: 7. [4 marks] Find all A , for which y = 6)“ satisfies the equation 1A. [Arz—Z only 7.: ()6qu _ _ ’1 )4 B.>\-—1and)\—4 yutae C. A=——2 and A=~4 D. A=0 only U 6 '+? E. There is no such A 7 4‘ Y 7 “X -: <1X‘:rb(>\+8)e 7’ o J‘uo so an. A «r? =>~ 8. [4 marks] If h(a;) = g(f(x)) and u = f(a:) , then h”(ac) = A. B. I/ (u) ' f’(w) + we > (f”(x))2 W) M >+g<u> f”(w) g-"<u> f (x) ' ' D we) (f (ea) +g'<u> f”(w) E 9%) WW . (x) , 9%) ‘n (x): j Eigggj? (09%)!) + 3 (-963) ‘ h" (>3 3 $0 ® 9 9 Page 5 of 11 NAME: ______________________________' STUDENT NO: 9. [4 marks] The derivative of f(x) = (2x2 — 3x + EMF-7+6“ _ is equal to A. (2:132 —— 311: + 5) “0 —1+em '1n(2m2 —- 33: + 5) B. (2:132_.__3‘,L,_+_5)Veg—1+5?J (( (I? + 6m)1n(2332 __ 311} + 5) +( /$2 _1+ 69:) 413 "- 3 ) «52—3-1 2:172 — 333 + 5 C. («iii—1 + a”) 1n(2932 —— 32: + 5)(2:1:2 -— 3x + 5)«/'£TZT+ew D. (251:2 — 3:2: + 5)m+em( $2 — 1 + 6m) (Jag—71 + em) E. é’é:)1n[f(x)] 3 x +5') 10. [4 marks] If y(ac) satisfies y‘” = my , What is the value of y’(x) when (m,y) = (4, 2) ? ~%+ln4 \/ . 2—1n2 D. “:34“ A6 X441): 3 1 ”£47 +1, , 1-11 , 2 E' 2—1:: 31’; + 21" 7 Page 6 of 11 NAME: STUDENT NO: PART B. Written-Answer Questions 1. [14 marks] [’7] (a) Solve the following inequality for a: [7] ( b ) If 7* = 300q2 — q3 is the total revenue when q units are sold then find when marginal revenue is positive. MR= .4; ... (:03— 3? d 6.2: C 200’?) Page 7 of 11 NAME: ' STUDENT NO: 2. [18 marks] . In order to sell q metres of photographic film per day (where q > 0) the manufacturer must 1 set its price at p(q) = (126— 12 dollars per metre sold. [4] (a) Find the manufacturer’s marginal revenue as a function of q. Olr A (P6,): Ol @58’%) [5] ( b ) Find the relative rate of change of revenue as a function of q . r a: %%%l% l 1 ’2; " T7. or a, /. l»X * x 1'"; [QUESTION-2 CONTINUES ON NEXT PAGE] Page 8 of 11 NAME: ' STUDENT NO: 2. [6] (c) Find the elasticity of demand as a function of q. 1235‘ [3/ (d) For which q is demand inelastic? Lelash}, Wm \72 l < | 2 ¢\e-e~~lwé"(5""‘71< I Page 9 of 11 NAME: ' - STUDENT NO: 3. [15 marks] [’7] (a) Given that m4 + y4 : 43;, find gig in terms of a: and y only. There is no need to d “simplify” . x 4x3+4/3y1147’ . I ‘ 3 6 )\ -' )q ’., 3 .. l " . d2 [8] ((2) Gwen that m4 + 314 2 4y, find EEC—g- in terms of a: and y only. There is no need to “simplify” . 5'l7Wlfl-j .0,” eke, answer ya wwmw ’ W , , (\ «— ),3) Page 10 of 11 Y ‘4 NAME: ’ . STUDENT N O: 4. [13 marks] Find a root of 2:5 + 433 +1 = O up>to _6—decimal-place—accuracy using Newton method and x1 = 0 as the initial estimate. Xh-n : X“ ‘I' '96“) M PW») : 4— X H “W 6%“ +4 H X. : Page 11 of 11 ...
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