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Test 2 - Solutions

# Test 2 - Solutions - MATH 1A03 Test 2 McMaster University...

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Unformatted text preview: MATH 1A03: Test 2 McMaster University Date: November 12, 2009 Duration: 90 minutes Instructors: D. Lozinski, D. Krepski, E. Martinez-Pedroza Student ID Num er: Instructions: This test paper is printed on both sides of the page There are 10 questions on pages 2 through 12, Pages 13 and 14 are blank for rough work. YOU ARE RESPONSIBLE FOR ENSURING THAT YOUR COPY OF THIS PAPER IS COMPLETE. BRING ANY DISCREAPANCIES TO THE ATTENTION OF THE INVIGILATOR. Only the McMaster standard calculator, the Casio fx 991, is permitted The test is out of 60 marks, For full credit you must show all your work. ANSWERS MUST BE IN PEN ONLY McMaster University MathlAOB Fall 2009 Page 1 of 14 McMaster University Math1A03 Fall 2009 Page 2 of 14 1 [4 marks] What is the area under the curve 3; = 1/2: between :1: =1 and :z: 2 e3 ? ’3 6 (3'5 ,3 j ‘/7: 5%” WA 2 ﬁne. ~54 ' -.— 350:3 McMaster University MathlAO3 Fall 2009 Page 2 of 14 McMaster University Math1A03 Fall 2009 Page 3 of 14 23 [2 marks] Determine lim .1: + sma: x—vo 41:3 - a: \_ ., t . ,ﬁ . m . 05% i%' We. we. IMOQPJJ laée ﬂ: 0% ”WW F5“ 6W? 0"” 7 A” l + 60.3% :— l{— I : #2— yoo Own—r 0 4 2b [3 marks] Determine 11m (3.2:2 + £8 + 1):?2 33—h“) :“ g1” 2x: HHJ lf’fosprd gm 6“ + l 10-7“) 6% ""' 50950 e); [WWW : W“ _G__ “a O \6—56‘0 67¢ 20 [3 marks] Determine li_r%(1 — Zxﬁ Tafw Z» Em g" [frgfjw 2. 5w 6.5M 9:20 >< x—ao v x? , g 1‘." ﬁfe. [ﬂ - (*2) “Kit? “(vigiljﬁwm :- “ 2 0 ‘ r‘x-‘Z ——2 ‘” “ Z {r- 29%.) “’ :: we is “2:2?! McMaster University MathlAO3 Fall 2009 Page 3 of 14 McMaster University MathlAOS Fall 2009 Page 4 of 14 3). [15 marks] Consider the function: :3 +1 me) = m 3.) Describe the domain of f (3). X215" -- 1 9r- fxéile ‘76—'11) or (-09, '4) U (”U +90) b) Describe the intercepts of the curve 3; = f (3:) t“ ‘7' O Whisk % o J j (:2: 04,9, iﬂ.“¥LUC‘3WL [(330) {9:0 04’7”) 0) Describe the vertical asymptote5_ \Jfﬁlﬂg: {Tap Q gyii-pz'oie 017‘ 3C: ..... “I (1) Describe the horizontal asymptotes. Y"? 00 2:511! i 3) On what intervals is f increasing or decreasing? 35K: (NH-vim «a ’ >0 (‘5; H)“ {if-"m .1» _.,...—~-—--——-...- I‘M-m. g WWM McMaster University MathlAO3 Fall 2009 Page 4 of 14 McMaster University MathlAO3 Fall 2009 Page 5 of 14 Question 3 continued“ f) Identify the local maximum and/or local minimum values of f, if any. “77W"? an! ﬁe: bf“? ”aw/Wm of mm’m‘mzj g) 011 What intervals is f oonvave upward or concave downward? IF”- #2 1(”>0 f x454 . ’3 (Kr—(3 <FH<O If X7"! (01’! r" {Ly-f ”134;“. x a: .w { [w 0:9} .-. () (“W "i " " #0:.- n. 1491’ E473 "‘ f“ 9|, 503) h) Sketch the graph of the function y = f (1:) below i ; s 4 I *- i McMaster University MathlAO3 Fall 2009 Page 5 of 14 McMaster University MathlAO3 Fall 2009 Page 6 of 14 4. a) [2 marks] If [3 [2 f (3:) + 39(3) +4h(m)]dx— 32 and we are given that [3 f(:::)d:z:= 3ancl [3 h(a:)d:s22, ﬁnd [3 g(x)dx. '3 QfFN-“JM BJ jﬂ‘b’e + ”J M)” 732 :2‘) 2(3) + Bj‘jmﬂw MM?) ’3"? k. :— ""2 x jji‘ 253“” ‘2 g 3 6 :23 3 4.13) [2 marks] If 33(3) = f q(s)ds + / q(s)ds, what is 39(2)? Show all your work. 3 22 - I his C) 63 j S: Ofa K7 L! _. r 1“ D _ (9:1 L _. 3 CL! )ﬂ ( if 2 McMaster University Math1A03 Fall 2009 Page 6 of 14 McMaster University Math1A03 Fall 2009 Page 7 of 14 5. [4 marks] The position of a moving particle at time t 2 O is given by 1(1) = 1 + 4013 — 3:5 Find the particle’s maadmum velocity. Vdcxb W/U': V206?" ’5” 7:) 8:149 {24‘ mﬁfxyman W Cf l_{..}%rbﬂ {301W {3 95%;" Any) 536%,»? ”is L” W (37M 1 17131ng \$0!“ '0 c201- (ox-t:- 1‘0 (2)115 C‘zwéﬂwﬂﬂ O (fiah({¢¢f 1/301“?ng CM!" (1C?) “'3'2 {3'2 m‘ﬁwjé’? 2mg - (9063 dowaglq. 011/6 are 05 #611sz (way; A; (Leg/MW, Hug ‘ WM (5;. 61.2"- f -2 1 3}" &W 1115+ a 2w! 0%.“, .1911 _. 753:1 ﬂag I {’30 xﬂzo {XIV >0 r"1__.t-‘.ur (O .... . ‘2 .. _, a“ j)”: rid-mwwam I/Mrﬁqé as WE’QK’Z) ~/§(/Z/I= Wig—20(1) McMaster University MathlAOS Fall 2009 Page 7 of 14 '2‘ 2 L/0_ McMaster University MathlAO3 Fall 2009 Page 8 of 14 6. [6 marks] You axe given that f (10) = 1.2 and f”(10) = -—5 3) Approximate f(10.5) 53”" O z Waoym on dFOJ d3: (‘5)(035): “'2; ‘9 PﬂOSU/‘v I»? "25/ : '4‘?) 00 OF]— ;(%)N €(IO)+ 1C/(10)(><~zo) :- A? afﬁx—,0) @005) N :32- 5"( 10133103 2' —z.3 b) Suppose we are using Newton‘s method to ﬁnd a. value of :1: for which In( f (3)) = 0, and our ﬁrst guess is \$1 = 10. What would 322 be? News MM ya: x. _ Mo WWW i ”757 is ﬁll-‘54) mm _ mm i[ﬁ(fm)] MM Sela/e Q): «CM—LI :X( __ j (57.) __ /0_, AZ(/'?) y; [we .4: {3w "ff—H ' I ——-"—' Flt/to) #5” l: J PM ”775 ‘KZZYCI ﬁ- CiCF{?<)"'> " /0 7L [2%{A1) N /&+ (Db/3% 5 ‘: IO __ L2 “I '5 :13 lo ~21 '5' Mc aster University Math1A03 F311 2009 Page 8 of 14 ;;0.OL/ 1 McMaster University Math1A03 Fall 2009 Page 9 of 14 7. [4 marks] Find the point on the line 3; = 4.1: + 7 that is closest to the origin. ‘7. ©5210“? :5 (>00? + [3’01 W/{M‘e (<3 7L/7c4'? c900 adj \( 7L0 ”MAM"? m My Mm) mm ﬁwmwggwﬁ tea a? M { _L hgﬂwﬁjg mmmmm .,. 3 0/ )5 ¢ g’é, 2 m w ~o :5 ><= 15}? 2 “'33 3‘1 /’—? w“ ‘13” -z, 25:? ﬂu {Ooh} rs ﬁ/ / ﬁxk?) McMaster University Math1A03 Fall 2009 Page 9 of 14 McMaster University MathlAO3 Fall 2009 8. a.) [3 marks] Determine / 12xeax2’1dx Page 10 of 14 L4 db? 3 61"” m : ZXZ-f q _. L4 J 8 gaff: : fzeoﬂa , 2.9 gm ‘7 2 6 4 +C b) {3 marks} Determine / f0 lsin(w)ldw _ Tr 3% J 3mg 0&4 + ‘SMK 019‘ O 11' ’Zn72 “(03* U + +50§”‘/7r 1* 1-6/0 + o~ M) Anew ﬂ 3 5 , m I 5/ d Qr'vﬂﬂ‘ﬂ O? [I 0 O McMaster University MathlAO3 Fall 2009 Page 10 of 14 McMaster University MathlAOS Fall 2009 1 I —1 + t ”2.1:, ﬁnd F (3). 9.3.) [2 marks] If F(:L') = [1 F73) : _J/_ I 4/34/31 H .L )3 9.1:.) [2 marks] If G(a:) = f: 1 1 dt, ﬁnd G’(3:). +t+ﬁ McMaster University MathlAO3 Fall 2009 Page 11 of 14 Page 11 of 14 McMaster University Math1A03 Fall 2009 Page 12 of 14 10.[5ma.rks] Find the value of Jim % (32!”+e4ln+eS/“+-~+32”’J"). Hint: Interpret the limit as a. Riemann sum 10 [+135 )( 2“ 5w 2; 6 .2 2 J 8 d9 : 6’7 ﬂ—wo ('1: ﬂ I .— 3 r. 6 ”'6 h—F—d—\v_ 9f n 3,; 2 x A 6 A 2 e «Z r— e e M X3300 LL: ﬂ 0 END OF TEST QUESTIONS McMaster University MathlAO3 Fall 2009 Page 12 of 14 ...
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