final-solns

final-solns - PRINT your last name 6 O L. l O N S PRINT...

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Unformatted text preview: PRINT your last name 6 O L. l O N S PRINT your first name Signature ID# UNIVERSITY OF WATERLOO MATH 137 Final Examination Calculus 1 Friday, December 11, 2009 9—11:30 a.m. CIRCLE your instructor’s name and your tutorial section number. Instructor Section Tutorials J. Nissen l 101 102 F. Zorzitto 2 103 104 A. Nica 3 105 106 J.Nissen 4 115 119 M. Eden 5 109 110 M. Eden 6 111 112' ED. Park 7 113 114 C. Struthers 9 St. Jerome’s S. Speziale 10 117 11-8- Instructions 1. Put your name, signature, and ID number at the top of this page. To prevent loss of your exam, circle your instructor’s name and your tutorial number up above. 2. NO ELECTRONIC DEVICES other than your ”Pink- Tie” Faculty Approved calculators are allowed at your examination desk. 3. Answer the questions in the spaces provided, using the backs of pages, or the blank page at the end, for over- flow or rough work. 4. Show all your work required to obtain answers. 5. Your grade will be influenced by how clearly you ex- press your ideas and organize your solutions. 6. If you are in Prof. Stastna’s Section 8, then you are writing the wrong exam. Go to Prof. Stastna’s exam in DC 1351. M137Final exam Page 2 of 10 Name: [3] 1- (a)1ff(93)= 1+x2, 2 2 Q. [4] (b) If g(x) = arcsin(e“”), specify the domain of g. Then find 9’ and specify the domain of g’ . W124 mum (9% 34—9 «exam-s; «e. o / I ’X $6,“):- 8 // Aft/«me, ‘w ‘M ’62; X77114) W ) so Q /_ .1 1‘ —,9,L:V\x + IWX V/ 92 fig. 3M 3 a S4wL(o,Q/COMWWMW 521.1—3— -4('WO)+5‘WO V H—o" M137Final exam Page 3 of 10 Name: [2] 2. (a) The displacement of a damped oscillator at time t is given by f(t) = e'tsint where 0 g t S 27r. By citing appropriate theorems, explain why f (t) has a maximum and a minimum value over the interval [0, 2a]. / p‘:F/\\Al wai—imw N fl [~0ij {MWI/ WW W {-0)le [4] (b) If f is the displacement function of part (a), find p in the interval [0, 2%] at which f (p) is maximum, and find q in the interval [0, 2%] at which f q is minimum. () re'tM‘t: Q-tKQANé‘W-b), / Neutral/(t) :0 75m mt : cm W T “mm «TV/4 \/ .7: 3 Q J3. \ ‘ f( ‘f ) / Ml WW -Q’fl'y— as) = “~- We 1+. [2] (c) Determine if g(:1:) : arctan has an absolute maximum over the interval (0, 1). (0,1)) 7r ' +08 #MO ‘4' WM; M137Final exam Page 4 of 10 Name: 3. Let f(a:) = 3x+sinas+ 1. [3] (a) Using appropriate theorems show that the equation f = 0 has a solution in [—1,0]. SAM ~\ : ~3+M(—t +\ <0 v 42 i ' I > 0 3 / + 0 ’ 4 .~ 2 w ” 6’ ‘ “ML W w-LM ctr W «as . WW 93M v to M ">03 [3] (b) Starting with the point (130 = —1, use Newton’s method along with your calculator (in radian mode) to find the next approximation $1 to the solution of f = 0 m A“: ’Xo. ‘ [72‘ / I 36(0) W): W01: (x): 3+ca/ox, / Sc («’X‘ZPI“W: 2.44424 _, 3 4' W C” '3 ‘5 +1.44) 1 /N O 7 9 Mata . ‘\ er [4] (c) Show that f has an inverse function, and state the domain of the inverse function. [2] (d) If g is the inverse of the function f, prove that f’g’ = 1 for all an in the domain of g. -_ r ' / 3, Mal/“MIL _ r__ I £(WxDSL/[O _._[;a3mfl/: ’X —] [2] (e) Find 9(1) , and using part (d), find g’(1). 9m ‘F[o)=1 We 3C')‘*O“" V @y W (at) agony/(I) :1 am I M 8/0): "I : , _' ’lo) 3mm) L+ - ’7 M137Fina1 exam Page 5 of 10 Name: 4. Consider the function f = a; In :1: for a: > 0. [3] (a) Find the intercepts of f, the intervals over which f is positive, and the intervals over which f is negative. \/o:€(x): o W X':\ x/o 12M 040%!) %l¥)<0 Va fix («<x )fi£(fi)>0 \ [2] (mandfiififlx). ‘WR w Mém ; [’f—T; » Vléwr “9‘54 ‘9 ya “53‘” M 2<~>OT UMjorl‘d ‘ \ LEM/g %(F):’&IW\—ZZ-—=’&4M+’X :0, «A0? rx-aol' ‘VX’L “’73 0 x \ (Eff-M W aim—9w. )ng-Q (Avg l’MbaL/J [4] (0) Determine the critical numbers of f, and the intervals on which f is increasing/decreasing. M. ’54): «.;L+€m<=1+~&m<- / “rod—HwQWM WWW (flfold’ WAX3fl6“ ‘/ ' Pmo<x<é)m ’Kw fltxocwa {em/W0 \/., {:Mégy )MMW/g’m 7095114) {QM [2] (d) Find f”(x), and use it to determine the concavity of f as well as any inflection points of f. I/ .u l/ 9 3f? )‘:) " .L X a l ‘ Mo Ala/tuggiht) >0) W£ M W Comm/t»? 3f! [2] (e) Sketch the graph of f on the axes below, indicating any possible intercepts, critical numbers or inflection points. Ml37Final exam Page 6 of 10 Name: [5] 5: If 1 — 005(332) fa"): x4 whenx7$0 , 0 when m = 0 determine if f is continuous at 0, and prove your conclusion. WI mud (fibc), mat) L/%w&t¢wH7—%ocv%q’*0“flmgo2 UMHM Q/‘W s ’2. ‘ ’L t K ‘ Aim/x ax ,_ ' 4W0 .___L—————’I€;:;OQ_ «Ao' "HO 4053’ 7‘4' . ‘ GA) \/W2, {:30 £57 3171’ 1—50 MOM/v £ «36 MOT Mmmd'o MW =flfl=0$ [/1' [5] 6. A quantity A(t) grows with time t in accordance with the law of exponential growth. If A(0) 2 2 and A(1) 2 3, how long does it take for the quantity A to double itself? \/ Ce“: W Cé/fi, k0 M137Finai exam Page 7 of 10 Name: : [3] (b) Find /x2\/1 + x3daz. Mzaxtoix :i F MM xdx>Jng,i/ 3’ :—L—27_. Z) . 1 1 [4] (c) Sketch the region bounded by y = —, y = 7, a: = 1 and a: = 2. .7: CE Then find the area of this region. M137Fina1 exam Page 8 of 10 Name: [4] 8. (a) Evaluate the Riemann sum for the function f = 3:2 taken over the interval [0, 1] with a uniform partition of the interval [0, 1] into 3 equal parts and taking sample points to be the mid—points of theintervals. In other words, do the mid-point rule calculation for this function using a uniform partition of [0, 1] into 3 subintervals. 0,5 % 1/3 E \ CD "‘ ‘ e (a ‘2. m Rh“; P): at; era-(46:3) V 1/ ' \ __ 35“ V I .:§%§E(t+6l+.2§)— [,Dg Cigar] [3] (b) Ifg(a:)=/1fl\/1+_tzdt, findg’(m)foralla:>0. P‘l‘c’t WEI/w? M W 1/2 1 ' [3] (c) Given that ln(1 + :12) 3 cc when x 2 0, show that ln(1 + t3) dt 3 » 0 M137Final exam Page 9 of 10 Name: [3] 9. (a) Give the precise statement of the Mean Value Theorem. 53 ifl LO mesmow [3,10 1/ ct» ;(L 2A OiFH—MLWHM m (Cub) / x/TLW Mmaap/fc) anCAOUE) low 7 [4] (b) If a function y = f has derivative f’ = 0 for all cc in some interval I, prove that f is a constant function on I . JM «WM éI, gags xxx», Wit Mat—a W M ginkgo». $3 an NV!“ 7% 15 was Mqu M \/ W {4%)‘abc0 :‘QK/[Q 75W c6 (’xb’xu)- ’h"’¥. v 6%ch) =0 Mylo“ MM» tMLQ WWW 72"?” [2] (c) Suppose that a function y = f defined on IR satisfies f (0) = 0 and f’ (0) = 1. Use the definition of derivatives to prove that x and f have the same sign when as is close to O. witmm ,‘M’aqmyao, ’erO x O) @M x at / M 1 new mtw ...
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final-solns - PRINT your last name 6 O L. l O N S PRINT...

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