MATH 201 Exam 2, Fall 2004

Perey S Heading Aoom oe ns eR Mein Qot Flo4-e5 Quiet Preece Klori-Maledis' 1. Multiple-choice questions (la-1f): CIRCLE YOUR ANSWERS. NO PARTIAL CREDIT, NO PENALTY FOR GUESSING. You do not have to justify your answers for the multiple-choice questions. Each of questions la—1f is worth 2 points. (i) Identify each equation below with ONE of figures A-E below. 3 Wis B Cc D E Circle your answers: la} -2P4+y4z2=1 A B Co) D E e+y=l (é>) B Cc oD E 1c) ytve=s KX B C OD Ee) (ii) Given a function f(x,y) satisfying f(1,2) = 6, VF la 7 (3, -2). Evaluate {in scratchwork) an approximation for each of f(1.01, 2.02) and (0.98, 1.99). Then use . your approximation to circle the correct answer in each question below: Ox - csc: LCP BA Geos) = Efiad eM laay sCOmdy) 2 Ce 3Ox-2ey -] rare Were Or: ate L 1d) The LARGEST number is: C#(.2 —f(1.01,2.02) _f (0.98, 1.99) ie) The SMALLEST number is : f(,2) — f(1.01,2.02) 0.98, 1.99) Ss S$ \¢ 6-~ 6.81 6-09.04 Note: comparing vaiues in this way is not actually justified unless we can also estimate the error in the approximation! But don’t let this worry you for the purposes of this exercise. (iii) The level curves of a function f(x,y) are shown below. Based on the picture, which of the two gradient vectors Vf l. and Vf lo has a larger magnitude (i.e., length)? Circle your answer below. y4 — ~ ds (Nye 1f) The LARGER magnitude is : Serabel, (vel,| = det, x = 1 for O in the when you mee in tha derechyn Linechen d- ot | Moximal incense maaximsl Increase , Using polar coordinates, sketch and clearly label the curves Ci, Cz given by Cy :r =cos8, Cy: r=1—cosd and find the area of the region that is inside C, and outside Co. y 2 Vy Tear 8 1-08 fe cesB= 4 Co SO here O-T ce: Ya. maa nies ay, [Rie Ret jie = 5 \ [ ate -C€ (-as0)" | le 6 Vv” “% wake. Us ic wt 0-1 + arb ace = -(r2osO we) 4G . 2 3. Consider the function f(x,y, z) = ye™ * and-the point Po(-1, 2,1). a) Find the gradient Vf | ~ : au a Vez CE ae %); xe ee CE, 55 (2xeye Je b) Find the equation of the tangent plane to the surface ye” * = 2e at the point Po. Q Gy) & berger glo oS) ee iL Vp, i) (xt, 4-2, a-1) (-4e,e,2e) ao | —4e (x4) + e (4-2) +e (2-1) = © ! “!) Yor my, bgev with, empl te Heat) y-2 +2U2-D=0 4 er tA ~ Hey Ze = c) Find the directional derivative of f at Po in the direction of the vector 7 = (1, 2,3). + The on Vecer in he directo, of Fig \ > eli29. 4. Given the parametrized curve in space: P(t) = (x(t), y(t), 2()) = (ee, V2 - t). a) Find the arclength of the part of the curve between the points P; = (1, 1,0) (corresponding to t; = 0) and P2 = (e, 1/e, /2) (corresponding to tg = 1). Note: you can, and should, smmputy the expression inside the integral to get rid of the square root. F() ORD = (eb,e%, Bt) © My Wloshy is G ~~ a i) wd Mu seeds Whe (edhe Cede +2 > Aha archer re) brs) 5 Nt VF iv de ce" 7 O + r eT eC-4.- b) (UNRELATED to part (a)) Use the same curve P(t) as above. Cn a functio! f(P) = f(x,y, z) with Vf | = (1,6,4), find the derivative <li [f (P@) | ase’ Va |n3) = -(3,% L Nel). our velociy ‘S$ \ “3 a a Vz) Goan above -i NZ), j = ( on. fol tele2 (3,1/3,v2 In 3) ak te Bor gettin ie ReW (08) = (3 eS ey 4 v4 7 / a é 5. a) Find the maximum and minimum values of the function f(P) = fle,y,2) a 5. é 2+y+z, under the constraint that the point P(x, y, 2) is restricted to lie on the ellipsoid / Qn? + y? + 2? = 1/2. THis ip = lagnage olin prrblew cite foe a) = Key (x a uu Me 7) i) tatey t \ @ a? ~ x +2cy + y*, and classify each point as a local minimum, a local maximum, or a saddle point. fx _- oss = L . ve (Boe l¥2y, 2ne2y) Gas a crtval pent EG, TFl|=o © PG) is al pen €& |: 8 Q~-l + -») +O od x JO a 3x -2y¥-1 =O au ys Gxt! x7!) \ac coals, = lar oie qeedeke Bonk) | = 7x J Ge $e “he criktal qeina ore (RCs), PRO). Cx Hession delermirot : Px s\ 3 ( eal Ay poly 5 ~ {6 2 a Ry A= " * | = 1242 §>0 ml fix (p= 60 4 ‘ local in 2 a 3 vA 6. a} Using the two-path test, show that on. jm, oz aA does NOT exist. Tele sgh PH) CK ct) 8. Gel s Cie 28.) bo = P()>(e,), o You much Poiel sthic ov The xy 2 bak t p(t) p tl a» afler cancelliy tt t We batts leas J4 ~ B® oid deperle 2 the deiee fog aot 4 Ee “ry PLE) \ha ie. on th choice at path, & P(E)—02,9) Se \ne oy —_—— doer nok exish : b) (Challenging) Show that lim oy? (x,y) (0,0) v4 + y4 exists CATION th iy NOT. neud bo Check Arab the liek t CN shea - Ve pith PE) ASE tn vel we - 2 Uke 4 2Se qt a. Nee wusk ack My Sow Teal | _ suas £- all wo) Thie amount be Shout ny That the numernter ae Lie. show The denen ts net +00 small. Eelen Yon the deneninahyr = 0. ¥ ey te 2xXy Then wyt | = [e+] z bey . = my rae 4 Kegt ~ 2x4 ya 2 Vee 7 s (a ) < 5 s vO S, as Ky) le °) (4) 2°) d\t2 OF Cte Vek on eacred) , by The Smaduyiels Yreoren Lim a 5. ° KD OD Wey" Blank sheet. 5 evaexe MelheL LL. Occ ee con/dinabes. . cor Q 4% eur ec? us sm@ c (one = ) = — = . . ae cat (ces rent) wt ® + sin [e) er cov sa 8 or © sa™O a’ (GN) ree) - Le BsinrO {- % Sin CA Mai ge . 4 Les Osi @ Aw Roca (<o,*0| <\ 2 b20\ <\ ) /| | —_— Se ure os OsitO | moo. | ae \- 4 sunt? a ° _ < se “er Ole fhe fondu The: ren f- rn ie {o) x £2, V O bate (Ks) os (22,9) K per Cone dinales dich is eae Ae uo rv J : are MefluL AZ Te ska Thee wey ome reo small in beme ab ex aa ; 1 Gy x valve wk 2a RK cancidee thw small circa Xry zat, a yteytt Than ve Lagoage vvltigien mmanae Foe x' sbjec the consra@ninT yy) = = Xo = ab Bon Cad ad) = We = (D424) Lid U0 3 Seo * (rat) J Ves cab By = YY => Ser “a re ae ear 4 Sys conser the grin (©, ta) ——e f= at (ta, > La at (% #%,) re a Oh min vale et Cig 97 4 here a eg 7 So a teem & woe ides AP back trys ct a sd ait — vy —— ley tes . . ond ae aa :\ <-' = ae - & \us 2 sw'o| SG ote Ua sluice Bearen agoin. =