MATH 201 Exam 2 With Solution, Fall 2010

GRAD Math 201, Fall 2010-11 Calculus and Analytic Geometry ITI, sections 1-4 Quiz 2, November 27 — Duration: 60 minutes _ ES: 1(/i2z) | 2(/14) | 3(/12) | 4(/14) | 5 (/22) 6 (/14) | TOTAL/78 YOUR YOUR Qroe sd ERG | x (orce ch ory NAME: AUB ID#: PLEASE CIRCLE YOUR SECTION: Section 1 Section 2 Section 3 Section 4 Lecture MWF 3 Lecture MWF 3 Lecture MWF 3 Lecture MWF 3 Pr ofessor Makdisi Professor Makdisi Professor Makdisi Professor Makdisi Recitation F 11 Recitation F 5 Recitation F 4 Recitation F 10 Ms. Mroue Ms. Mroue Ms. Mroue Ms. Mroue INSTRUCTIONS: 1 2. 3. . Write your NAME and AUB ID number, and circle your SECTION above. Solve the problems inside the booklet. Explain your steps precisely and clearly to ensure full credit. Partial solutions will receive partial credit. Each problem on this exam counts for 12 or 14 points. THE TOTAL NUMBER OF POINTS ON THE EXAMIS 78. You may use the back of each page for scratchwork OR for solutions. There are three extra blank sheets at the end, for extra scratchwork or solutions. If you need to continue a solution on another page, INDICATE CLEARLY WHERE THE GRADER SHOULD CONTINUE READING. Closed book and notes. NO CALCULATORS ALLOWED. Turn OFF and put away any cell phones. GOOD LUCK! An overview of the exam problems. THE TOTAL NUMBER OF POINTS ON THIS EXAM IS 78. Take a minute to look at all the questions, THEN solve each problem on its corresponding page INSIDE the booklet. 1. (12 pts total) Two UNRELATED questions about polar coordinates. a) (2 pts) Given the curve C = {(x,y) € R? | zr?+y* =32+4y}. Write the equation of Cin polar coordinates. (Cultural note: C is a circle.) b) (10 pts) Let C’ be the curve in polar coordinates whose equation is r = 1+¥2 cosé. Make a table of values, and draw the curve C’. Make sure to label: (i) the intercepts on both «- and y-axes, (ii) the tangent lines at the origin. 2. (14 pts total) Consider the moving point in space given by the position vector P(t) = yt pt |. 3" 3° a) (8 pts) Find the arclength for the parametrized curve P(t) between t = 0 and¢ = 1. Hint: Vat? + bt? = tat? +0. b) (4 pts) Give an approximation for the displacement Af and the distance As traveled by the point P(t) in the short time interval between t = 1 and ¢ = 1.01. c) (2 pts) Is your approximation for As from part (b) larger or smaller than the true distance traveled between t = 1 and t = 1.01? Explain why. 3. (12 pts total) Given the function f(z, y,2) = aV1t y? + 22°. a) (2 pts) Find the gradient Vi. b) (6 pts) A point moves in R3 with position vector 7(¢). We know that the position iF and velocity of the point at time ¢ = 0 are r(t) | = (1,1,1) and a = (3,4,5). Your t=0 dt lt=o job is to find —firt . c) (4 pts) Starting from the point Py = (1,1,1), in which direction does f decrease most rapidly? Your answer should be a unit vector. 4. (14 pts total) Consider the function f : R* — R and its level set 5, given by f(a,y,2) = 2? +42", S={(2,y,z) € RB? | 2? +42 = 8}. a) (3 pts) Sketch the surface S. b) (6 pts) Find the equation of the tangent plane to S at the point Po(2,—1, 1). c) pts) Starting from the point Po above, we move 0.1 units in the direction of the = (3, 4,0). Approximately how much is the resulting change in f? 5. (12 pts) Find the maximum and minimum values of the function f(z, y) = ry on the elliptical disk D = {(z,y) € R? | x? + 2y? < 1}. 6. (14 pts total) Suggestion for both parts: use Taylor series and O(-) notation. 1 a) (7 pts) Show that — lim In(1 + zy) does NOT exist. (ay) (0,0) 2? + y? 24,2 b) (7 pts) Show that Jim G+ 2'y") (@y)—(0,0) 22+ y? DOES exist. ij 1. (12 pts total) Two UNRELATED questions about polar coordinates. a) (2 pts) Given the curve C = {(z,y) € R? | a2+y? = 32+ 4y}. Write the equation of C in polar coordinates. (Cultural note: C is a circle.) b) (10 pts) Let C’ be the curve in polar coordinates whose equation is r = 1+ V2 cos 6. Make a table of values, and draw the curve C’. Make sure to label: (i) the intercepts on both z- and y-axes, (ii) the tangent lines at the origin. RY) X= COsO > J =o gn ce SrasO * tro ee 7 Saye 6 ™ can cancel in this ense +o aek ips SesO+4s © >» 6 ag ce lyitcerO he ~ ‘ “ay 1 her. on 9) 1 jata eo 24 “eagle te 4 & 2 at, Cs V2 “45 *% ° \ ig r20 yo AS O=% ure 3 ~% wre ane 6:34 5621 A a o — wk lines oe) v= 4% -\ IVa 4 ae origin S$ _u w 9.3% “a TL o <___/ oO: 53, Ba A BOR \ Cogerly ot 2 wee me 2n5 ny her ~) eyed ; bine Ona A aN \ \ va oa i i a mA 4 ~ angt Vine . Ox sy H ! 2. (14 pts total) Consider the moving point in space given by the position vector a) (8 pts) Find the arclength for the parametrized curve P(t) between t = Oandt=1. Hint: Vat? + bt? = tV/at? +d. b) (4 pts) Give an approximation for the displacement Ar’ and the distance As traveled by the point P(¢) in the short time interval between = 1 andt = 1.01. c) (2 pts) Is your approximation for As from part (b) larger or smaller than the true distance traveled between t = 1 and t= 1.01? Explain why. > fal 3 = med a P(E eee) 0. oe (me wee 2b) veloc: 4 Tt b) t=, Ak 0,01 ar x ve Ak = (reset ut 8] Ak = (31,2).00' + loa sa, oO. 201) As =(n7\ ro \s ., \ot = \ (3,42 | (0.01) =(Wiy)(o wl) = [Fe |oenay joo PVPs c) Ks \ “ncrenses Peons \ 4o Lot, He speed \O\- (zea zee increases Lom roe via at f=) + Sond slat \ Ph ae “ws over the 0.c\ seconde a Avene speed ta s\i ary _loeger er than iy QA fre dichnce davelel ig salts leer tan Vat: Vik Too 3. (12 pts total) Given the function f(x,y, 2) = a/1+y? + 22%. a) (2 pts) Find the gradient Vf. b) (6 pts) A point moves in R® with position vector 7(t). We know that the position dr’ and velocity of the point at time t = 0 are 7(¢) | = (1,1,1) and ail = (3,4,5). Your d t=0 dt leo™ job is to find 5 (Fé) |, c) (4 pts) Starting from the point Py = (1,1, 1), in which direction does f decrease most rapidly? Your answer should bea unit vector. J4 DW (GGG) = OS, oR Boa os -|(Raw, Pe =e) rey at a the og wl, . 2 ‘ \ ie) x ad vo =o Sten a | (xn 2, os (3.4.5) - AB bavess Cotes of] We kaw Atak £ deckERsES Most rapi ay c) " uher we MEE tn a creck © Pe oeos te to VE» his iu becuse peklp= Vel 3 AMI, |) os uid 16 moet nage ha dem Beca= -1,) Ts G2 — MeL Lak) Lak (Sely,| fee ea Ww 4. (14 pts total) Consider the function f :R° — R and its level set S, given by F(z, yz) = 2 + 427, S = {(z,y,2) € R? | 2? + 42? = 8}. a) (3 pts) Sketch the surface S. b) (6 pts) Find the equation of the tangent plane to S at the point Po(2, —1, 1). c) (5 pts) Starting from the point Py above, we move 0.1 units in the direction of the v ] = (3.4, 4,0). Approximately how much i l the resulting change in f? oo c [ alli thes\ oy Kader) | 7) jf oh : iw ~y be he cresg Ce chen L. Qred 4 | ls a WS oan lkpse b) Sis alerel ot <p | cep = oe ond Mua Tengen ah lane. bo S ot PLES ‘s pereuieder tv VF |, = ( & 9, fe) | TO, ) Po{2,-1,4) “Pe ton gl ev Pp =o ; 7 yD e, ; . ies ae =) oO e -2, 441, z4)cO RN ON = x2) + OKs + Cato Sy > enw use . inet ons, Ss Ose wwe C= Sit GER = are) UMIT vedere inthe dinedion dt Y Wet ae eo 0 (SEs) OR) AP) AC pe (40,8) * (SF St) 423 5 oro 2 a) (or & we 0s) = .S aft yo M2 “mela DeGle = HELP = Gas CeO” 6 faye ie (dst \, Ss (2) = Atk Soe value a olwove. 5. (12 pts) Find the maximum and minimuin values of the function f(z, y) = zy on the elliptical disk D = {(2,y) € R? | 2? + 2y? < 1}. \yo wrdany & D MS ea 24 & \ <5 L RS | ede =| Note D is closed ond bovnded ad c is Conkinv ews, ge atlas a May, Q min, onl) Sep\ Check. for cc Kea| pene we Aa intenor ff B Ve = A) yee v rs ar oe ce Pez Pe (o, v) ‘s Ang, oy en keel peiak ot R eD to we must Cons ‘ber ih, Step “2 Checde Ang foovedary points VS Lejroang. moliplierg . man /min Loe aj 2 »\ s yer 4 he Con thrari nb along) = He Dy 2, Necessary condthwas, Ae Mais are r Vl, - > O%Gle Sees ~ (Zp, 4 4) @: c x bay af cet a OT Cae a (=) i 4 & wee 4x4 kms 5 2y ot Cathe elimina S) eae) \ ade | ce *F tte} ye St sy “Thus we ah & cardilale ersten |e leguderty ot, on Gon *, ? Me Level, m4: % a » p Cf Be) Pa (-s 4 “i AY Telsde of valves Cad hohe aa Phew) FM aoe Ps (a, ») ° Re, 4) Wo we bo Vl %& ; ~y Ker See ig Ye smaed valve. Pr ( “HD ~ he the isthe lar \ va Wwe. . \ ae I Pale 74) re. te ee 8 ttn i 1 . , wo . . , 1 \ x é ; are praximum yale ot a on Das * es Ye os atberned ch P, ( Kh: a) ; ° ee om om a4 Pe [-& et [Al mareinvnn velue HEAD Is: iS 5 5 oe Wikia allatned oR, oa 4) od Pal ~ 3 Fos SN set 6. (14 pts total) Suggestion for both parts: use Taylor series and O(-) notation. a) (7 pts) Show that — lim n(l + u) does NOT exist. (ey) (0,0) £2 + y? b) (7 pts) Show that — lim In(l + 2°y?) DOES exist. (z,y)7(0,0) oe + y¥ % 4 are a Crem Vor GoM, gets: In (tae 0-7 Beeb — ‘Anus sto) cippten’ Nehian w hv - Ot (a \s coarrer— Beh remebont (atte BA uv smell. 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