MATH 201 Exam 2, Fall 2019

Problem 1 a. (8 pts) Find a polar equation for the circle (x-1P +y’=1. (x -4] Pyle f¥ ~ NCO RY=0 EL. RN-lea Soe 7 Vide = &(2/0(%e) . fo (rel = (0, cihy,he2] c= 0 . SeO Earlene. 2 Sarre loa Gu O praber ofl of ewok (a4/ ty os? b. (8 pts) Find a Cartesian equation for the curve r=~——_—,__- 3cos0-2sin® Identify the curve. x =: the 2ceo G-lane Got S$nee ~Irane= J av Sh Se &~Umo~o S coe Of La Gack z ots aD ao \ Brame flame Bat dy 3a-1 £0 Oh Arla ee Eee LN Problem 2 Consider the function f(x, y)=9-x? - y” a. (6 pts) Find the domain and range of the functi Rly) % dered Bor J. a?-y2> 6 b. (6 pts) Determine if the domain of f is an open region, a closed region, or neither and decide if the domain of f is pounded or unbounded. 1 fo haiti dh domven WN om anck eobett pei is ) 4 2 . re A « . a6 y PRS Orie f hy r i a\ ; f ; f . } ; j A a a a) d. (6 pts) The plane x = 2 intersects the graph of fat a curve C. Find the slope of the line L tangent to C in the plane x = 2 at the point (2,1,2). Tot bey %) = Yn Ayr} T ktongat Pome b(sabetites VE] TA uve Lt nyt He Cy oe CVE) 28 | YU) és L CG Nave a= of merndd lav E(4oo So t. ded ry So Khe churrdiond spelen i ofr VE ley mall ~- i - 4 : + } mM aSg-ayt 99-5 ~1.44 WF boyy O° af : ox (3,44) “yeah Be Nye pope Problem 3 a. (8 pts) Does x’y lim &.») > 0.0) x4 +|y| exist? Justify your answer. 06 fe fete | C fie bee KS lu]] ~ (Ads) (y] OCD. a! < Dx. ala alte, Aly) — ey fag (“J be (e o) Problem 3 a. (8 pts) Does 2 K x lim : = (x,y) > (0,0) x" + |y| exist? Justify your answer. of b.. ies CS Yim = [a] (aPC Yat ely]] “(4a ss) (gy) i | 0) le Ga | Cal, «) aly] ~ Malfos) (at of Me Ally) <2 On <a 2 (ay) 2©) ah (y | (44/4/00) e G Cf QQ. allyl <e (Ag) $60] aksly) Cy cll Brom b. (8 pts) What about lim ri 2 (x,y) > (0,0) ¥ +Y Justify your answer. je whe nek 2 ae ."s~. Problem 4 (14 pts) Find the tangent plane and normal line of the surface 5x? +4e” =12-In(z>x) at the point (1,0,e). P (Ace ) é "4gd aot 4.0, (g 3a} <0 WILE om Yo _ xh = lo \ Fy yh = L3f¥— 3 : 3 dX eA x3 x Ni ‘ ‘13 ~ 1044 44 dA ee) ( ‘ - ok | ~&e? <4 dm (A,oe) , Fad ap < 14, & 3 : i> Whey CYS) is, n VE hey ) Mle morn tabie BP Oe Ainge pl k at l4 re} 4(0-y/B) => pa.vg] x0 Problem 5 (14 pts) Let f(x, y,z) be a differentiable function of three variables. Suppose that £(8,3,4)=4, V/(8,3,4)=4i-3j-7k, ind 28 an Oat (r,s) = (2,4). a “¥ or Os ee a - —— ; Problem 6 2 (18 pts) Consider the function (x)= (i - *) Find the Taylor polynomials p,(x) and p>(x) generated by f at the center a= 0. Then use Taylor’s theorem to estimate the error resulting from the approximation f (x) = p, (x) for O<x<0.1. oe Ua)” atten oe ~¥, Pol). ete : .3 ? A“(als =r (4-2) - -( ta | a |. : Leb <n hacks (af ob asO = o Pe 4™ Roo nm ( fF Oty on F hw lina & ~ =| a: “i as a Z, (mey- (ae