Name: NetID: UID: /43Section:ED1: Nathan Dunfield (8am) ED3: Ping Hu (9am) ED5: Ping Hu (10am)ED2: Boonrod Yuttanan (8am) ED4: Jeff Mudrock (9am) ED6: Boonrod Y. (10am)Instructions: Take care to note that problems are not weighted equally. Calculators, books, notes, andsuchlike aids to gracious living are not permitted. Show all your work as credit will not be given forcorrect answers without proper justification, except for the “circle your answer” questions.Important note: There are problems on the back of each sheet.Scratch Space: Below.Good luck!Problem Score Out of1 52 113 104 115 6Total 431. Consider the function f = x3+ y3+ 3xy.(a) It turns out the critical points of f are (0, 0) and (−1, −1). Classify them into local mins,local maxes, and saddles. (4 points)(b) Based on your answer in (a), circle the correct contour diagram of f . (1 point)-1-0.5 0.5-1-0.50.5-1-0.5 0.5-1-0.50.5-1-0.5 0.5-1-0.50.5-1-0.5 0.5-1-0.50.52. Consider the function f : R2→ R given by f (x, y) = x2− 2x + y2− 2y.(a) Use Lagrange multipliers to find the max and min of f on the circle x2+ y2= 8. (6 points)(b) Consider the region D where x2+ y2≤ 8. Explain why f must have a global min and maxon D. (2 points)(c) Find the global min and max of f on D. (3 points)3. Let C be the portion of a helix parameterized byr(t) =cos(2t), − sin(2t), 9 − tfor 0 ≤ t ≤ 2π.(a) Circle the correct sketch of C below: (2 points)(b) Compute the length of C. (5 points)(c) Suppose C is made of material with density given by ρ(x, y, z) = x + z. Give a line integralfor the mass of C, and reduce it to an ordinary definite integral (something likeR10t2sin t dt).(3 points)4. Let C be the curve parameterized by r(t) = (et, t) for 0 ≤ t ≤ 1, and consider the vector fieldF = (1, 2y).(a) Circle the picture of F below: (2 points)-1-0.5 0.51-1-0.50.51-1-0.5 0.51-1-0.50.51-1-0.5 0.51-1-0.50.51(b) Directly computeZCF · dr. (5 points)(c) The vector field F is conservative. Find f : R2→ R so that ∇f = F. (2 points)(d) Use your answer in (c) to check your answer in (b). (2 points)5. Let C be the indicated portion of the ellipsex24+ y2= 1 between A = (0, −1) and B = (0, 1).(a) Give a parameterization r of C, indicating the domain so that it traces out precisely thesegment indicated. (3 points)(b) Let L be the line segment joining B to A. Give a parameterization f: [0, 1] → R2of L so thatf(0) = B and f(1) = A. (2 points)(c) Suppose g : R2→ R is a function whose level sets are indicated below. Circle the sign ofRCg ds (1 point)positive negative 0g = 3g = 4g = 5g = 6g =
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