Name Section NetID ED1 Nathan Dunfield 8am ED2 Boonrod Yuttanan 8am 43 UID ED3 Ping Hu 9am ED4 Jeff Mudrock 9am ED5 Ping Hu 10am ED6 Boonrod Y 10am Instructions Take care to note that problems are not weighted equally Calculators books notes and suchlike aids to gracious living are not permitted Show all your work as credit will not be given for correct 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 1 2 3 4 5 Total Score Out of 5 11 10 11 6 43 1 Consider the function f x 3 y 3 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 0 5 1 1 0 5 1 point 0 5 0 5 1 0 5 0 5 0 5 0 5 1 1 0 5 0 5 0 5 0 5 1 0 5 0 5 0 5 0 5 1 1 2 Consider the function f R2 R given by f x y x 2 2x y 2 2y a Use Lagrange multipliers to find the max and min of f on the circle x 2 y 2 8 6 points b Consider the region D where x 2 y 2 8 Explain why f must have a global min and max on 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 by r t cos 2t sin 2t 9 t a Circle the correct sketch of C below b Compute the length of C for 0 t 2 2 points 5 points c Suppose C is made of material with density given by x y z x z Give a line integral R1 for the mass of C and reduce it to an ordinary definite integral something like 0 t 2 sin t dt 3 points 4 Let C be the curve parameterized by r t et t for 0 t 1 and consider the vector field F 1 2y a Circle the picture of F below 1 2 points 1 1 1 0 5 0 5 0 5 0 5 0 5 1 1 0 5 1 0 5 0 5 1 1 0 5 0 5 0 5 1 1 Z F dr b Directly compute 0 5 5 points C c The vector field F is conservative Find f R2 R so that f F d Use your answer in c to check your answer in b 2 points 2 points 1 5 Let C be the indicated portion of the ellipse x2 y 2 1 between A 0 1 and B 0 1 4 a Give a parameterization r of C indicating the domain so that it traces out precisely the segment indicated 3 points b Let L be the line segment joining B to A Give a parameterization f 0 1 R2 of L so that f 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 of R C g ds 1 point positive negative 0 g 3 g 4 g 5 g 6 g 7
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