thq_08-key

thq_08-key - Exam Course Due Date Instructor 16s MATH 2451...

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Exam : 16s MATH 2451 thq 08 Course : Multivariable Calculus Due Date : 2016-03-08 (Tue) Instructor : McCary T.C. Use (First Name) (Last Name) 2016-03-01 16:10

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1. (10 points) Show that the set X = �� x y R 2 x + x 2 + y 2 = 2 is a smooth curve, using the following procedure: fi nd a C 1 function f : R 2 R such that X is a level set of f and [D f ] is onto.
2. (10 points) Let X a = x y z R 3 x 2 + y 3 + z = a and Y b = x y z R 3 x + y + z = b . (a) Show X a and Y b are smooth curves (same procedure as previous exercise). (b) Determine when X a Y b is a smooth curve, using the following procedure: fi nd a function F : R 3 R 2 such that X a Y b is a level set of F and show [D F ] is onto. Hint: if you used f 1 : R 3 R and f 2 : R 3 R in the previous part then you can use F = f 1 f 2 for this part (think about why!).

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3. (10 points) Let X c be the locus of the equation x 2 + y 3 = c .
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