Math32a 1 R Kozhan Midterm 2 Nov 21 2011 Name UID Circle your TA and discussion session 1A Tues Mike O Brien 1C Tues Jeff Lin 1E Tues Jordy Greenblatt 1B Thur Mike O Brien 1D Thur Jeff Lin 1F Thur Jordy Greenblatt Instructions If you get stuck move on to the next question You don t have a lot of time Show all work if you want to get full credit I reserve the right to take off points if I cannot see how you arrived at your answer even if your final answer is correct No books notes electronics incl calculators and cell phones are allowed Good luck Question Max Your score 1 6 2 14 3 10 4 10 5 10 Total 50 1 Problem 1 a 2 points Sketch the curve y 2 9x2 9 0 in R2 b 2 points Write the definition of the partial derivative fx a b of a function f at a point a b c 2 points Write what it means for a function of two variables f to be differentiable at a point a b 2 Problem 2 If you don t know how to solve part of this problem you may skip it and solve the other parts a 3 points Classify and sketch very roughly the surface given by the equation x2 2y 2 z 2 2x 6z 12 0 b 2 points Find the parametric equation of its axis of symmetry c 3 points Describe all possible traces of this surface in planes parallel to xz plane d 4 points Viewing z as a function of independent variables x and y find the derivative z z and y at the point 1 3 1 x e 2 points Find the tangent plane at the point 1 3 1 3 Problem 3 10 points If you weigh a solid body in air m1 and when submerged in water m2 then basic Archimedean principle tells you that the volume of the body is m1 m2 and therefore its density is 1 s m1m m 2 Suppose you obtained measurements m1 100 gram and m2 90 gram and you know that the maximal error is no more than 0 01 in absolute value when measuring m1 and no more than 0 02 in absolute value when measuring m2 Estimate the maximal error of s 4 Problem 4 a 5 points Does there exist a number c that makes the following function continuous everywhere on R2 Justify your answer 2 x2 y if x y 6 0 0 2 x 1 y 2 1 f x y c if x y 0 0 b 5 points Find the limit if it exists or show that the limit does not exist x2 y 1006 x y 0 0 x4 y 2012 lim Hint does this function remind you xy x2 y 2 5 Problem 5 10 points Suppose that w f x y where x uv y 2u 3v Find 2w u v in terms of partial derivatives of w with respect to x and y 6 Extra scratch paper 7
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