Math32a 1 R Kozhan Midterm 2 Mar 09 2012 Name UID Circle your TA and discussion session 1A Tues Anthony Ruozzi 1C Tues Xiaokui Yang 1B Thur Anthony Ruozzi 1D Thur Xiaokui Yang 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 You have an extra scratch paper on the last page If you need more raise your hand and ask No books notes electronics incl calculators and cell phones are allowed Good luck and have fun Question Max Your score 1 11 2 15 3 12 4 12 Total 50 1 Problem 1 a 3 points Classify the surface and find parametric equation of its line of symmetry x2 y 2 z 2 2x 2y 0 b 3 points Find an equation of the ellipse that lies in second quadrant and touches x axis at 3 0 and y axis at 0 1 see the picture c 3 points Suppose r 0 t i 2 j 10t k and r 1 4 i 4 j Find r t for any t d 2 points State Clairaut s theorem with all the conditions 2 Problem 2 Note parts a b aren t required to solve parts c d Consider a particle moving along the following curve r t h3 sin t 5 cos t 4 sin ti a 3 points Find the length of the curve between times t 4 and t 2 Hint sin 2s 2 sin s cos s in case you need it b 5 points Find the arc length function s t and reparametrize the curve with respect to arc length i e find natural parametrization as I called it in class starting from t 0 c 2 points Write the parametric equation of the tangent line to the curve at time t 2 d 5 points Suppose another particle is constrained to move along the same trajectory for the initial 2 seconds but at t 2 it is released from the constraint and continues moving along the tangent line Assume that the speed of the particle when it moves along the tangent line is constant and equal to the speed at the moment of release Find the coordinates of the particle at time t 3 Problem 3 Let z f x y where f x y x3 y xex 1 y 2 a 2 points Compute partial derivatives fx x y and fy x y b 2 points Is f x y differentiable at 1 1 Justify your answer Hint if your justification is long then you re doing it wrong c 5 points Find the differential dz of z at 1 1 Find the linearization function L x y of f x y at 1 1 Find the equation of the tangent plane of z f x y at 1 1 2 These three questions are of course very closely related to each other d 3 points Estimate the change in z when x y changes from 1 1 to 0 98 0 98 4 Problem 4 a 1 point What is the natural maximal domain of the function x sin x2 y 2 x2 y 2 b 3 points What is the domain of continuity of this function Carefully justify your answer c 4 points Compute the following limit or prove that it doesn t exist p x sin x2 y 2 lim x2 y 2 x y 0 0 d 4 points Compute the following limit or prove that it doesn t exist p x2 sin x2 y 2 lim x2 y 2 x y 0 0 5 Extra scratch paper 6
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