Math32a 2 R Kozhan Midterm2 Nov 19 2012 Name UID Circle your TA and discussion session 2A Tues Jordan Greenblatt 2C Tues William Rosenbaum 2E Tues Silas Richelson 2B Thur Jordan Greenblatt 2D Thur William Rosenbaum 2F Tues Silas Richelson 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 12 3 8 4 12 Total 43 1 Problem 1 a 2 points Suppose the set of all points x y that satisfy the equation F x y 0 is as on the picture Sketch the set of all points x y that satisfy the equation F x 2 y 0 b 3 points Find the center asymptotes and sketch the hyperbola x 2 2 y 1 2 4 4 c 2 points Complete the definition z f x y is called differentiable at a b if d 4 points Is the following statement necessarily TRUE of could it be FALSE No justification is needed i If f x y L as x y a b along every straight line through a b then lim f x y L x y a b ii If both fx a b and fy a b exist then f is differentiable at a b iii limy b f a y f a b y b fy a b iv If r t is the law of motion of a particle in space then r 0 t is its speed and r 0 t is its velocity 2 Problem 2 Compute the following limits or prove that they do not exist Justify all the steps a 4 points x y x y 0 0 x y lim b 4 points ex y x y 0 0 x y 1 lim 3 c 4 points lim x sin x2012 x y 0 0 4 1 y 2013 Problem 3 8 points Use linear approximation to approximate 3 8 04 8 94 Hint 3 x y 5 Problem 4 Consider the equation z 2 y x2 2x a 3 points Classify and sketch the surface b 3 points Find its center and the parametric equation of its axis of symmetry c 2 points Classify all horizontal traces of this surface i e traces parallel to the xy coordinate plane d 4 points Consider the horizontal trace of this surface that passes through the point 0 1 1 Find the parametric equation of the tangent line to this trace at the point 0 1 1 6 Extra scratch paper 7
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