M IDTERM 1 May 20 2002 Instructions Please show your work You will receive little or no credit for an answer not accompanied by appropriate explanations even if the answer is correct If you have a question about a particular problem please raise your hand and one of the proctors will come and talk to you Calculators or computers of any kind are not allowed You are not allowed to consult any other materials of any kind including books notes and your neighbors You will find a list of some useful formulas on page 2 of the exam At the end of the exam please hand the exam paper to your TA Please be prepared to show your university ID upon request If you have a question about the grading of a particular problem please come and see me or one of the TAs within 14 days of the exam Student ID Name Section 1 2 3 4 5 Total 2 cos2 t sin2 t 1 sin 2t 2 sin t cos t 2 t 1 cos t t sin 2 2 cos2 2t 1 cos 2 f x x y f x y f cos 2t cos2 t sin2 t lim x 0 x x Rb d L a k r kdt hf g hi hf 0 g 0 h0 i dt v w k v kk w k cos k v w k k v kk w k sin 3 Problem 1 Let f x y xy a Determine the domain and range of f b Sketch enough level lines of f to give an idea of how the level lines look like c Does the limit lim x y 0 0 f x y exist Solution a Since the formula defining f makes sense when y 6 0 the domain of f is all points x y for which y 6 0 In other words these are all points of the plane not lying on the x axis b The level line correspoding to the value c is the set of solutions to the equation f x y c Thus we solve x c y to get x cy Remembering that y 6 0 we find that the level lines are the pairs of rays eminating from the origin but not containing the origin with slopes 1 c for all c c Set x t y t Then t lim f x y lim 1 t 0 t 0 t Set instead x 0 y t Then 0 lim f x y lim 0 t 0 t 0 t So the limit does not exist 4 Problem 2 Evaluate the following limit or show that the limit does not exist sin x y lim x y 0 0 x y Solution Since sin t 1 t 0 t we know that given 0 there is a 0 so that if 0 t we have that sin t 1 t lim Thus if 0 x y we know that sin x y 1 x y The conditions of the definition of the limit are satisfied so we know that the limit exists and equals 1 5 Problem 3 A ball is thrown from the ground at the angle of 6 with the ground at the initial velocity of 10m s The acceleration of gravity is g 9 8m s2 Find how long it takes before the ball hits the ground again Solution Choose a coordinate system for which the ball is initially at the origin the x axis is parallel to the ground and the y axis is perpendicular to the ground pointing up Let r t denote the position of the ball at time t The acceleration of the ball is then r 00 t Since we are in a free fall the acceleration is equal to the acceleration of gravity which is a vector pointing downward and of magnitude g So we know that r 00 t g Integrating we get r 0 t v0 gt j 1 r t r0 v0 t gt2 j 2 The initial conditions imply that r0 0 and that v0 is the initial velocity Thus we know that the magnitude of v0 is 10 and that it makes the angle of 6 with the x axis Thus v0 h10 cos 6 10 sin 6i h5 3 5i It follows that 1 r t h5 3 t 5t gt2 i 2 1 2 The ball hits the ground again when 5t 2 gt 0 i e 5 12 gt so that t 10 g 1 02s 6 Problem 4 Sketch the curves obtained as intersections of the quadratic surface x2 3y 2 z 0 with the planes z const Name and sketch the quadratic surface Solution If z c we get the equation x2 3y 2 c For c 0 this is a hyperbola oriented along the y axis For c 0this is a hyperbola oriented along the x axis The surface is a hyperbolic parabaloid It is sketched on p 858 of the book 7 2 2 Problem 5 Let f x y e x y Compute the following partial derivatives of f f f 2 f x y y x Solution We have f f 2 2 2 2 2xe x y 2ye x y x y Also 2f 2 2 2 2 2 2 2xe x y 2 2y e x y 4xye x y y x y
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