1 Consider the vectors a h1 0 1i b h1 1 1i c h 1 1 0i a Compute a b 3 points a b D E b Compute the volume of the parallelepiped determined by the vectors a b and c 2 points Volume 2 Given that u and v in the picture at left have length 1 compute u v u w and projv w 1 point each w u v u v u w projv w 3 A particle moves with constant velocity h3 1 1i starting from the point 3 2 4 at time t 0 When and where will it cross the x y plane 3 points When t Where 4 Let A be the plane given by x z 1 and B the plane given by x y z 2 a Find a normal vector n for the plane A 1 points n D E b Find the angle between the two planes 2 points c Find the equation of a plane C which is perpendicular to both A and B 3 points Equation x y z 5 Exactly one of the following two limits exists Circle the one that exists and justify your answer 5 points x2 y 2 xy lim lim p x y 0 0 x y 0 0 x 2 y 2 2 x2 y 2 6 For each function label its graph from among the options below 3 points each a x y 2 b x cos y 7 Circle the equation for the quadratic surface shown at right 3 points a x 2 y 2 z 2 1 2 2 z 2 b x y z 1 c x 2 y 2 z 2 1 d x 2 y 2 z 2 1 e x y 2 z 2 1 x y 8 Is the function f R2 R given at right continuous at 0 0 Justify your answer 2 points x 2 y 1 if x y 6 0 0 f x y 0 if x y 0 0 9 Let f x y be a function with values and derivatives in the table Use linear approximation to estimate f 2 1 0 9 3 points x y f x y f x y x f x y y 1 3 2 1 2 4 3 6 0 2 3 1 4 1 7 3 4 3 7 5 f 2 1 0 9 10 Suppose f x y has the contour plot below right with points labelled Circle the best answer to each of the following questions 1 point each y 0 a f a is b f b is x 2 f c 2 c is y positive negative 0 positive negative 0 2 3 4 2 b 3 positive negative 0 c 5 d Circle one f f a b x x 1 1 f f a b x x 4 a x 11 An exceptionally tiny spaceship positioned as shown is travelling so that its x coordinate increases at a rate of 1 2 m s and y coordinate increases at a rate of 1 3 m s Use the Chain Rule to calculate the rate at which the distance between the spaceship and the point 0 0 is increasing 6 points y 4 3 2 1 x 1 2 3 4 5 Distances in meters Rocket courtesy of xkcd com rate m s 12 Let f R2 R be the function whose graph is shown at right a Find the equation of the tangent plane to the graph at 0 0 0 2 points z y 2 b The partial derivative positive negative f 0 0 is circle your answer x y 0 1 point x 13 Extra Credit Problem Suppose f R2 R is continuous at 0 0 with f 0 0 2 and partial derivatives f x 0 0 1 and f y 0 0 1 If in addition lim t 0 f t t 2 1 t can f be differentiable at 0 0 Carefully justify your answer 3 points Scratch work may go below and on the back of this sheet
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