Practice Exam1

Practice Exam1 - , 3 , 2). 6. Determine the points (if...

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Math 226 - Fall 2008 Exam 1 Practice Problems 1. Let P = (1 , 2 , - 1), Q = (4 , - 2 , 1) and R = (3 , 5 , - 2). (a) Find the equation of the plane containing the points P , Q and R . (b) Find the area of the triangle formed by the three points. (c) Find the orthogonal projection proj ~ PQ ( ~ PR ) of the vector ~ PR onto the vector ~ PQ . 2. Suppose ~u is a unit vector, and ~v and ~w are two more vectors that are not necessarily unit vectors. Simplify the following expression as much as possible: (( ~v · ~u ) ~u ) · ( ~v × ~w ) - ( ~w × ~v ) · ( ~v - ( ~u · ~v ) ~u ) 3. We say two planes are perpendicular if their normal vectors are perpendicular. Given the following two planes (which are not perpendicular): x + 2 y + 4 z = 1 x + y - 2 z = 5 Find the equation of a plane that is perpendicular to both of these planes, and that contains the point (3, 2, 1). 4. Let f ( x,y ) = ( x - y ) 3 +2 xy + x 2 - y . Find the linear approximation L ( x,y ) near the point (1 , 2). 5. Given the plane x + y + z = 1, find the point in the plane that is closest to the point P = (3
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Unformatted text preview: , 3 , 2). 6. Determine the points (if there are any) where the following functions are not continuous. Justify your answers. (a) f ( x,y ) = sin( x + y ) x-y (b) g ( x,y ) = 1 x 2 + y 2 + 1 (c) h ( x,y ) = ( xy x 2 + y 2 , ( x,y ) 6 = (0 , 0) , ( x,y ) = (0 , 0) 7. Explain why the limit of f ( x,y ) = 3 x 2 y 2 2 x 4 + y 4 does not exist as ( x,y ) approaches (0 , 0). 8. Two masses travel through space along space curve described by the two vector functions ~ r 1 ( t ) = < t, 1-t, 3 + t 2 > ~ r 2 ( s ) = < 3-s,s-2 ,s 2 > where t and s are two independent real parameters. (a) Show that the two space curves intersect by nding the point of intersection and the param-eter values where this occurs. (b) Find parametric equations for the tangent line to each of the two space curves at the inter-section point....
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