MAT 237 Y MULTIVARIABLE CALCULUS COURSE 2005-2006 |
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SAMPLES OF PREVIOUS YEAR'S TESTS
TEST #1. COURSE 2004-2005.
1. Consider the points A ( 1 , 1 , 2 ) , B ( 3 , 2 , 0 ) , C ( 0 , 2 , 1 ) and D ( 5 , 6 , 7 ) . Let L be the line that passes through the points A and B . Let S be the plane that passes through the points A , B , and C . a) (10 marks) Find the coordinates of all the points P on the line L , if any, for which . b) (10 marks) Find the coordinates of the point E such that the points E and D are symmetric with respect to the plane S . Note: Two points are said to be symmetric with respect to a plane if the plane is perpendicular to and bisects the segment joining the two points.
2. a) (10 marks) Let V denote the volume of the parallelepiped generated by the vectors a = ( a , – 1 , 1 ) , b = ( b , 0 , – 2 ) and c = ( c , 3 , 0 ) , where a 2 + b 2 + c 2 = 4 . Show that V ≤ 14 . Hint: Use the Cauchy-Schwarz inequality. b) (10 marks) Let S be the surface described in Cartesian coordinates as the set consisting of all points ( x , y , z ) such that x 2 + y 2 + ( z – 1 ) 2 = 1 with y ≥ 0 and z ≥ 1 . Describe the surface S using spherical coordinates.
3. a) (10 marks) Consider the function .
Find the values of k , if any, for which this function is continuous at ( 0 , 1 ) . b) (10 marks) Let f : A Ì R 2 ® R 3 , where A is an open set and let ( 1 , 2 ) be in A or be a boundary point of A . Use d ¢s and e ¢s to describe the meaning of the equation .
4. a) (10 marks) Let f ( u , v , w ) = ( sin u + sin ( π v ) + w , 3 u + v + w 2 ) . Suppose that g : R 2 ® R 3 is a function such that g ( 1 , 2 ) = ( 0 , 1 , – 1 ) and D g ( 1 , 2 ) = . Compute D ( f ë g ) ( 1 , 2 ) .
b) (10 marks) Let P be the plane that is tangent to the surface x 2 + 2 y 2 – z 2 + x y z = 3 at the point ( 1 , – 1 , 0 ) . Find the coordinates of all the points Q in the path c ( t ) = ( 1 + t , t + t 2 , 2 t 3 ) , if any, for which the tangent line to the path c ( t ) at the point Q is parallel to the plane P .
5. a) (10 marks) Let r = x i + y j + z k , r = and w = r 3 . Prove or disprove: = 3 r 2 . b) (10 marks) Let f ( x , y ) = . Compute the directional derivative of the function f at the point ( – 1 , 2 ) in the direction of the unit vector ( 1 , 2 ) .
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