MAT 237 Y

MULTIVARIABLE CALCULUS

COURSE 2005-2006

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  2y  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 2z 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 , tt 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 kr =   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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