MAT 237 Y

MULTIVARIABLE CALCULUS

COURSE 2005-2006

SAMPLES OF PREVIOUS YEAR'S TESTS

TEST #1. COURSE 2002-2003.

1. a)  (10 marks) Find an equation for the plane that passes through the point  A ( 1 , 2 , 3 )  and contains the line  ( x , y , z ) = ( 1 , 0 , 1 ) + t ( 1 , 1 , 1 ).

b)   (10 marks) Let  l1  be the line that passes through the origin and through the point with spherical coordinates  ( r , q , f ) = ( 2 , 0 , p / 4 ) , and let  l2  be the line that passes through the origin and through the point with spherical coordinates  ( r , q , f ) = ( 2 , p / 4 , p / 2 ) . Find the angle between the lines  l1  and  l2  .

2.   a)  (10 marks) Find the distance from the point  P ( 0 , - 2 , 6 )  to the plane that passes through the points  A ( 1 , 0 , - 1 ) ,  B ( 1 , 2 , 0 ) , and  C ( 0 , 2 , - 1 ) .

# b) ( 10 marks) Find the coordinates of the point of the line  ( x , y , z ) = ( 1 , - 2 , - 2 ) + t ( 1 , 1 , 0 )  which is closest to the line  ( x , y , z ) = ( 1 , 3 , 2 ) + t ( 1 , 0 , 1 ) .

3. a)  (10 marks) Consider the function  . Show that   does not exist.

b) (10 marks) Consider the function   . Compute   .

4. a)  (10 marks) Find an equation of the plane that is tangent to the surface   ( 2 xy ) 3 + ( 2 yz ) 3 = 2  at the point  ( 1 , 1 , 1 ) .

b) (10 marks) Let   f ( x , y ) = ( 3 yx , 2 yx y , y 2 ) , and let  D g ( u , v , w ) =   .

Compute   D ( g o f ) ( 2 , 1 ) .

5. a)  (10 marks) Suppose that a particle follows the path  c ( t ) = ( e 1 - t , e t - 1 3 , t 2 )  until it flies off on a tangent at  t = 1 . Determine the position of the particle at  t = 3 .

b) (10 marks) Let   f : R 2 ® R   denote a function which is differentiable at the point   P Î R 2 . Suppose that it is known that the directional derivative of the function   f   at the point   P   along the vector   ( 4 / 5 , 3 / 5 )   is   1 , and the directional derivative of the function   f   at the point   P   along the vector   ( 3 / 5 , - 4 / 5 )   is   2 . Give the unit vector   u   that indicates the direction of fastest increase of the given function   f   at the point   P .

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