MAT 237 Y

MULTIVARIABLE CALCULUS

COURSE 2005-2006

 

 

SAMPLES OF PREVIOUS YEAR'S TESTS

 

TEST #2. COURSE 2004-2005.

 

1. (10 marks) Find and classify the critical points of the function

f ( x , y ) = x 3 + 3 x 2 + 6 x + y 2 y 3 x y  .

 

2. (10 marks) Show that the equation

F ( x , y , z ) = x e x + x y z + y 2 = 3

defines  z  as a function of  x  and  y  in a neighbourhood of the point  ( x , y , z ) = ( 2 , 1 , 0 )  and find

z x  and  z x y  at this point.

 

3. (15 marks) Mr. Worm lives in an apple occupying the region  { ( x , y , z ) x 2 + y 2 + z 2 ≤ 8 }  in space.

The temperature at a point  ( x , y , z )  in the apple is given by  T ( x , y , z ) = x 2 + y 2 2 x + 2 y + z 2 + 4  .

Mr. Worm lives at the coldest point in the apple and vacations at the hottest point in the apple. Where does Mr. Worm lives and where does he vacation?

 

4. (13 marks) Consider a particle travelling in space, with its position at time  t  given by

c ( t ) = t 3 i + t 2 j + k .

 

a) Find the distance the particle has travelled at time  t  (assuming it started at time zero).

b) What is the position of the particle after it has travelled    units of distance?

 

5. (15 marks)

a) If  r = x i + y j + z k  and  r = || r || 0 , then  div ( ln ( r ) r ) = a ln ( r ) + b  for some constants

a  and  b . Find  a  and  b .

b) Show that the vector field  F ( x , y , z ) = y z i + ( x z 1 ) j + x y k  is irrotational and find a potential function. Recall that a potential function for a vector field  F  is a real-valued function  f  such that  F = f .

 

6. (15 marks) Let  f : R R2 R , where  R = [ 0 , 1 ] [ 0 , 1 ] , be defined by

Evaluate  .

 

7. (12 marks) Find the volume of the portion of the first octant bounded by the planes  z = y  , x y = 2

and the surface  y =  .

 

8. Let  f : R R2 R , where  R = [ 1 , 1 ] [ 0 , 1 ] , be given by

a) (5 marks) Show that   exists and compute it.

 

b) (5 marks) Show that  f  is not integrable over  R .

If you wish, you may use the formula  .

 

 

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