MAT 237 Y

MULTIVARIABLE CALCULUS

COURSE 2005-2006

 

 

SAMPLES OF PREVIOUS YEAR'S TESTS

 

TEST #2. COURSE 2003-2004.

 

 

1. (15 marks) Find all the critical points of the function  f ( x , y ) = x 2 y + 2 x y 2 4 x y 6  and use the second-derivative test to classify each of these critical points as a local minimum, a local maximum or a saddle point.

 

2. (15 marks) Find the absolute minimum and maximum values of the function  f ( x , y , z ) = 2 x y + z 3

over the region  x 2 + 2 y 2 + 3 z 2 12 ,  z 0 .

 

3. (10 marks) Show that the system     can be solved for  u  and  v

 

as functions of  x  and  y  near  ( u , v ) = ( 1 , 2 ) ,  ( x , y ) = ( 3 , 4 ) .

Find the value of    when  u = 1 ,  v = 2 ,  x = 3  and  y = 4 .

 

4. (10 marks) Let  C  denote the curve of intersection of the surfaces  x 2 + y = 1  and  2 x 3 3 z = 27 .

Compute the length of the curve  C  from the point  ( 0 , 1 , 9 )  to the point  ( 3 , 8 , 9 ) .

 

5. (10 marks) Let  F ( x , y , z ) = .

Let  G = curl Ff = F G  and  g = 2 f . Compute  g ( 1 / 2 , 1 / 2 , 1 / 2 ) .

 

6. (15 marks) Change the order of integration and evaluate the integral   .  

 

7. (15 marks) Evaluate the triple integral  , where  W  is the solid region in the first octant

( x 0 ,  y 0  and  z 0 )  bounded by the surfaces  y = 3 x z 2y = 2 x  and  x = 0 .

 

8. (10 marks) Let  f ( t ) =  . Compute  f  ( 1 ) ,  f  ( 1 )  and  f  ( 1 ) .

    

 

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