MAT 237 Y

MULTIVARIABLE CALCULUS

COURSE 2005-2006

SAMPLES OF PREVIOUS YEAR'S

FINAL EXAMS TESTS

COURSE 2003-2004.

1. a) (5 marks) Let  f ( x , y ) =  .

Find the value of  a , if any, for which the function  f  is continuous at  ( 0 , 0 ) .

b) (5 marks) Let  g ( x , y ) =  .

Compute   .

2. a) (5 marks) Find an equation of the plane that is tangent to the surface  x z – 2 yz  2 = 3

and is perpendicular to the line  ( x , y , z ) = ( 1 – 2 t , 3 + 4 t , 5 ) .

b) (10 marks) Let  f ( x , y ) = ( 1 + x , 2 + y 2 , 3 x 2 + y ) ,  g ( u , v , w ) = ( v w , u 2 + v ) ,

h = and  A = D h ( 1 , 1 ) . Compute  det A .

3. (15 marks) Find the absolute minimum and the absolute  maximum of   f ( x , y , z ) = 3 x – 6 y + 2 z

under the constraint  3 ( x – 1 ) 2 + 2 ( y + 1 ) 2 + z 2 = 25 .

4. (10 marks) Calculate the total mass of the solid region in the first octant, bounded by the surfaces

x = 0 ,  z = 0 ,  y = 2 x  and  z = 2 yy 2  if the density is  d ( x , y , z ) = x y .

5. (10 marks) Calculate the average value of the function  f ( x , y , z ) = x y + z  along the path

parametrized by  c ( t ) = ( 2 t , 3 t 2 , 3 t 3 ) ,  0 £ t £ 1 .

6.   (15 marks) Verify Stokes’ theorem for the surface  S = { ( x , y , z ) ½ z = 8 – 2 x 2y 2  and  z ³ y 2 }

and the vector field  F ( x , y , z ) = y i + z j + x k .

7.   (15 marks) Calculate the flux of the vector field  F ( x , y , z ) = 2 x y 2 i + 4 y z 2 j + ( x – 1 ) 2 z k

out of the ellipsoid  ( x – 1 ) 2  + 2 y  2 + 4 z 2 = 4 .

8.   (10 marks) Let  D Ì R 2  be a region to which Green’s theorem applies and let  D  be its boundary,

positively oriented. Let  n  denote the outward pointing unit normal vector to the curve  D .

Suppose that  f : D ® R  is a  C 2  function and that  g ( x , y )  denotes the directional derivative of the

function  f  at the point  ( x , yin the direction of the unit normal vector  n .

Show that   . Hint: Compare  Ñ × ( f Ñ f )  and  f Ñ 2 f  .

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