MAT 237 Y

MULTIVARIABLE CALCULUS

COURSE 2005-2006

SAMPLES OF PREVIOUS YEAR'S

FINAL EXAMS TESTS

COURSE 2004-2005.

1. a) (5 marks) Show that   f ( x , y ) =    is not continuous at  ( 0 , 0 ) .

b) (7 marks) Let  r = x i + y j + z kr =   and  w = r 3 .

Find the value of the constant  k , if any, for which  Ρ 2 w = k r .

2. a) (6 marks) Suppose that the position of a bug moving in space at any time  t ³ 0  is given by the path

c ( t ) = ( 2 t  3 , t 2  2 , 7  t 2 ) . Suppose also that the temperature at any point  ( x , y , z )  is given

by the function  T ( x , y , z ) = x z  y + z .

What is the rate of change of temperature that the bug experiences at the point  ( 1 , 2 , 3 ) ?

b) (7 marks) Suppose that the equation  z 3 = 2 + x 2   3 y + z  is used to implicitly defines  z  as a

function of  ( x , y )  near  ( 1 , 1 , 1 ) . Compute    at  ( x , y ) = ( 1 , 1 ) .

3.  (10 marks) Find all the critical points of the function  f ( x , y ) = x 3  4 x y + 2 y 2 + y + 5  and use

the second-derivative test to classify each of them as a local maximum, a local minimum or a saddle point.

4. (10 marks) Determine the absolute minimum and absolute maximum values for   f ( x , y ) = x 2  x + 3 y 2

on the elliptic region  x 2 + 2 y 2 ≤ 4 . Find the coordinates of all the points on the given region where the

function reaches its absolute maximum or its absolute minimum.

5. Evaluate each of the following integrals:

a) (7 marks)   .

b) (8 marks)   .

6.   (15 marks) Let  F ( x , y , z ) = y i  2 x z j + x y k  and let  S  be the part of the parabolic cylinder

z = y 2  that lies inside the elliptic cylinder  x 2 + 4 y 2 = 4 .

Verify Stokes theorem for the  vector field  F  and the surface  S .

7.   (15 marks) Let  W  be the solid region consisting of all points  ( x , y , z )  for which  x 2 + y 2 ≤ 4 ,

x ³ 1  and  0 ≤ zy . Let  Ά W  denote the closed surface that bounds the region  W , oriented by the

outward pointing normal vector. Suppose that  g = div F , where  F  denotes any  C 1 vector field such

that  g ( x , y , z ) ≤ 6  for every  point  ( x , y , z )  in the region  W . What is the minimum value of the

constant  M  for which  M ?

8.   (10 marks) Let  d ( x , y ) = x 2  be the density at any point  ( x , y )  of the region  D Μ R 2 .

Suppose that the boundary of the region  D  is a simple closed curve  C , positively oriented and such

that   , for any two positive integer numbers  m ≤ 2  and  n ≤ 2 .

Find the coordinates  of the centroid of the region  D .

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