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 k ,  r =   and  w = r ³ .

Find the value of the constant  k , if any, for which  Ñ 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 ≤ z ≤ y . Let  ¶ 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 Ì 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 .