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 y – z
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 y
– y 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 2
– y 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 , y )
in the direction of the unit normal vector
n .
Show that
.
Hint: Compare
Ñ
× ( f
Ñ
f ) and f
Ñ
2 f .