Sample Exercises for the Final Examination
1.
Given the surface
,
)
(
2
/
)
(
2
2
2
2
y
x
e
y
x
z
−
−
−
=
(a)
write the equation of the tangent plane at an arbitrary point (x,y,z) on the
surface.
(b)
find the maxima, minima and the saddle points of the surface and sketch
the surface.
(c)
using a double integral, compute the volume under the surface and over
the square formed by the points (-2,-2), (2,-2), (2,2) and (-2,2).
2.
(a)
Compute using cylindrical or spherical coordinates, the volume of the
solid enclosed by the cones
2
2
2
2
2
1
z
and
y
x
y
x
z
+
−
=
+
=
.
Hint: divide the solid in two regions.
(b)
Use the divergence theorem to find the outward flux of the vector field
F
=x
2
i
+y
2
j
+z
2
k
across the pyramid with basis the square formed by the
points (0,0,0), (2,0,0), (2,2,0) and (0,2,0) and vertex at (1,1,2).
3.
Show that the line integral
∫
−
+
)
,
1
(
)
0
,
1
(
cos
sin
π
ydy
x
ydx
is independent of the path and evaluate the integral by:
(a)
finding a potential function for the integrand (this means finding