Department of Mathematics The City College of NY
Math 20300 Final Exam Spring 2006
No calculators permitted. Answers may be left in terms of radicals,
!
, e, etc. and do not
need to be simplified unless stated otherwise. Show all work.
Part I. Answer all 7 questions. Each is 10 points.
1.
Given the points
P
(1,1,0)
,
Q
(2,
"
1,2)
, and
R
(
"
2,2,1)
.
a)
Find the equation of the plane determined by the given points
P
,
Q
,
R
.
b)
Find the equation of the line, in symmetric form, which is perpendicular to
the plane in (a) at the point
P
.
c)
Find the area of the triangle
PQR
.
2.
Given that
u
=
f
(
x
,
y
,
z
)
=
x
3
y
+
y
2
e
z
.
a)
Evaluate
grad u
.
b)
Find the rate of change of
u
at
(1,1,0)
in the direction of
(3,
"
2,6)
.
c)
Find the equation of the tangent plane to the surface
x
3
y
+
y
2
e
z
=
2
at the
point
(1,1,0)
.
3.