Department of Mathematics
The City College of New York
Math 203
Final Exam
Fall 2006
No calculators permitted.
Answers may be left in terms of radicals,
π
,
e
, etc and do not
need to be simplified unless stated otherwise. Each question is worth 10 points.
Part I. Answer all 7 questions
1.
a) Compute an equation for the plane which contains the point (1,0,1) and the line given
parametrically by the equations
x
= 2
t
;
y
= 2 +
t
;
z
= 2

t
.
b) Compute the directional derivative of the function
F
(
x, y, z
) =
xz
2

(2
y

1)
2
at the
point (1
,
2
,
3) in the direction of the vector
<
1
,
0
,

1
>
.
2.
For parts (a) and (b) let
f
(
x, y
) =
e
x
y

3 tan(
x
).
a) Compute the unit vector pointing in the direction of greatest increase of the function
f
at the point (0
,

1) and compute the rate of increase in that direction.
b) Compute an equation for the plane tangent to the surface given by the equation
z
=
f
(
x, y
)
at the point in space with
x
= 0 and
y
=

1.
3.
Evaluate
Z Z
T
x
2
y dA
, where
T
is the first quadrant region bounded by the curve
with equations
y
=
x
3
and lines
y
= 8 and
y
= 8
x
. Include a labeled sketch of the region
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 Ocken
 Radicals, Power Series, Taylor Series, Mathematical analysis, ﬁrst quadrant region, compute parametric equations

Click to edit the document details