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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

3tan(
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.
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 Spring '11
 Ocken
 Radicals

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