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MATH 203 Final Exam
May 23, 2007
PART I: Answer all parts of Questions 18 (points as indicated).
Question 1
(8 points) Given the two lines
x

2
3
=
y
4
=
z
+ 1
5
and
x
+ 2
6
=
y

1
8
=
z
10
(a) Are they parallel? Explain.
(b) Find an equation of the plane containing the two lines.
Question 2
(8 points) (a) Find an equation of the tangent plane at (2
,
1
,
3) to the
surface
z
=
xy
+
√
y
.
(b) Find the parametric equations of the normal line; that is, the line perpendicular to
the tangent plane, passing through (2
,
1
,
3).
Question 3
(8 points) Consider the function
T
(
x,y,z
) =
9
1 +
x
2
+
y
2

3
z
.
(a) Find the rate of change of
T
at the point (1
,
1
,
1) in the direction towards the point
(11
,
21
,
6).
(b) In what direction does
T
increase most rapidly at the point (1
,
1
,
1)?
(c) At what rate is
T
increasing in the direction given by the answer to part (b)?
Question 4
(8 points) Let
f
(
x,y,z
) =
√
x
yz
2
. Use linear approximation (diﬀerentials) to
approximate
f
(
.
98
,
1
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 Spring '11
 Ocken
 Math

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