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MATH 21 D
(Section A)
NAME:
Quarter
: Spring
SECTION:
Date:
June 08, 2002
1
2
3
4
5
total
6
Final
Problem 1.
(30 pts;
estimated time:
20 mn
)
Find the dimensions of the box of
largest volume
whose surface area is to be 6
square inches (see ﬁgure 1).
Problem 2.
(40 pts;
estimated time:
25 mn
)
Denote by
S
the surface of equation:
z
=

1
2
x
2
+
y
2
+ 2
x
+ 1
.
(a) Verify that the point (0
,
1
,
2) belongs to the surface
S
.
(b) Verify that the vector
i
+
j
+ 4
k
is tangent to the surface
S
at the point
(0
,
1
,
2).
(c) Find the parametric equations of the
vertical plane
passing through the
point (0
,
1
,
2) and containing the vector
i
+
j
+ 4
k
.
(d) Find the parametric equations of a
curve drawn on the surface
S
whose
tangent at the point (0
,
1
,
2) is the vector
i
+
j
+ 4
k
.
Problem 3.
(40 pts;
estimated time:
25 mn
)
(a) Show that the curve parameterized by
G
(
t
) =
e
t
+
e

t
2
i
+
e
t

e

t
2
j
= cosh
t
i
+ sinh
t
j
,
for 0
≤
t
≤
a
, with
a >
0, lies on the parabola
x
2

y
2
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 Spring '07
 DianwenZhu

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