(a)
div
F
=
(b)
curl
F
=
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Final Exam/MAC2313 Page 7 of 8
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13. (10 pts.) (a)
Obtain an equation for the plane that is tangent to the graph of the function
f
(
x
,
y
) =
x
2
+
y
2
at the point (1,2,5) in 3-space.
(b)
Does the plane in part (a) that is tangent at (1,2,5) to the graph of
f
(
x
,
y
) =
x
2
+
y
2
intersect the yz-plane? If the answer is "yes", obtain a set of parametric equations in 3-space for the line of
intersection.
______________________________________________________________________
14. (10 pts.)
Evaluate the surface integral,
⌡
⌠
⌡
⌠
σ
f
(
x
,
y
,
z
)
dS
where
f
(
x
,
y
,
z
) =
x
-
y
-
z
and
σ
is the portion of the plane
x
+
y
= 1 in the first octant between
z
= 0 and
z
= 1.
⌡
⌠
⌡
⌠
σ
f
(
x
,
y
,
z
)
dS
Name:
Final Exam/MAC2313 Page 8 of 8
______________________________________________________________________
15. (10 pts.)
Let R be the triangular region enclosed by the lines
y
= 0,
y
=
x
, and
x
+
y
=
π
/4. Use a suitable transformation to compute the following integral:
⌡
⌠
⌡
⌠
R
sin(
x
y
)
cos(
x
y
)
dA
______________________________________________________________________
Silly 10 Point Bonus:
It turns out that routine computations reveal that
⌡
⌠
∞
0
e
xy
dy
1
x
for
x
> 0 and
⌡
⌠
∞
0
sin(
x
)
e
xy
dx
1
1
y
2
for
y
> 0. One interesting consequence is that, although there is no elementary antiderivative for the function
f
(
x
) =
sin(
x
)/
x
, one can completely evaluate the following improper integral:
⌡
⌠
∞
0
sin(
x
)
x
dx
.
Show how this is done.
Note: There is no problem with
f
at
x
= 0 since we "plug the hole" by setting
f
(0) = 1, the obvious limit value. Say where
your work is for it won’t fit here.
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- Spring '06
- GRANTCHAROV
- Multivariable Calculus, Vector Calculus, Line integral, Vector field, Stokes' theorem
-
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