A div f b curl f name final exammac2313 page 7 of 8

(a)
div F =
(b)
curl F =

Name: 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.
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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
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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
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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.