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

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(a) div F = (b) curl F =
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Name: Final Exam/MAC2313 Page 7 of 8 ______________________________________________________________________ 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
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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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