# Name final exammac2313 page 5 of 7 9 12 pts let

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Name: Final Exam/MAC2313 Page 5 of 7 ______________________________________________________________________ 9. (12 pts.) Let . Compute the divergence and the curl of the vector field F . (a) div F = (b) curl F = ______________________________________________________________________ 10. (16 pts.) Use the substitution u = x + y , v = x - y to evaluate the integral where R is the region enclosed by the lines x + y = 0, x + y = 1, x - y = 1, and x - y = 4

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Name: Final Exam/MAC2313 Page 6 of 7 ______________________________________________________________________ 11. (12 pts.) A particle, starting at (0,0) moves to the point (1,1) along the parabola y = x 2 and then returns to (0,0) along the parabola x = y 2 . Use Green’s Theorem to compute the work done on the particle by the force field F ( x , y ) 8 xy ,4 x 2 10 x . ______________________________________________________________________ 12. (16 pts.) Evaluate the surface integral, where f ( x , y , z ) = x 2 z and σ is the portion of the cone z = ( x 2 + y 2 ) 1/2 between between the planes z = 1 and z = 2.
Name: Final Exam/MAC2313 Page 7 of 7 ______________________________________________________________________ 13. (12 pts.) (a) If f is differentiable at ( x 0 , y 0 ), then the local linear approximation to f at ( x 0 , y 0 ) is (b) Assume f (1,-2) = 4 and f ( x , y ) is differentiable at (1,-2) with f x (1,-2) = 2 and f y (1,-2) = -3. Using an appropriate local linear approximation, estimate the value of f (0.9,-1.950). f (0.9,-1.950) ______________________________________________________________________ 14. (18 pts.) Compute the unit vectors T (t) and N (t) and the curvature κ (t) for the "hot-rod" helix defined by ______________________________________________________________________ Silly 10 Point Bonus: Show how to use the Fundamental Theorem of Calculus to prove the Fundamental Theorem of Line Integrals. Say where your work is, for it won’t fit here. [If you intend to seriously attempt this, you might begin by actually stating the Fundamental Theorem of Line Integrals.]
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