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8 15 pts a 12 pts compute the unit vectors t t and n

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______________________________________________________________________ 8. (15 pts.) (a) ( 12 pts.) Compute the unit vectors T (t) and N (t) and the curvature κ (t) for the helix defined by r (t) = < cos(t), t , sin(t) >. (b) (3 pts.) Locate the center, C(x 0 ,y 0 ,z 0 ), of the circle of curvature when t = π /2.
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Name: Final Exam/MAC2313 Page 5 of 8 ______________________________________________________________________ 9. (15 pts.) Obtain a set of parametric equations for the line through the point (0, 0, 0) that is perpendicular to the plane that contains the three points (1,0,0), (0,2,0), and (0,0,3). Then locate the point in 3-space where the line intersects the plane. ______________________________________________________________________ 10. (15 pts.) Locate and classify the critical points of the function f ( x , y ) = x 3 - 3 xy - y 3 . Use the second partials test in making your classification. (Fill in the table below after you locate all the critical points. It always helps to factor. How does a product equal zero??) Crit.Pt. f xx @ cp f yy @ cp f xy @ cp D @ cp Conclusion
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Name: Final Exam/MAC2313 Page 6 of 8 ______________________________________________________________________ 11. (10 pts.) Let F ( x , y ) = < 6 x + y , 4 y + x >. (a) Show that the field is conservative by producing a potential function φ (x,y) so that ∇φ ( x , y ) = F ( x , y ) for all ( x , y ) in the plane. (b) Working within the influence of the force field from part (a), you move a particle along an ellipse C given by r ( t ) <3cos( t ),4sin( t )> from time t 0 = π /2 to time t 1 = π . How much work did you do?? ______________________________________________________________________ 12. (10 pts.) Let F ( x , y , z ) = < x 2 y , 2 y 3 z , 3 z >. Compute the divergence and the curl of the vector field F .
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