In terms of the initial point cos1 1 sin1 which is

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in terms of the initial point (cos(1), 1, sin(1)) which is the terminal point of r (0) in standard position. Rather than overloading the symbol r , write this new parameterization as R ( s ). How are R and r related? ______________________________________________________________________ 4. (16 pts.) Locate and classify the critical points of the function Use the second partials test in making your classification. Crit.Pt. f xx @ c.p. f yy @ c.p. f xy @ c.p. D @ c.p. Conclusion
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Name: Final Exam/MAC2313 Page 3 of 7 ______________________________________________________________________ 5. (16 pts.) Find the absolute extrema of in the region f ( x , y ) x 2 2 y 2 x D ( x , y ) : x 2 y 2 4 . ______________________________________________________________________ 6. (12 pts.) Evaluate the following line integral, where C is the path from (1,-1) to (1,1) along the curve x = y 4 .
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Name: Final Exam/MAC2313 Page 4 of 7 ______________________________________________________________________ 7. (10 pts.) (a) Show that the vector field F ( x , y ) < cos( x ) e y 2 x , sin( x ) e y > is actually a gradient field by producing a function φ ( x , y ) such that ∇φ ( x , y ) = F ( x , y ) for all ( x , y ) in the plane. (b) Using the Fundamental Theorem of Line Integrals, evaluate the path integral below, where C is any smooth path from the origin to the point ( π /2,ln(2)). [ WARNING: You must use the theorem to get credit here.] ______________________________________________________________________ 8. (18 pts.) Write down but do not attempt to evaluate the iterated triple integrals in (a) rectangular, (b) cylindrical, and (c) spherical coordinates that would be used to compute the volume of the sphere with a radius of 1 centered at the origin. [For rectangular, there are many correct variants.] (a) V = (b) V = (c) V =
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