# In terms of the initial point cos1 1 sin1 which is

This preview shows pages 2–5. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 .
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 =

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern