Sample Exam 3 Solution on Calculus III

Z f x x z f z x z f z f x if we use subscript

This preview shows page 4 - 5 out of 5 pages.

z 0 F x x z F z x z F / z F / x . If we use subscript notation for the partial derivatives, we set g(y,z) = F(h(y,z), y, z). Then 0 = g z (y,z) = F x (h(y,z), y, z)h z (y, z) + F z (h(y,z), y, z), which implies that x/ z = h z (y,z) = - F z (h(y,z), y, z)/F x (h(y,z), y, z).
Image of page 4

Subscribe to view the full document.

TEST3/MAC2313 Page 5 of 5 _________________________________________________________________ 9. (10 pts.) It turns out that h(y) = sin(y)/y does not have an elementary antiderivative. Despite that you can evaluate the following iterated integral. Do this by reversing the order of integration and then evaluating the new iterated integral you obtain. [Hint: It helps to sketch the region of integration, R.] π 0 π x sin( y ) y dydx π 0 y 0 sin( y ) y dxdy π 0 sin( y ) y y 0 1 dx dy π 0 y sin( y ) y dy π 0 sin( y ) dy cos( π ) ( cos(0)) 2. Note: Although the integral appears to be improper, it really isn’t. Why?? _________________________________________________________________ 10. (10 pts.) (a) Let R be the region bounded by the curves defined by y = x 2 and y = x + 2. Write an iterated double integral that gives the area of the region, but do not attempt to evaluate the iterated integral. area ( R ) R 1 dA 2 1 x 2 x 2 1 dydx (b) Set up, but do not attempt to evaluate the iterated double integral that will give the numerical value for the volume of the solid bounded by the cylinders in 3-space defined by x 2 + y 2 = 1 and y 2 + z 2 = 1. [Hint: You should only need to sketch the xy-cylindrical stuff to obtain the limits of integration.] // Here the top surface is z = (1 - y 2 ) 1/2 and the bottom surface is z = -(1 - y 2 ) 1/2 . V R (1 y 2 ) 1/2 ( (1 y 2 ) 1/2 ) dA 1 1 (1 x 2 ) 1/2 (1 x 2 ) 1/2 2(1 y 2 ) 1/2 dydx 1 1 (1 y 2 ) 1/2 (1 y 2 ) 1/2 2(1 y 2 ) 1/2 dxdy _________________________________________________________________ Silly 10 Point Bonus: Suppose f(x,y) is differentiable at an interior point (x 0 ,y 0 ) in its domain. Pretend there are at least three distinct unit vectors u satisfying the following equation: D u f(x 0 ,y 0 ) = 0. Does it follow as a consequence that this equation must be true for all unit vectors? Proof?? Where???
Image of page 5

{[ snackBarMessage ]}

Get FREE access by uploading your study materials

Upload your study materials now and get free access to over 25 million documents.

Upload now for FREE access Or pay now for instant access
Christopher Reinemann
"Before using Course Hero my grade was at 78%. By the end of the semester my grade was at 90%. I could not have done it without all the class material I found."
— Christopher R., University of Rhode Island '15, Course Hero Intern

Ask a question for free

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern