math318.midterm2 Spring 2013 - Math 318 Krantz Spring 2013...

This preview shows page 1 - 7 out of 7 pages.

Math 318 Spring, 2013 Krantz April 10, 2013 Second Midterm General Instructions: Read the statement of each problem carefully. Do only what is requested—nothing more and nothing less. Provide a complete solution to each problem. If you only write the answer then you will not get full credit. If you need extra room for your work then use the backs of the pages. Be sure to ask questions if anything is unclear. (10 points) 1. Let U R n be an open set and f : U R m be a function. Let v be a vector in R n . Let a R n . Give a rigorous definition of the directional derivative D v f ( a ). (10 points) 2. Let U R n be open and a U . Let f : U R m be a function. Give 1
Image of page 1

Subscribe to view the full document.

the precise definition of what it means for f to be differentiable at a . (10 points) 3. Show that the function f ( x, y ) = xy x 2 + y 2 if ( x, y ) 6 = 0 , 0 if ( x, y ) = 0 is not differentiable at the origin in R 2 . 2
Image of page 2
(10 points) 4.Define 5. Write down (but do not evaluate) the integral that represents the arc
Image of page 3

Subscribe to view the full document.

Image of page 4
Image of page 5

Subscribe to view the full document.

Image of page 6
Image of page 7

Unformatted text preview: length of the curve γ ( t ) = ± e t , sin 2 t,t 2 ² 3 between t = 1 and t = 4. (10 points) 6. Use Gaussian elimination to completely solve the linear system 2 x 1-x 3 + x 4 = 1 x 1 + x 2-x 3 +2 x 4 = 0 2 x 2 + x 3-x 4 = 2 4 (10 points) 7. Use Gaussian elimination to find the inverse of the matrix 3 0 1 0 1 0 1 1 0 5 (10 points) 8. Calculate the curvature κ of the curve g : R → R 3 given by g ( t ) = cos 2 t sin 2 t 2 t at the point g (0) = (1 , , 0). (10 points) 9. At the point (1 , 1), what is the direction of greatest increase of the function f ( x,y ) = x 2 + y 2 ? Give a rigorous mathematical explanation for your answer. 6 (10 points) 10. What is the equation of the tangent plane to the graph of f ( x,y ) = x 2-y 2 at the point (1 , 1 , 0)? 7...
View Full Document

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