practice_midterm2_solutions.pdf - Sample Midterm Exam 2...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Sample Midterm Exam 2 — Solutions Math 32A/3, Fall 2014 This is a collection of problems that would not be unreasonable for a real midterm exam. WARNING: the inclusion (or exclusion) of a certain topic or type of problem on this sample exam does not guarantee its inclusion (or exclusion) on the actual exam! 1. Let f ( x, y ) = 1 p x - y 2 . (a) Describe the domain of f (as a set). In the box below, indicate the domain and sketch at least three level curves. [Your sketch only needs to be qualitatively correct.] (b) Let ( a, b ) = (10 , 3). Show that f is differentiable at ( a, b ) and find an equation for the tangent plane above that point. Solution: (a) (Compare with § 15.1 # 5-12, 29-36) The domain is { ( x, y ) | x - y 2 > 0 } , with strict inequality to avoid division by zero. The function is always positive, and we can make it as big as we want by taking y = 0 and x arbitrarily small. Then level curves are of the form c = 1 x - y 2 , where c > 0. This can be rewritten as x = y 2 + 1 c 2 . These curves are therefore parabolas, opening towards the x -axis, shifted to the right by 1 /c 2 . (b) (Compare with § 15.4 # 3-10) The partial derivatives of f are f x ( x, y ) = - 1 2 ( x - y 2 ) - 3 / 2 , f y ( x, y ) = y ( x - y 2 ) - 3 / 2 .
Image of page 1

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

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

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