320final - Stony Brook University MAT 320 Introduction to...

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

View Full Document Right Arrow Icon
Stony Brook University MAT 320 Introduction to Analysis Final Examination with Solutions December 21, 2007 Work any five problems. Tell us clearly which ones you have chosen. 1. Show how the Bolzano-Weierstrass Theorem (Every bounded sequence contains a con- vergent subsequence) follows from the Least Upper Bound Axiom (Every bounded set of real numbers has a least upper bound). (Solution) This is from the book. You need to show first that every sequence contains a monotone subsequence. Then show that a bounded monotone sequence converges. 2. Prove: If f : R R is positive, bounded away from zero (i.e. K > 0 such that | f ( x ) | ≥ K for all x ) and differentiable, then g ( x ) = q f ( x ) is also differentiable. (Solution) The Chain rule tells us that if f is differentiable at x and if H is differentiable at f ( x ) then H f is differentiable at x (and in fact ( H f ) 0 ( x ) = H 0 ( f ( x )) f 0 ( x )). Here H is the square-root function, which is differentiable at all x > 0. (“Bounded away from zero” was not necessary for this argument.) 3. Prove: if f 0 is continuous and f 0 ( x ) > 0, then f is increasing on a neighborhood of x . Show by example that the hypothesis “ f 0 is continuous” is necessary.
Image of page 1

Info iconThis 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