104_Fall09_review_Final

# 104_Fall09_review_Final - Review Problems for Final Review...

This preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: Review Problems for Final Review the definitions and theorems covered in Chapter 2 and 3. You will be asked to state a few of them precisely. Also, review homework problems and problems from the first two midterms. 1. For f : [0, 1] R, define 1 1 2 f 2 = 0 f (x) dx 2 . For f, g C 0 [0, 1], define d2 (f, g) = f - g 2 . Prove that d2 gives a metric on C 0 [0, 1] (you may use the Cauchy-Schwarz inequality, which is proven exactly the same as for Euclidean space - see p. 22 of Pugh): 1 f (x)g(x)dx f 0 2 g 2 2. Let (M, d) be a metric space. Given a set S M , define the characteristic function S : M {0, 1} of S as 1 if x S S (x) = 0 if x S. / (a) Prove that S is discontinuous at x if and only if x S. (b) Prove that for the Cantor set C (as a subset of the metric space [0, 1]), C is Riemann integrable. 3. Determine if the following series converge or diverge. (a) (b) log(k+1) k=0 k2 +1 k k=0 (-1) ( k +1- k) k k=0 cos(k)x (c) Find the radius of convergence. 4. Does -x2 dx 0 e exist? 5. Let f : [-1, 1] R be continuous. What is lim 1 x2 x x0 xf (x)dx? 0 6. Suppose f : [a, b] [0, ), and the discontinuity set D(f ) is a zero set. Show that g = f /(1 + f ) is Riemann integrable. ...
View Full Document

Ask a homework question - tutors are online