You must complete a total of 7 problems, at least two of which are from problems. 1-3 (you can do all three if you want). Before you hand in the exam, please use my purple pen to circle the problem numbers you wish to be graded. Use induction to prove your formula. (b) consider the function f : n n de ned by f (n) = an. Brie y explain: consider the function f (x) =(cid:26) |x| |x| if x rational if x irrational (a) calculate z 3. 1 f using the de nition of the upper integral. You may use evenly-spaced partitions. (b) recall: f is said to be lipschitz on an interval i if there exists k > 0 such that for all x, x0 i. Either way, prove it using the de nition of lipschitz. |f (x) f (x0)| k|x x0: note: the parts of this problem are not related. (a) can the series 1 1.
