MATH 3A03 Assignment #5 Due: Thursday, November 19, in class 1. Let f (x) be the function dened on the interval [0, 2] such that f (x) = 1 for 0 x 1 and f (x) = 2 for 1 < x 2. Using the denition of the Riemann Integral, show that f is Riemann Integrable over the interval [0, 2]. 2. Let f (x) = 1/x for 1 x 3 and let be the partition {1, 2, 3} of [1, 3]. Find m() and M (), the lower and upper sums for f using , respectively. 3. Let f (x) be the function with domain [0, 1] dened by f (x) = x if x is rational and f (x) = 0 if x is irrational. Determine if f is Riemann Integrable. Justify your answer. 4. Let f be continuous on the interval [a, b]. Show that there is some b number c (a, b) with f (x)dx = f (c)(b a). a 5. Let g(x) be a continuous function on the interval [0, 1] such that g(x) 1 0 for all x [0, 1]. Show that if g(x)dx = 0 then g(x) = 0 for all 0 x [0, 1]. Is this conclusion still valid if g is Riemann Integrable on [0, 1] but it is not assumed to be continuous on [0, 1]? Justify your answer. 1