MATH 210 Exam 2, Fall 2010

Math 210 - Quiz 2 (Fall 2010) T. Tlas 1. Consider the following: i- (15 points) Suppose f(z) is a continous function on (0, 1], such that supjo,1(f) = M@. Assume Jo f(x)de = M. Prove that f(x) = M, Vz € (0, 1). ii- (15 points) Suppose f(x) is a continous function on (0, 2], such that supj,.(/) = M. Assume iF f(x)dx = 2M. Prove that f(z) = M, Vz € [0,2]. 2. Consider the following: i- (15 points) Prove that a continuous function on a compact set is uniformly continuous. ii- (15 points) Assume f(a) is a function differentiable everywhere on R, and that |f’(x)| < A, Va. Prove that f is uniformly continuous. 3. (20 points) Assume f(z) is a function which is three times differentiable (ie. f’”(x) exists at every x). Moreover, you are given that f(0) = f’(0) = f”(0) = 1 and that |f’”"(x)| <1, Vx. Find (with careful justification) the following limit lim. f(z)-l-2 aie 20 x 4. (20 points) Suppose f and g are differentiable on [a,b] and that f’ and g' are Riemann integrable on [a,b]. Prove the following formula: b b [seal erate = F098) - soala) = ff F'a)g(\ae The fact that the above integrals exist should be proven as well.