EE 518 Homework 6 Due on Monday 2 00pm October 10 2016 Problem 1 Suppose f g a b R is differentiable on its domain and g 0 6 0 Show that c a b such that f a f c f 0 c 0 g c g b g c Hint Construct proper function to apply mean value theorem Problem 2 If f x g x on a b and f g are Riemann integrable on a b show that Z b Z b f x dx g x dx a a Hint Use definition of Riemann integral Problem 3 Let f 0 denote the Gamma Function i e Z f x 1 e x dx 0 i Show that f is well defined for any 0 i e show that 0 both Z 1 Z 1 1 x x e dx lim x 1 e x dx t 0 0 and Z x 1 e x dx lim t 1 t Z t x 1 e x dx 1 exists ii Show that f 1 f 1 1 and consequently f n n 1 n Z Problem 4 Calculate the following integrals i Z e x ln x xex 1 ii Z 2 sinn xdx n 0 1 2 0 hint consider both n is even and odd number Page 1 of 2 EE 518 Homework 6 Due on Monday 2 00pm October 10 2016 Problem 5 Suppose a function f is continuous on a c and differentiable on a c some b a c Show that there exists a a c such that f 0 0 Rb a f x dx Rc b f x dx 0 for Problem 6 For the following parts submit your code and your results R 1 Calculate sin x x cos x dx R 10 2 Calculate x dx R 10 3 Calculate 0 x2x 5 dx using some numerical method Page 2 of 2