Prove that if f(x) = x^alpha, where alpha = 1/n for some n in N (the natural numbers), then y = f(x) is differentiable and f'(x) = alpha x^(alpha - 1). Progress I have made so far: I have managed to prove, (x^n)' = n x^(n - 1) for n in N and x in R both from the definition of differentiation involving the limit and ...continues
For which real values alpha does lim {x -> 0+} x^alpha sin(1/x) exist? It is easy to show using the epsilon - delta definition below that this limit exists for all real alpha >= 1. In fact the limit is zero in this case. The case alpha equals zero is also quite simple and the limit does not exist. Consider the two sequence ...continues
Prove that if f(x) = x^alpha, where alpha = 1/n for some n in N (the natural numbers), then y = f(x) is differentiable and f'(x) = alpha x^(alpha - 1). Progress I have made so far: I have managed to prove, (x^n)' = n x^(n - 1) for n in N and x in R both from the definition of differentiation involving the limit and the binomial ...continues
Let n be a natural number, and let f(x) = x^n for all x are members of R. 1) If n is even, then f is strictly increasing hence one-to-one, on [0,infinity) and f([0,infinity)) = [0,infinity). 2) If n is odd, then f is strictly increasing, hence one-to-one, on R and f(R) = R. "This needs to be a proof by induction, provin ...continues
Real Analysis: Mean Value Theorem
Please see the attached file for the fully formatted problem. Let f(x) = ˆ e− 1/x 2 x 6 = 0 0 x = 0 Show that the nth derivative of f(x) exists for all n 2 N . Please justify all steps and be rigorous because it is an analysis problem. (Note: The problem falls under the chapter on Differentiability on R in the s ...continues
Real Analysis : Proof of a Constant
Please see the attached file for the fully formatted problem. Let > 0. Prove that log x x for x large. Prove that there exists a constant C such that log x C x for all x 2 [1, 1 ), C ! 1 as ! 0+, and C ! 0 as ! 1 Please justify all steps and be rigorous because it is an analysis problem. (Note: The probl ...continues
Real Analysis: Riemann Integrability
Please see the attached file for the fully formatted problem. Let E = { 1/n : n 2 N } . Prove that the function f(x) = ˆ 1 x 2 E 0 otherwise is integrable on [0,1]. What is the value of R 1 0 f(x)dx? Since this problem is an analysis problem, please be sure to be rigorous. It falls under the chapter on Integrabili ...continues
Real Analysis : Riemann Integrability
Please see the attached file for the fully formatted problem. Prove that if f is integrable on [0, 1] and > 0, then lim n !1 n Z 1/n 0 f(x)dx = 0 for all < . What happens for ? Since this problem is an analysis problem, please be sure to be rigorous. It falls under the chapter on Integrability on R , ...continues
Real Analysis: Riemann Integrability
Please see the attached file for the fully formatted problems. Prove that if f is integrable on [0, 1], then lim n !1 Z 1 0 x n f(x)dx = 0 Since this problem is an analysis problem, please be sure to be rigorous. It falls under the chapter on Integrability on R , where they define partition, refinement of a partition, ...continues
Real Analysis: Riemann Integrability
Please see the attached file for the fully formatted problems. Let f be continuous on a closed, bounded, nondegenerate interval [a, b] and set M = sup x 2 [a,b] | f(x) | . a) Prove that if M > 0, then for every > 0 there is a nondegenerate interval I ˆ [a, b] such that (M − )n | I | Z b a | f(x) | n dx ...continues
