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Real Analysis

Real Analysis : Young's Inequality

Note: * = infinite Suppose that the function f:[0,*)->R is continuous and strictly increasing, with f(0) = 0 and f([0,*)) = [0,*). Then define F(x) = the integral from 0 to x of f and G(x) = the integral from 0 to x of f^-1 for all x>=0 (a) Prove Young's Inequality: ab <= F(a) + G(b) for all a >= 0 and b >= 0 (b) N

Real Analysis: Geometric Interpretation in Terms of Areas

Note: * = infinite Suppose that the function f:[0,*) -> R is continuous and strictly increasing, and that f:(0,*) -> R is differentiable. Moreover, assume f(0) = 0. Consider the formula: the integral from 0 to x of f + the integral from 0 to f(x) of f^-1 =xf(x) for all x>= 0. How can I provide a geometric interpretation

Real Analysis: Criteria for Integrability

Suppose the continuous function f:[a,b]->R has the property that: The integral from c to d f<=0 whenever a<=c<d<=b Prove that f(x)<=0 for all x in [a,b]. Is this true if we require only integrability of the function?

Real Analysis: Criteria for Integrability

Define f(x) = x^2 for all x in [0,1]. For each natural number n, compute L(f,Pn) and U(f,Pn), where Pn is the regular partition of [0,1] into n subintervals.Then use the Integrability Criterion to show that the function f:[0,1]->R is integrable.

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,

MacLaurin Series And Laplace Transforms : Absolute Convergence

Find MacLaurin Series for the given function f. Use the linearity of the Laplace Transform to obtain a series representation L(f)=F(s) Determine 5 values for which the series converges absolutley (and uniformly). Also show the Laplace transform exists, i.e. that it has exponential order alpha. Here are the functions. A) f

Real Analysis : Derivatives

If I say that the function f:R->R has two derivatives, with f(0) = f'(0) = 0 and the absolute value of f"(x) is less than or equal to one, if the absolute value of x is less than or equal to 1. How can I prove that: f(x) <= 1/2 if x <= 1

Real Analysis: Differentiation

If I let the function f:R->R have two derivatives with f(0) = 0 and f'(x) <= f(x) for all x in R. Is f(x) = 0 for all x in R

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

Real Analysis: Mean Value Theorem

Let r e-1/x2 i(x) = i 0 x740 x = 0 Show that the nth derivative of 1(x) exists for all n E N. Please justify all steps and be rigorous because it is an analysis problem. (Note: The problem falls under the chapter on Differentiability on IR in the section entitled The Mean Value Theorem.)

Real Analysis Problem

The inverse cosine function has domain [-1,1]and range [0, pi]. Prove that (cos^-1)'(x) = -1/ sqrt(1-x^2). This needs to be proved from a real analysis point of view not a calculus.

Composition Series

Write a composition series for the rotation group of the cube and show that it is indeed a composition series.

Series Convergence: Two-sided Estimator

Please see the attached file for the fully formatted problems. A simple technique to estimate values of the finite sums of reciprocals to natural numbers raised to positive power and to define if corresponding infinite series converge. Deduce two-sided estimator for the sums of the positive powers (p) of reciprocals to nat

Limits of Functions

Evaluate the following limits using the epsilon - delta definition and the limit theorems: a) lim {x -> 0} (x^2 + cos x)/(2 - tan x) b) lim {x -> sqrt(pi)} ((pi - x^2)^(1/3))/(x + pi)

Functions: Limits

Using the definition of a limit (rather than the limit theorems) prove that lim {x -> a+} f(x) exists and find the limit in each of the following cases a) f(x) = x/|x|, a = 0. b) f(x) = x + |x|, a = -1. c) f(x) = (x - 1)/(x^2 - 1), a = 1. In which cases do lim {x -> a-} f(x) and lim {x -> a} f(

Real Analysis Problem

I need a correct and concise solution. If the answer is not 100% correct, I will ask for my money back! We just finished integration and are done with a first course in analysis, i.e. chapters 1-6 of Rudin. We are also using the Ross and the Morrey/Protter book. The Problem: f : R --> R , f ' ' ' ' continous.

Real Analysis Problem

We have just finished up integration and are done with a first course in analysis, so chapters 1-6 of Rudin. We are also using the Ross and Morrey/Protter book. Please answer question fully and clearly explaining every step. Any solution short of perfect is useless to me. So if you are not 100% sure whether your answer is right,

Real Analysis Problem

We have learned Rolle, Lagrange, Fermat, Taylor Theorems in our Real Analysis class and we have finished differentiation. We just started integration. In this problem we are not supposed to use any material we haven't learned, ie integration. We are using the books by Rudin, Ross, Morrey/Protter. ****************************