Let f(x) be integrable on [a,b], and let g(x) be nondecreasing and continuously differentiable on [a,b]. Let {p be element of P} be a partition of [a,b], and define

U(f,g,p) = SIGMA (Mi(g(the ith term of x) - g(the (i-1)th term of x))) as i=1 to n
L(f,g,p) = SIGMA (Ni(g(the ith term of x)-g(the (i-1)th term of x))) as i=1 to n

Use mean value theorem to prove that (inf U(f,g,p), for p is element of P) = (sup L(f,g,p), for p is element of P) = ( INTEGRAL f(x)g'(x)dx, as x from a to b)

Solution Preview

Please see the attached file for the full solution.

Thanks for using BrainMass.

Let f(x) be integrable on [a,b], and let g(x) be nondecreasing and continuously differentiable on [a,b]. Let {p be element of P} be a partition of [a,b], and define

Use mean value theorem to prove that

Proof. Given a partition P: of [a,b]. Since g(x) is nondecreasing and continuously differentiable on ...

Solution Summary

The Mean Value Theorem is used to prove an an integral relation. The mean value theorem to prove a function is found.

... problems are real analysis related problems. One of these is Liptuaz continuity/ holder condition. It would be nice if you use mean value theorem for solving ...

Real Analysis : Differentiable and Increasing Functions. ... on (a,b). It suffice to prove that for any x<y in (a,b), f(x)<=f(y). By Mean Value Theorem , we have f(x ...

... A real analysis for textbook metric spaces is examined ... sequence of open neighborhoods of the real number such ... Since , the system is centered by Definition 2 on ...

... rise in price due to fall in real income or ... increased supply of a particular product means surplus of ... to the common practice, economic analysis is divided into ...

... begin to see its manifestations in real life ... A brief analysis of your completed Ethics Awareness ... had the opportunity of understanding the meaning and importance ...

... to predict reasonably well if one confines the analysis to internationally ... br> <br>4.Question: A real depreciation of ... The low cost of the dollar meant that the ...

... measurements is more important, the mode, the mean, or the ... much dependent on how well the real issue was ... related to the issue, on the analysis utilized for ...

... d) Give a real life example of when you'd use ... However, correlation does not mean causation ... Correlation analysis is a group of techniques to measure the strength ...

... What does that mean in practical terms ... copy of Excel and you decide to learn regression analysis. ... non-statistical person can understand (a very real and common ...