Recent content by y_lindsay

can anyone tell me how to prove the 1st mean-value theorem for integral \int^{b}_{a}f(x)g(x)dx=f(\xi)\int^{b}_{a}g(x)dx by applying Lagrange mean-value theorem to an integral with variable upper limit? thanks a lot.

thanks, sutupidmath. but there is a question about the Cauchy's theorem for convergence. as you've mentioned, N(e) is determined by an arbitrary e>0, which we take here e=1/n, so N(e)+1 need not to be necessarily smaller than 1/e=n. thus the fact |x_{N(e)+p+1}-x_{N(e)+1}|>\frac{1}{N(e)+1} does...

if the real sequence {x_n} satisfies that for every m>n, |x_n-x_m|>\frac{1}{n}, can we prove it's unbounded? here is what i thought: let's suppose the sequence is bounded, then there is its sub-sequence who converges to some real number L. now the problem is converted to this: can a sequence...

here is the problem: we know that (1) f(x) is defined on (-\infty, +\infty), and f(x) has the second derivative everywhere, (2) lim_{|x|\rightarrow+\infty}(f(x)-|x|)=0, (3) there is x_0\in R such that f(x_0)\leq0 how do we prove that f"(x) changes sign on (-\infty, +\infty)? I can...

to answer EnumaElish's question: because exp(-x) is 1-1, and if f(x) is not, i.e. there are distinct x1 and x2 such that f(x1)=f(x2)=y, then f(y)=exp(-x1)=exp(-x2), obviously this is impossible since f(x) has to be a function. I think continuity is needed here to show that f is strictly...

Is there a continuous function f(x) defined on (-\infty,+\infty) such that f(f(x))=e^{-x}? My opinion is "no", and here is how i think: first of all if such a function exists, it should be a "one-to-one" function, that is for every y>0, there should be exactly one x such that f(x)=y. Thus by...

How to find a polynomial P(x) of the smallest degree such that sin(x-x^2)=P(x)+o(x) as x->0? Do I have to calculate the first six derivatives of f(x)=sin(x-x^2) to get Taylor polynomial approximation?

hi, lazypast, actually the original problem does not become \int\frac{sec^{2}\theta}{tan\theta}d\thetaby using x=sec(theta), it becomes \int sec\theta d\theta however anyway, lazypast's little mistake has led to an inspiring discussion on how to deal with integration problems. and I really...

thanks mjsd. i know that the method itself is correct, but it just seems a little tedious, especially when we need to write the final answer in variable x. is there any other substitution we could use to attack this integral? or any alternative methods rather than the routine process to...

another question is how to evaluate the integral \int\frac{\sqrt{2-x-x^2}}{x^2}dx. i used the method of integration by parts, anyone knows some smarter way to do it?

i'm trapped with a problem: \int\frac{dx}{x\sqrt{2-x-x^2}}. i think this problem could be solved by subtitutions: \ x+\frac{1}{2}=\frac{3}{2}sint and \ u=tan\frac{t}{2}. and finally we would get an expression in \ u: \frac{\sqrt{2}}{4} log\left|\frac{2\sqrt{2}+u-3}{2\sqrt{2}-u+3}\right| (am...

We have X\in R^{s\times n}, A\in R^{n\times s}, I\in R^{s\times s}, where I stands for the indentity matrix. Now if we assume that rank(A)=s, can we conclude that there must exist a solution X to matrix equation XA=I? For me the answer is obviously "yes" if we think this problem in the...

Thanks arildno, i haven't thought out the hyperbolic solution to the first problem. yet to solve the 2nd problem, is there any other way rather than using substitution x=sin u ? I've thought out another possible substitution u=1/x, and then we could use the result of the firtst problem...

Actually i have solved this problem by using the substitution x=sec u. I'm just thinking if there is any other way to solve this problem? and another question is how can I delete my post if I find it not worth discussion any more? (I'm new to this forum) Thanks a lot.

how to evaluate the indefinite integral \int \frac{1}{\sqrt{x^2-1}} dx