Recent content by klondike

Do you remember the derivative of of $$x^n$$?

How about this: $$\int_{x_0-\varepsilon}^{x_0+\varepsilon} g'(x)\varphi(x)dx = g(x)\varphi(x)\vert_{x_0-\varepsilon}^{x_0+\varepsilon}-\int_{x_0-\varepsilon}^{x_0+\varepsilon} g(x)\varphi'(x)dx$$ and $$\int_{x_0-\varepsilon}^{x_0+\varepsilon} g(x)\varphi'(x)dx=0$$ if $$\varphi$$ is...

[/PLAIN] Just follow the questions in sequence. Use Gradient theorem. Like Gauss's and Stoke's theorems, they are fancy version of FTC in Calc.

(1+\frac{1}{n})^{n}<e for n=1 \frac{2}{1}<e for n =2 \frac{3^2}{2^2}<e for n =3 \frac{4^3}{3^3}<e and so on so forth... Multiply them together, we then have \frac{(n+1)^n}{n!}<e^n

Dick, my original proposal is a dead end cause even I managed to prove it monotonic, I won't be able to evaluate its limit -- FAILED ATTEMPT:cry:. Let me try again. It's possible to prove (1+\frac{1}{n})^n monotonic increase. As a fact, it converges to e. Hence, (1+\frac{1}{n})^{n}<e After...

Perhaps we can prove the sequence monotonic so that we at least get the converge part. but I don't know how off top of my head.-:(

How about using inequality of AM-GM means in conjunction with Sandwich theorem?

I would show a_{n+1}-a_{n}>0 for all n which is pretty obvious in your case.

The integral is very nasty if it were to be done by hand. Is it a homework question?

Use the definition of "strictly increasing" to show part 1 and use the definition of "continuous" for part 2.

Exam problems are usually 0/0, ∞/∞, 1∞, 0∞, ∞0 which you can't just simply plug and chug.:devil:

Not necessary. Good start:) What does it look like after factoring out the term? It just means -x, or (-1)x. And I'm sure you know -b+a=a-b. Do you remember how to add/subtract fractions with different denominators?

You are welcome! This happens a lot in self-studying. We don't actually integrate the functional because u,v, and their partial derivatives are unknown. Or did I miss your question? We are dealing with the terms related to first variation of \frac{\partial u}{\partial x}. y is treated as constant.

You might not have *fully* understood the derivation of Euler-Lagrange equation for the classic example of S(t)=\int_{a}^{b}L(t,x,\dot{x})dt. I'm using the classical mechanics convention here where x is function of t, and \dot{x}=dx/dt, L is Lagrangian. The integration by part of the selected...

Had you made only one mistake, you won't get the right answer. Kinda like (-1)*(-1)=1. You have made TWO errors. 1. antiderivative of sin(t2) is wrong as Mark suggested. 2. \frac{d}{dx}\left(\frac{cos(x^2)}{2x}\right)\neq -\sin x^2. Review quotient rule.