## Monday, May 15, 2006

### I Hope That's All. Topic: Calculus. Level: AIME.

Problem: (Problem-Solving Through Problems - 6.7.6) Calculate

$\displaystyle \lim_{x \to \infty} \left(x \int_0^x e^{t^2-x^2} dt \right)$.

Solution: First, we want to simplify it. Note that $x$ is not the variable of integration, which means it can be regarded as a constant. In particular, we have

$\displaystyle x \int_0^x e^{t^2-x^2} dt = xe^{-x^2} \int_0^x e^{t^2} dt$.

We notice that $e^{t^2}$ is not fun to integrate, so we think L'Hopital's Rule. Rewrite the limit as

$\lim_{x \to \infty} \left(\frac{\int_0^x e^{t^2} dt}{\frac{e^{x^2}}{x}} \right)$.

This is a $\frac{\infty}{\infty}$ indeterminate form, so we can apply L'Hopital's Rule. Our limit becomes (after differentiating the top and the bottom)

$\displaystyle \lim_{x \to \infty} \left(\frac{e^{x^2}}{\frac{2x^2e^{x^2}-e^{x^2}}{x^2}}\right) = \lim_{x \to \infty} \left(\frac{x^2}{2x^2-1}\right) = \frac{1}{2}$.

QED.

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Practice Problem: (Problem-Solving Through Problems - 6.7.2) Suppose that $f$ is a function with two continuous derivatives and $f(0) = 0$. Prove that the function $g$ defined by

$g(0) = f^{\prime}(0)$, $g(x) = \frac{f(x)}{x}$ for $x \neq 0$

has a continuous derivative.

1. I didn't get the first part but..
if g(0) is equal to 0, then f ^1 would also be 0, cuz they're equal. Then that means f(x) is also equal to 0, cuz anything ^1 is equal to the same thing. that leaves g(x)=0/x. 0 divided by anything is 0, therefore g(x)=0. And since g(0)=f(0), then f(0) is also equal to 0.
.. well, i tried.....

2. That's actually f'(0) as in f-prime, not f^1. It's a derivative, aka calculus.

3. yay for trying ^^ that's more than i do....

4. although i get it... i could never do that but i get it....

5. lim_{x \to 0} f(x) / x = f'(0)

Thus g(x) is continuous. g'(x) has a removable singularity at 0, and since f is twice differentiable it follows g'(x) is continuous everywhere.

6. Well, whoops, that proof's not really complete.

g'(x) = - f(x) / x^2 + f'(x) / x = ( x f'(x) - f(x) ) / x^2

\lim_{x \to 0} g'(x) = (x f''(x) + f'(x) - f'(x) ) / 2x = f''(x) / 2

So the limit is obviously defined on both sides and hence g'(x) is continuous.

7. AIME level? I thought calculus wasn't on AIME ... And if it was ... would this be question 1 or something?

8. Calculus isn't on the AIME... but I don't have another system for difficulty. It'd probably be an easy AIME question, yeah.

9. i don't get the title... :?