# Thread: Find the maximum over a given range?

1. ## Find the maximum over a given range?

Hello,

I've been searching through my book and all over to try to find out how to do this, I feel like there is some easy way I'm just forgetting so I'm hoping someone can help me

Lets say I'm given the following integral:

$\displaystyle \int_0^1 \sin{\frac {\Theta^2}{2}}d\Theta$
(First time using LaTex)

And I need to find the approximation of the area using Simpson's Rule.
I can do all that fine but then the problem comes when finding the error.

I know the error's formula and all but how do I find the value of K, where K is the maximum value the function reaches over a given range [0,1]?

What methods do we have to find the value of K (over the specified range), especially when it comes to more complicated integrals? My professor never really went over that and I'm just having a tough time getting the questions done.

Thanks!

2. Originally Posted by trigoon
Hello,

I've been searching through my book and all over to try to find out how to do this, I feel like there is some easy way I'm just forgetting so I'm hoping someone can help me

Lets say I'm given the following integral:

$\displaystyle \int_0^1 \sin{\frac {\Theta^2}{2}}d\Theta$
(First time using LaTex)

And I need to find the approximation of the area using Simpson's Rule.
I can do all that fine but then the problem comes when finding the error.

I know the error's formula and all but how do I find the value of K, where K is the maximum value the function reaches over a given range [0,1]?

What methods do we have to find the value of K (over the specified range), especially when it comes to more complicated integrals? My professor never really went over that and I'm just having a tough time getting the questions done.

Thanks!
the max possible value for $\displaystyle \sin\left(\frac{\theta^2}{2}\right)$ is 1, and occurs when $\displaystyle \frac{\theta^2}{2} = \frac{\pi}{2}$ , or when $\displaystyle \theta = \sqrt{\pi} \approx 1.772$ , outside the given interval $\displaystyle [0,1]$.

since $\displaystyle \sin\left(\frac{\theta^2}{2}\right)$ is strictly increasing on $\displaystyle \left[0,\frac{\pi}{2}\right]$ , then the max value would occur when $\displaystyle \theta = 1$ on the interval $\displaystyle [0,1]$.

3. Ah maybe that wasnt the best example, what about when I have something like this:

$\displaystyle \int_1^2 \frac {\ln(x)}{1+x}dx$

Are there any methods to do these types?