Error bounds - Simpson, Trap, and Midpoint

1. Jun 6, 2004

Math Is Hard

Staff Emeritus
I always have trouble finding my "k" value for error bounds when doing approximation of integrals.
With Trapeziodal and Midpoint error bounds, I take the second derivative of my function. Then I find the number on the interval (between limits of integration that I am given) that will give me the biggest output when plugged into f ''(x).
I run that through the f '' (x) function and the number that results is my "k".

With Simpson's, I know the 4rth derivative is used - but is it the same technique? Am I looking for the maximum output I can get from the 4rth derivative using a value from my limits of integration? In which case, should I be taking the 5th derivative as well to determine maxima on the interval for my fourth derivative function?

I hope this makes sense. My brain is starting to meltdown from studying for midterms. :yuck:

2. Jun 6, 2004

Tom Mattson

Staff Emeritus
Right.

Exactly.

In the book I'm currently teaching from, it doesn't even use "k". It states the error formulae in the more suggestive form:

Simpson Error Formula (sorry, haven't got LaTeX down yet)
E<=[(b-a)5/180n4][max|f(4)(x)|],

which more clearly tells you what to do: find the critical numbers of the 4th derivative, via the 5th derivative.

edit: fixed color bracket

3. Jun 6, 2004

Parth Dave

does anyone know where those formulas came from? I've never read a proof for them anywhere. I asked my calculus teacher and he said he had no idea either.

4. Jun 6, 2004

Math Is Hard

Staff Emeritus
Tom,
as always - my eternal gratitude! I jumped for joy when immediately after posting I saw you online. I just knew you'd respond. Stewart's Calculus does a really bad job of explaining error bounds for Simpson's. Your formula makes it clear.
Getting the 5th derivative for some of these problems to find the max is going to be excruciating, but I'll muddle through.
Thanks so much!

5. Jun 7, 2004

Tom Mattson

Staff Emeritus
It's nice to feel needed--thanks.

I agree. The book I'm teaching from is Calculus by Larson, Hostedler, and Edwards. The book I learned it from was not as clear either. Just keep tuning in to PF for more helpful info!

6. Jun 7, 2004

Gokul43201

Staff Emeritus
I have a little red book by Joseph Edwards which I think is dandy.