Error bounds - Simpson, Trap, and Midpoint

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Discussion Overview

The discussion revolves around finding error bounds for integral approximations using Trapezoidal, Midpoint, and Simpson's methods. Participants explore the process of determining the "k" value, which relates to the derivatives of the function involved in the approximation.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant describes their method for finding the "k" value using the second derivative for Trapezoidal and Midpoint error bounds, seeking clarification on whether the same approach applies to Simpson's method using the fourth derivative.
  • Another participant confirms that the maximum output from the fourth derivative should be found, suggesting that the fifth derivative may also be necessary to determine critical points for the fourth derivative.
  • A participant mentions that their teaching material does not use "k" and presents an alternative error formula for Simpson's method, which emphasizes finding critical numbers of the fourth derivative through the fifth derivative.
  • One participant expresses gratitude for the clarification provided by another, noting that their current textbook does not adequately explain error bounds for Simpson's method.
  • Another participant shares their positive experience with a different calculus book, indicating varying levels of clarity among different educational resources.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and clarity regarding the error bounds and the use of derivatives, indicating that there is no consensus on the best approach or the clarity of existing educational materials.

Contextual Notes

Some participants mention difficulties in understanding the derivation of the error formulas, highlighting a potential gap in foundational knowledge or instructional clarity.

Who May Find This Useful

Students studying calculus, educators seeking clarification on error bounds in integral approximations, and individuals interested in the nuances of mathematical error analysis.

Math Is Hard
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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.

Thanks in advance.
 
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Math Is Hard said:
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".

Right.

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?

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)|[/color]],

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

edit: fixed color bracket
 
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.
 
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!
 
Math Is Hard said:
Tom,
as always - my eternal gratitude! I jumped for joy when immediately after posting I saw you online. I just knew you'd respond.

It's nice to feel needed--thanks. :smile:

Stewart's Calculus does a really bad job of explaining error bounds for Simpson's. Your formula makes it clear.

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 into PF for more helpful info!
 
I have a little red book by Joseph Edwards which I think is dandy.
 

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