Error bounds - Simpson, Trap, and Midpoint

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SUMMARY

This discussion focuses on determining the "k" value for error bounds in integral approximations using the Trapezoidal, Midpoint, and Simpson's methods. For the Trapezoidal and Midpoint methods, the second derivative, f''(x), is evaluated to find the maximum value within the integration limits, which serves as "k". In contrast, Simpson's method utilizes the fourth derivative, f(4)(x), and requires finding critical points using the fifth derivative, f(5)(x), to determine maxima. The error formula for Simpson's is presented as E ≤ [(b-a)5/180n4][max|f(4)(x)|], emphasizing the importance of the fourth and fifth derivatives in error estimation.

PREREQUISITES
  • Understanding of integral approximation methods: Trapezoidal, Midpoint, and Simpson's.
  • Knowledge of derivatives, specifically second, fourth, and fifth derivatives.
  • Familiarity with error analysis in numerical methods.
  • Basic calculus concepts, including critical points and maxima.
NEXT STEPS
  • Study the derivation and application of the Simpson Error Formula in detail.
  • Learn how to find critical points using the fifth derivative in calculus.
  • Explore numerical integration techniques and their error bounds in various textbooks.
  • Practice problems involving the calculation of error bounds for different integral approximation methods.
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Students preparing for calculus exams, educators teaching integral approximation methods, and anyone interested in mastering error analysis in numerical integration.

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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