Error Bound Formulas for midpoint and trapezoid rules, sect7.7#19b | Summary and Q&A

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March 14, 2017
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Error Bound Formulas for midpoint and trapezoid rules, sect7.7#19b

TL;DR

Learn how to use error formulas to approximate integrals using the trapezoid and midpoint rules, with examples and calculations.

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Questions & Answers

Q: What are the formulas for estimating the error in the trapezoid and midpoint rules?

The formula for the trapezoid rule is E(T) = -((b-a)^3/(12n^2)) * K, while the formula for the midpoint rule is E(M) = -((b-a)^3/(24n^2)) * K, where K is the maximum value of the absolute value of the second derivative.

Q: How do you determine the K value?

To determine the K value, you need to find the maximum value of the absolute value of the second derivative of the function being integrated. This can be done by graphing the function on a graphing calculator and using the calculator's maximum function.

Q: What are A and B in the error formulas?

A and B represent the limits of integration. A is the starting value, while B is the ending value. In the provided example, A is 0 and B is 1.

Q: Why is it important to find the K value accurately?

Finding the K value accurately ensures a more precise estimation of the error in the approximation. Using a more accurate K value improves the reliability of the calculated error in both the trapezoid and midpoint rules.

Summary & Key Takeaways

  • The content explains how to use error formulas to approximate integrals using the trapezoid and midpoint rules.

  • It discusses the formulas for estimating the error in both methods, emphasizing their similarities and differences.

  • The video demonstrates step-by-step calculations and provides explanations for finding the maximum value of the second derivative to determine the K value.

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