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

TL;DR

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

Key Insights

• 📏 The formulas for estimating error in the trapezoid and midpoint rules are similar, with the only difference being the coefficient (12 in trapezoid rule, 24 in midpoint rule).
• 👌 The K value is the maximum value of the absolute value of the second derivative, which can be found by graphing the function and using a graphing calculator.
• 👌 Accurately determining the K value is crucial for obtaining reliable error estimations.
• 📏 The trapezoid and midpoint rules are useful methods for approximating integrals when an exact solution is challenging to find.
• ❓ The error formulas provide a quantitative measure of the accuracy of the approximation.
• 😉 The provided example demonstrates the step-by-step calculations for finding the K value and estimating the errors using the given formulas.
• 🥺 Using a more accurate K value leads to a more precise estimation of the error.

Transcript

that's how to use the error formulas for using the trapezoid and also the mid point rules when we're trying to approximate integrals and this is the integral that we're trying to approximate the integral from 0 to 1 cosine of x² DX and in this case we're trying to use n is equal to 8 and as you can see I put on the formulas on the board right here ... Read More

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.