In Exercises 51 and 52, use a computer algebra system to...

In Exercises 51 and $52,$ use a computer algebra system to find the maximum value of $\left|f^{(4)}(x)\right|$ on the closed interval. (This value is used in the error estimate for Simpson's Rule, as discussed in Section 4.6.)
$$
f(x)=\frac{1}{x^{2}+1}, \quad[-1,1]
$$

Key Concepts

Fourth Derivative

The fourth derivative of a function captures the rate of change of the third derivative and plays an essential role in error analysis for numerical methods such as Simpson's Rule. Its maximum absolute value on a given interval is used to estimate how much the approximation might deviate from the true integral.

Computer Algebra Systems (CAS)

A Computer Algebra System is a tool that performs symbolic mathematics, including differentiation and solving equations. It is particularly useful in computing higher-order derivatives and finding their maximum values on closed intervals, which would be tedious and error-prone if done manually.

Error Estimation for Simpson's Rule

This concept focuses on controlling the error in Simpson's Rule, a numerical integration method. Specifically, the error bound for Simpson's Rule involves the fourth derivative of the function being integrated; knowing or estimating the maximum absolute value of this derivative over the interval of integration is key to determining the accuracy of the approximation.

Maximum Value on a Closed Interval

This concept involves finding the highest value that a function achieves within a specified closed interval. In the context of error estimation, it is necessary to determine the point at which the absolute value of the fourth derivative is greatest, ensuring that the error bound in Simpson's Rule is as tight as possible.

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