Question

In estimating cos(3x)dx using Trapezoidal and Simpson's rule with n = 6, we can estimate the error involved in the approximation using the Error Bound formulas. " For Trapezoidal rule, the error will be less than________.

Answer 1

To estimate the error for the Trapezoidal rule used in the approximation of an integral, the sample size and the maximum absolute value of the function's second derivative on the interval are taken into account. With an increase in sample size, the error typically decreases, indicating a more precise approximation.

Step-by-step explanation:

For estimating cos(3x)dx using the Trapezoidal and Simpson's rule with n = 6, the error involved in the approximation can be estimated using Error Bound formulas. Specifically, for the Trapezoidal rule, if we want to discuss the error bound which will be less than a certain amount, it is linked to a function's second derivative's maximum absolute value over the interval of integration.

Based on the reference information provided, it could be interpreted that as the sample size increases, the error bound decreases. However, the exact error bound for the Trapezoidal rule applied to evaluating the integral under question cannot be provided without additional information, such as the interval of integration and the maximum absolute value of the second derivative of the function on that interval.

In general, we could say that the error bound would decrease with a larger sample size because the variability decreases with more data points which makes the approximation more reliable.