Final answer:
To guarantee the accuracy of Simpson's Rule approximation, we need to determine the value of n. The error bound formula is used to calculate the number of intervals necessary to achieve a desired accuracy. The maximum value of the fourth derivative needs to be evaluated for the calculation.
Step-by-step explanation:
The question is asking for the value of n such that the Simpson's Rule approximation on the integral of 19eˣ² from 0 to 1 is accurate to within 0.0001.
Simpson's Rule is an approximation method for finding the definite integral of a function. To ensure the accuracy of the approximation, we need to consider the error bound formula:
Error Bound = ((b - a)³ / (180n²)) * max |f''''(x)|
where a and b are the limits of integration, and n is the number of intervals.
In this case, a = 0 and b = 1. To find the maximum value of the fourth derivative |f''''(x)|, we need to evaluate the function's derivatives and find the maximum among them.
Once we have the maximum value of |f''''(x)|, we can use the error bound formula to solve for n by setting the error bound to be less than or equal to 0.0001.
Let me do the calculations for you.