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Chapter 7: Problem 6

State how to compute the Simpson's Rule approximation \(S(2 n)\) if the Trapezoid Rule approximations \(T(2 n)\) and \(T(n)\) are known.

Short Answer

Expert verified
Answer: The Simpson's Rule approximation, \(S(2n)\), can be computed using the Trapezoid Rule approximations \(T(2n)\) and \(T(n)\) by plugging their values into the following relationship: \(S(2n) = \frac{4 T(2n) - T(n)}{3}\).

Step by step solution

01

Understanding the Trapezoid Rule

The Trapezoid Rule is a method for approximating definite integrals of functions by dividing the area under the curve into trapezoids. The formula for the Trapezoid Rule approximation, \(T(n)\), of the integral of function \(f(x)\) over an interval \([a, b]\) is: \(T(n) = \frac{b-a}{2n} \left[f(a) + 2\sum_{k=1}^{n-1} f\left(a+k\frac{b-a}{n}\right) + f(b)\right]\)
02

Understanding the Simpson's Rule

Simpson's Rule is a method for approximating definite integrals of functions by dividing the area under the curve into parabolic sections. The formula for the Simpson's Rule approximation, \(S(2n)\), of the integral of the function \(f(x)\) over an interval \([a, b]\) is: \(S(2n) = \frac{b-a}{6n} \left[f(a) + 4\sum_{k=1}^{n} f\left(a+(2k-1)\frac{b-a}{2n}\right) + 2\sum_{k=1}^{n-1} f\left(a+(2k)\frac{b-a}{2n}\right) + f(b)\right]\)
03

Connecting Trapezoid Rule and Simpson's Rule

To find the Simpson's Rule approximation \(S(2n)\) using the Trapezoid Rule approximations \(T(2n)\) and \(T(n)\), we'll use their respective formulas and the following relationship: \(S(2n) = \frac{4 T(2n) - T(n)}{3}\)
04

Computing the Simpson's Rule approximation

Once the Trapezoid Rule approximations \(T(2n)\) and \(T(n)\) are known, we simply plug their values into the relationship derived above to compute the Simpson's Rule approximation \(S(2n)\): \(S(2n) = \frac{4 T(2n) - T(n)}{3}\)

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