1. Let SN be the approximated value of the integral of f(x) over an interval [a, b] obtained withSimpson's rule using N subintervals,SN -| f(x) dz.(1)Here, the domain [a, b] is evenly divided into N subintervals with endpoints {ro, x1,..., IN} witha = xo < x1 <.…< ¤N = b. The distance between subinterval endpoints is xk – *k-1 = Ax(b – a)/N. The absolute error in approximating the integral with Simpson's method is bounded,K(b – a)5SN- |f (x) dx <(2)180N4approximation errorwhere K is the maximum of the absolute value of the fourth derivative of f(x) over the interval[a, b], i.e., K = max |f(4) (x)|.1. Prove that Simpson's method is exact for cubic polynomials over any interval [a, 6]. That is,show the approximation error in equation (2) is zero for such functions.2. Demonstrate this principle for the integral of f(x) = 4x3 – 20x? + 12x + 37 over the domain[0, 4] using N = 2 subintervals for Simpson's rule.

Answered: Prove that Simpson's method is exact…

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

6th Edition

ISBN:9781337111348

Author:Bruce Crauder, Benny Evans, Alan Noell

Publisher:Bruce Crauder, Benny Evans, Alan Noell

ChapterA: Appendix

SectionA.2: Geometric Constructions

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Related questions

Question

1. Let SN be the approximated value of the integral of f(x) over an interval [a, b] obtained with
Simpson's rule using N subintervals,
SN -
| f(x) dz.
(1)
Here, the domain [a, b] is evenly divided into N subintervals with endpoints {ro, x1,..., IN} with
a = xo < x1 <.…< ¤N = b. The distance between subinterval endpoints is xk – *k-1 = Ax
(b – a)/N. The absolute error in approximating the integral with Simpson's method is bounded,
K(b – a)5
SN
- |
f (x) dx <
(2)
180N4
approximation error
where K is the maximum of the absolute value of the fourth derivative of f(x) over the interval
[a, b], i.e., K = max |f(4) (x)|.
1. Prove that Simpson's method is exact for cubic polynomials over any interval [a, 6]. That is,
show the approximation error in equation (2) is zero for such functions.
2. Demonstrate this principle for the integral of f(x) = 4x3 – 20x? + 12x + 37 over the domain
[0, 4] using N = 2 subintervals for Simpson's rule.

Expert Solution

Step 1

(1).

Let $f (x) = a_{1} x^{3} + b_{1} x^{2} + c_{1} x + d_{1}$ be an arbitrary cubic polynomial.

$S_{N}$ be the approximated value of the integral of $f (x) = a_{1} x^{3} + b_{1} x^{2} + c_{1} x + d_{1}$ over an arbitrary interval $[a, b]$ obtained with Simpson's rule using $N$ intervals. That is:

$S_{N} \approx \int_{a}^{b} f (x) d x$ .

We know that maximum error using Simpson's rule with $N$ intervals can be calculated as:

$|S_{N} - \int_{a}^{b} f (x) d x| \leq \frac{K {(b - a)}^{5}}{180 N^{4}}$ , where $K = max |f^{(4)} (x)|$ on $[a, b]$ .

Since $f (x) = a_{1} x^{3} + b_{1} x^{2} + c_{1} x + d_{1}$ is a cubic polynomial, therefore its fourth and higher derivative will be zero. That is:

$f^{(4)} (x) = 0$ , therefore $max |f^{(4)} (x)| = 0 = K$ over $[a, b]$ .

Hence the maximum error while calculating $S_{N} \approx \int_{a}^{b} f (x) d x$ is:

$\begin{array}{rcl} |S_{N} - \int_{a}^{b} f (x) d x| & \leq & \frac{K {(b - a)}^{5}}{180 N^{4}} \\ = & \frac{0 {(b - a)}^{5}}{180 N^{4}} \\ = & 0 \end{array}$

Hence the maximum error is zero. Therefore Simpson's rule is exact for the cubic polynomial $f (x) = a_{1} x^{3} + b_{1} x^{2} + c_{1} x + d_{1}$ over arbitrary interval $[a, b]$ .