Explanation Determine the end points and use those to determine the value of M 10 and S 10 for the integral. ∫ 0 π 1 − x 2 d x Given: Sometimes Simpson’s Rule performs poorly – calculate M 10 and S 10 for the integral ∫ 0 π 1 − x 2 d x whose value we know to be π 4 (one quarter of the area of the unit circle). Calculation: Given ∫ 0 π 1 − x 2 d x Let f ( x ) = 1 − x 2 The value of Δ x = 0.1 M 10 = 1 10 [ 1 − 0.05 2 + 1 − 0... To determine (b) How to explain the result of part (a) using the hint that the error bounds are not valid because | f ' ' ( x ) | and | f ( 4 ) ( x ) | tend to ∞ as x → 1 but | f ( 4 ) ( x ) | goes to infinity more quickly.

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Numerical Integration

In a mathematical investigation, numerical integration comprises a wide group of calculations for computing the mathematical estimation of definite integral, and likewise, the term is moreover in some cases used to depict the numerical solution of differ…

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Chapter 8.9, Problem 64E

To determine

(a)

We usually expect the value of $S_{N}$ to be more accurate than $M_{N}$ . Which of $M_{10}$ and $S_{10}$ is more accurate in this case?

To determine

(b)

How to explain the result of part (a) using the hint that the error bounds are not valid because $| f'' (x) |$ and $| f^{(4)} (x) |$ tend to $\infty$ as $x \to 1$ but $| f^{(4)} (x) |$ goes to infinity more quickly.

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Chapter 8.9, Problem 63E

Chapter 8, Problem 1CRE

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(a) We usually expect the value of S N to be more accurate than M N . Which of M 10 and S 10 is more accurate in this case?

Solution Summary: The author concludes that M 10 and S 10 are more accurate in this case.

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(a)

We usually expect the value of $S_{N}$ to be more accurate than $M_{N}$ . Which of $M_{10}$ and $S_{10}$ is more accurate in this case?

(b)

How to explain the result of part (a) using the hint that the error bounds are not valid because $| f'' (x) |$ and $| f^{(4)} (x) |$ tend to $\infty$ as $x \to 1$ but $| f^{(4)} (x) |$ goes to infinity more quickly.

Want to see the full answer?

Chapter 8 Solutions

(a) We usually expect the value of S N to be more accurate than M N . Which of M 10 and S 10 is more accurate in this case?

Solution Summary: The author concludes that M 10 and S 10 are more accurate in this case.

Concept explainers

Videos

(a) We usually expect the value of SNto be more accurate than MN. Which of M10 and S10 is more accurate in this case?

(b) How to explain the result of part (a) using the hint that the error bounds are not valid because |f''(x)| and |f(4)(x)| tend to ∞ as x→1 but |f(4)(x)| goes to infinity more quickly.

Want to see the full answer?

Chapter 8 Solutions

(a)

We usually expect the value of $S_{N}$ to be more accurate than $M_{N}$ . Which of $M_{10}$ and $S_{10}$ is more accurate in this case?

(b)

How to explain the result of part (a) using the hint that the error bounds are not valid because $| f'' (x) |$ and $| f^{(4)} (x) |$ tend to $\infty$ as $x \to 1$ but $| f^{(4)} (x) |$ goes to infinity more quickly.