Book Problem 17 (a) Use the Midpoint Rule, with n = 4, to approximate the integral f5e-² dx. M4 = (Round your answers to six decimal places.) (b) Compute the value of the definite integral in part (a) using your calculator, such as MATH 9 on the T183/84 or 2ND 7 on the TI-89. f5e-² dr = (c) The error involved in the approximation of part (a) is EM = f 5e-² dx - M₁ = (d) The second derivative f"(x) = The value of K = max |f"(x)| on the interval [0, 4] = (e) Find a sharp upper bound for the error in the approximation of part (a) using the Error Bound Formula EM≤ = (where a and b are the lower and upper limits of integration, n the number of partitions used in part a). 24n2 (f) Find the smallest number of partitions n so that the approximation M, to the integral is guaranteed to be accurate to within 0.001. n=
Book Problem 17 (a) Use the Midpoint Rule, with n = 4, to approximate the integral f5e-² dx. M4 = (Round your answers to six decimal places.) (b) Compute the value of the definite integral in part (a) using your calculator, such as MATH 9 on the T183/84 or 2ND 7 on the TI-89. f5e-² dr = (c) The error involved in the approximation of part (a) is EM = f 5e-² dx - M₁ = (d) The second derivative f"(x) = The value of K = max |f"(x)| on the interval [0, 4] = (e) Find a sharp upper bound for the error in the approximation of part (a) using the Error Bound Formula EM≤ = (where a and b are the lower and upper limits of integration, n the number of partitions used in part a). 24n2 (f) Find the smallest number of partitions n so that the approximation M, to the integral is guaranteed to be accurate to within 0.001. n=
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Book Problem 17**
(a) **Midpoint Rule Approximation:**
Use the Midpoint Rule, with \( n = 4 \), to approximate the integral \( \int_{0}^{4} 5e^{-x^2} \, dx \).
\[ M_4 = \_\_\_ \]
*(Round your answers to six decimal places.)*
(b) **Definite Integral Calculation:**
Compute the value of the definite integral in part (a) using your calculator, such as MATH 9 on the TI83/84 or 2ND 7 on the TI-89.
\[ \int_{0}^{4} 5e^{-x^2} \, dx = \_\_\_ \]
(c) **Error Calculation:**
The error involved in the approximation of part (a) is
\[ E_M = \int_{0}^{4} 5e^{-x^2} \, dx - M_4 = \_\_\_ \]
(d) **Second Derivative Calculation:**
The second derivative \( f''(x) = \_\_\_ \)
The value of \( K = \max |f''(x)| \) on the interval \([0, 4]\) is \( \_\_\_ \)
(e) **Sharp Upper Bound for Error:**
Find a sharp upper bound for the error in the approximation of part (a) using the Error Bound Formula
\[ |E_M| \leq \frac{K(b-a)^3}{24n^2} = \_\_\_ \]
*(where \( a \) and \( b \) are the lower and upper limits of integration, \( n \) is the number of partitions used in part (a)).*
(f) **Number of Partitions Required for Desired Accuracy:**
Find the smallest number of partitions \( n \) so that the approximation \( M_n \) to the integral is guaranteed to be accurate to within 0.001.
\[ n = \_\_\_ \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe8fabc15-e5db-480c-94d9-e42b67b656a8%2Fce0f6917-2619-44ba-b4a7-879938d5b98e%2Fe54sjt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Book Problem 17**
(a) **Midpoint Rule Approximation:**
Use the Midpoint Rule, with \( n = 4 \), to approximate the integral \( \int_{0}^{4} 5e^{-x^2} \, dx \).
\[ M_4 = \_\_\_ \]
*(Round your answers to six decimal places.)*
(b) **Definite Integral Calculation:**
Compute the value of the definite integral in part (a) using your calculator, such as MATH 9 on the TI83/84 or 2ND 7 on the TI-89.
\[ \int_{0}^{4} 5e^{-x^2} \, dx = \_\_\_ \]
(c) **Error Calculation:**
The error involved in the approximation of part (a) is
\[ E_M = \int_{0}^{4} 5e^{-x^2} \, dx - M_4 = \_\_\_ \]
(d) **Second Derivative Calculation:**
The second derivative \( f''(x) = \_\_\_ \)
The value of \( K = \max |f''(x)| \) on the interval \([0, 4]\) is \( \_\_\_ \)
(e) **Sharp Upper Bound for Error:**
Find a sharp upper bound for the error in the approximation of part (a) using the Error Bound Formula
\[ |E_M| \leq \frac{K(b-a)^3}{24n^2} = \_\_\_ \]
*(where \( a \) and \( b \) are the lower and upper limits of integration, \( n \) is the number of partitions used in part (a)).*
(f) **Number of Partitions Required for Desired Accuracy:**
Find the smallest number of partitions \( n \) so that the approximation \( M_n \) to the integral is guaranteed to be accurate to within 0.001.
\[ n = \_\_\_ \]
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