If f has a continuous second derivative on [a, b], then the error E in approximating f(x) dx by the Trapezoidal Rule is (b − a)³ |E| ≤ 12n² Moreover, if f has a continuous fourth derivative on [a, b], then the error E in approximating [ f(x) dx by Simpson's Rule is (b-a) 180n |Ε| = -[max (4)(x)], a ≤x≤ b. Use these to find the minimum n such that the error in the approximation of the definite integral is less than or equal to 0.00001 using the Trapezoidal Rule and Simpson's Rule. e²x dx n -[max IF"(x)], a ≤ x ≤ b. (a) the Trapezoidal Rule n= (b) Simpson's Rule
If f has a continuous second derivative on [a, b], then the error E in approximating f(x) dx by the Trapezoidal Rule is (b − a)³ |E| ≤ 12n² Moreover, if f has a continuous fourth derivative on [a, b], then the error E in approximating [ f(x) dx by Simpson's Rule is (b-a) 180n |Ε| = -[max (4)(x)], a ≤x≤ b. Use these to find the minimum n such that the error in the approximation of the definite integral is less than or equal to 0.00001 using the Trapezoidal Rule and Simpson's Rule. e²x dx n -[max IF"(x)], a ≤ x ≤ b. (a) the Trapezoidal Rule n= (b) Simpson's Rule
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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![If \( f \) has a continuous second derivative on \([a, b]\), then the error \( E \) in approximating \(\int_a^b f(x) \, dx\) by the Trapezoidal Rule is
\[
|E| \leq \frac{(b-a)^3}{12n^2} \left[\max |f''(x)|\right], \quad a \leq x \leq b.
\]
Moreover, if \( f \) has a continuous fourth derivative on \([a, b]\), then the error \( E \) in approximating \(\int_a^b f(x) \, dx\) by Simpson's Rule is
\[
|E| \leq \frac{(b-a)^5}{180n^4} \left[\max |f^{(4)}(x)|\right], \quad a \leq x \leq b.
\]
Use these to find the minimum \( n \) such that the error in the approximation of the definite integral is less than or equal to 0.00001 using the Trapezoidal Rule and Simpson's Rule.
\[
\int_1^3 e^{2x} \, dx
\]
(a) the Trapezoidal Rule
\[ n = \text{ } \]
(b) Simpson's Rule
\[ n = \text{ } \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3738e16a-10fd-4378-b4a3-b8733d2d5bb5%2F72a77aa9-7606-4e7c-b611-f8c7cf09adc9%2Fxhy0dmk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:If \( f \) has a continuous second derivative on \([a, b]\), then the error \( E \) in approximating \(\int_a^b f(x) \, dx\) by the Trapezoidal Rule is
\[
|E| \leq \frac{(b-a)^3}{12n^2} \left[\max |f''(x)|\right], \quad a \leq x \leq b.
\]
Moreover, if \( f \) has a continuous fourth derivative on \([a, b]\), then the error \( E \) in approximating \(\int_a^b f(x) \, dx\) by Simpson's Rule is
\[
|E| \leq \frac{(b-a)^5}{180n^4} \left[\max |f^{(4)}(x)|\right], \quad a \leq x \leq b.
\]
Use these to find the minimum \( n \) such that the error in the approximation of the definite integral is less than or equal to 0.00001 using the Trapezoidal Rule and Simpson's Rule.
\[
\int_1^3 e^{2x} \, dx
\]
(a) the Trapezoidal Rule
\[ n = \text{ } \]
(b) Simpson's Rule
\[ n = \text{ } \]
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