If f has a continuous second derivative on [a, b], then the error E in approximating IEI ≤ (b-a)³ 12n² n = Moreover, if f has a continuous fourth derivative on [a, b], then the error E in approximating |E| ≤ n = (b-a)5 180n4 (a) the Trapezoidal Rule [max [f"(x)], a ≤x≤ b. e²x dx (b) Simpson's Rule fºrex [max f(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. [³₁² f(x) dx by the Trapezoidal Rule is Sº f(x) dx by Simpson's Rule is.
If f has a continuous second derivative on [a, b], then the error E in approximating IEI ≤ (b-a)³ 12n² n = Moreover, if f has a continuous fourth derivative on [a, b], then the error E in approximating |E| ≤ n = (b-a)5 180n4 (a) the Trapezoidal Rule [max [f"(x)], a ≤x≤ b. e²x dx (b) Simpson's Rule fºrex [max f(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. [³₁² f(x) dx by the Trapezoidal Rule is Sº f(x) dx by Simpson's Rule is.
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
|E| ≤
n =
(b − a) ³
12n²
|E| ≤
Moreover, if f has a continuous fourth derivative on [a, b], then the error E in approximating
n =
(b − a) 5
180n4
(a) the Trapezoidal Rule
-[max [f"(x)|], a ≤ x ≤ b.
2x dx
(b) Simpson's Rule
fºr(x:
-[max [ƒ(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.
3
L³e²x
f(x) dx by the Trapezoidal Rule is
b
forex
f(x) dx by Simpson's Rule is](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb898cad9-5347-4e0a-a74d-32f84bfad0f6%2F78b745e2-eaf2-4aa5-a698-752b8c481392%2F88qb0o_processed.png&w=3840&q=75)
Transcribed Image Text:If f has a continuous second derivative on [a, b], then the error E in approximating
|E| ≤
n =
(b − a) ³
12n²
|E| ≤
Moreover, if f has a continuous fourth derivative on [a, b], then the error E in approximating
n =
(b − a) 5
180n4
(a) the Trapezoidal Rule
-[max [f"(x)|], a ≤ x ≤ b.
2x dx
(b) Simpson's Rule
fºr(x:
-[max [ƒ(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.
3
L³e²x
f(x) dx by the Trapezoidal Rule is
b
forex
f(x) dx by Simpson's Rule is
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