If f has a continuous second derivative on [a, b], then the error E in approximating ["F(x)(b − a)³ [max [ƒ"(x)\], a≤x≤ b.12n²Moreover, if f has a continuous fourth derivative on [a, b], then the error E in approximatingis|E| ≤(b-a)5180n44[²1+x1 + xdxf(x) dx by the Trapezoidal Rule is|E| ≤—[max [ƒ(4)(x)|], a ≤x≤b.Use these to find the minimum integer n such that the error in the approximation of the definite integral is less than orequal to 0.00001 using the Trapezoidal Rule and Simpson's Rule.(a) Trapezoidal Rulen =(b) Simpson's Rulen =[°F(X)f(x) dx by Simpson's Rule

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If f has a continuous second derivative on [a, b], then the error E in approximating is |E| ≤ (b-a)³ 12n² Moreover, if f has a continuous fourth derivative on [a, b], then the error E in approximating |E| ≤ -[max [f"(x)], a≤ x ≤ b. (b - a) 5 180n4 1 1 + x [max [f(4)(x)], a ≤x≤ b. dx ["F(x) (a) Trapezoidal Rule n = f(x) dx by the Trapezoidal Rule is Use these to find the minimum integer 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. 4 (b) Simpson's Rule n = [ F(x) f(x) dx by Simpson's Rule

If f has a continuous second derivative on [a, b], then the error E in approximating is |E| ≤ (b-a)³ 12n² Moreover, if f has a continuous fourth derivative on [a, b], then the error E in approximating |E| ≤ -[max [f"(x)], a≤ x ≤ b. (b - a) 5 180n4 1 1 + x [max [f(4)(x)], a ≤x≤ b. dx ["F(x) (a) Trapezoidal Rule n = f(x) dx by the Trapezoidal Rule is Use these to find the minimum integer 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. 4 (b) Simpson's Rule n = [ F(x) f(x) dx by 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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Question

If f has a continuous second derivative on [a, b], then the error E in approximating ["F(x)
(b − a)³ [max [ƒ"(x)\], a≤x≤ b.
12n²
Moreover, if f has a continuous fourth derivative on [a, b], then the error E in approximating
is
|E| ≤
(b-a)5
180n4
4
[²1+x
1 + x
dx
f(x) dx by the Trapezoidal Rule is
|E| ≤
—[max [ƒ(4)(x)|], a ≤x≤b.
Use these to find the minimum integer 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.
(a) Trapezoidal Rule
n =
(b) Simpson's Rule
n =
[°F(X)
f(x) dx by Simpson's Rule

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Follow-up Question

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
is
|E| ≤
-[max [f"(x)|], a ≤x≤ b.
(b- - a) 5₁
180n4
-[max |ƒ(4)(x)|], a≤ x ≤ b.
4
1
bitra
dx
1 + x
Use these to find the minimum integer 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.
(a) Trapezoidal Rule
n = 1033
(b) Simpson's Rule
n = 61
[° F(X).
X
f(x) dx by Simpson's Rule

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