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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, 12n² Moreover, if f has a continuous fourth derivative on [a, b], then the error E in approximating is |E| ≤ [ -[max |f"(x)1], a ≤x≤ b. (b-a)5 180n4 3 dx X |E| ≤ [max [f(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 = [ºr(x) (b) Simpson's Rule n = f(x) dx by 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, 12n² Moreover, if f has a continuous fourth derivative on [a, b], then the error E in approximating is |E| ≤ [ -[max |f"(x)1], a ≤x≤ b. (b-a)5 180n4 3 dx X |E| ≤ [max [f(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 = [ºr(x) (b) Simpson's Rule n = f(x) dx by Simpson's Rule

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter4: Calculating The Derivative
Section4.2: Derivatives Of Products And Quotients
Problem 35E
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If f has a continuous second derivative on [a, b], then the error E in approximating [ f(x (b − a)³ |E| ≤ 12n² Moreover, if f has a continuous fourth derivative on [a, b], then the error E in approximating is -[max [f"(x)|], a ≤ x ≤ b. |E| ≤ (b-a)5 180n4 f(x) dx by the Trapezoidal Rule is (a) Trapezoidal Rule n = -[max [f(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. [²³2/dx (b) Simpson's Rule n = [°F(x) f(x) dx by Simpson's Rule
Transcribed Image Text:If f has a continuous second derivative on [a, b], then the error E in approximating [ f(x (b − a)³ |E| ≤ 12n² Moreover, if f has a continuous fourth derivative on [a, b], then the error E in approximating is -[max [f"(x)|], a ≤ x ≤ b. |E| ≤ (b-a)5 180n4 f(x) dx by the Trapezoidal Rule is (a) Trapezoidal Rule n = -[max [f(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. [²³2/dx (b) Simpson's Rule n = [°F(x) f(x) dx by Simpson's Rule
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