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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Error Analysis In Exercises 37-40, use the error formulas to find n such that the error in the approximation of the definite integral is less than 0.0001 using (a) the Trapezoidal Rule and (b) Simpson’s Rule. See Example 3.

0 2 x 4 d x

(a)

To determine

To calculate: The value of n such that the error in the approximation of the definite integral 02x4dx is less than 0.0001 by the error formula in the Trapezoidal Rule.

Explanation

Given Information:

The definite integral is 02x4dx.

Formula used:

The simple power rule of differentiation,

ddx(xn)=nxn1

The chain rule of differentiation for differential function f(g(x)).

ddx(f(g(x)))=(f(g(x)))(ddx(g(x)))

The quotient rule of differentiation,

ddx(u(x)v(x))=v(x)du(x)dxu(x)dv(x)dx[v(x)]2

The approximate value of integral abf(x)dx for n equal subdivision of closed interval [a,b] by Trapezoidal Rule is,

abf(x)dx(ba2n)[f(x0)+2f(x1)+2f(x2)+2f(x3)++f(xn)]

The approximate error in Trapezoidal Rule for integral abf(x)dx is,

|E|(ba)312n2[max|fn(x)|], axb

The steps to choose the value of n subdivision to calculate error:

Step:1. Find second derivative of function f(x).

Step:2. Find the maximum of |f(x)| on interval [a,b].

Step:3. The inequality for the error as,

|E|(ba)312n2[max|fn(x)|]

Step:4

(b)

To determine

To calculate: The value of n such that the error in the approximation of the definite integral 02x4dx is less than 0.0001 by the error formula in the Simpson’s Rule.

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