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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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Use the error formulas to find n such that the error in the approximations of

0 1 ( 2 x 6 + 1 ) d x is less than 0.01 using (a) the Trapezoidal Rule and (b) Simpson’s Rule.

(a)

To determine

To calculate: The value of n such that the error in the approximation of the definite integral 01(2x6+1)dx is less than 0.01 by using the error formula in the Trapezoidal Rule.

Explanation

Given Information:

The definite integral is 01(2x6+1)dx.

Formula used:

According to Trapezoidal Rule the error E in approximating abf(x)dx is as shown,

|E|(ba)312n2[max|f(x)|],axb

Calculation:

Consider the definite integral 01(2x6+1)dx.

Steps to determine the value of n are as follows:

(1) Begin by finding the second derivative of f(x)=(2x6+1).

f(x)=(2x6+1)f1(x)=12x5f2(x)=60x4

(2) Find the maximum of |f2(x)| on the interval [a,b]

(b)

To determine

To calculate: The value of n such that the error in the approximation of the definite integral 01(2x6+1)dx is less than 0.01 by using the error formula in the Simpson’s Rule.

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