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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 49 and 50, 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.

0 5 e x / 5 d x

(a)

To determine

To calculate: The value of n such that the error in the approximation of the definite integral 05ex5dx is less than 0.0001 by using the error formula in the Trapezoidal Rule.

Explanation

Given Information:

The definite integral is 05ex5dx.

Formula used:

The errors E in approximating abf(x)dx is shown below:

Trapezoidal Rule: |E|(ba)312n2[max|f"(x)|],axb

Calculation:

Consider the definite integral 05ex5dx.

Steps to determine the value of n to choose, are as follows:

1. Begin by finding the second derivative of f(x)=ex5.

f(x)=ex5f(x)=15ex5f(x)=125ex5

(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 05ex5dx is less than 0.0001 by using the error formula in the Simpson’s Rule.

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