   Chapter 7.7, Problem 22E

Chapter
Section
Textbook Problem

# How large should n be to guarantee that the Simpson’s Rule approximation to ∫ 0 1 x x 2 d x is accurate to within 0.00001?

To determine

To find: n such that Simpson’s rule approximation 01ex2dx accurately to within 105

Explanation

The provided function is:

f(x)=ex2

Find the fourth derivative of the above function f(x)=ex2 as follows:

f(x)=2xex2f(x)=2ex2+4x2ex2

Calculate f(x) as follows:

f(x)=2(2xex2(2x2+1)+ex2(4x))=4ex2(2x3+3x)

Calculate f(4)(x) as follows:

f(4)(x)=4(2xex2(2x3+3x)+ex2(6x2+3))=4ex2(4x4+12x2+3)

Now, the fifth derivative will be non-negative for the interval [0,1]

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