   Chapter 6.3, Problem 48E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Using Simpson’s Rule Prove that Simpson’s Rule is exact when approximating the integral of a cubic polynomial function, and demonstrate the result with n   = 2 for ∫ 0 1 x 3 d x

To determine

To prove: The Simpson’s Rule is exact when approximating the integral of a cubic polynomial function, and also demonstrates the result with n=2 for the function 01x3dx.

Explanation

Given Information:

The given definite integral is 01x3dx.

Formula used:

Simpson’s Rule:

If f is continuous on [a,b] and n is an even integer, then

abf(x)dx,n=(ba3n)[f(x0)+4f(x1)+2f(x2)+4f(x3)++4f(xn1)+f(xn)].

The errors E in approximating abf(x)dx by the Simpson’s Rule:

|E|(ba)5180n2[max|f4(x)|],axb

Proof:

Consider the definite integral 01x3dx.

To prove Simpson’s Rule is exact when approximating the integral of a cubic polynomial function.

Let P3(x)=ax3+bx2+cx+d is a cubic polynomial function.

The first derivative of P3(x)=ax3+bx2+cx+d is,

P31(x)=ddx(ax3+bx2+cx+d)=3ax2+2bx+c

The second derivative of the function is,

P32(x)=ddx(3ax2+2bx+c)=6ax+2b

The third derivative of the function is,

P33<

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