$\"\"$

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Consider the cubic function $\"\"$ on $\"\"$ .

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The integral is $\"\"$ .

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Approximate error in Simpsons rule $\"\"$ .

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The first derivative is $\"\"$ .

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The second derivative is $\"\"$ .

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The third derivative is $\"\"$ .

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The fourth derivative is $\"\"$ .

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The maximum value of $\"\"$ on the interval $\"\"$ is $\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$ .

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The value of an error is zero.

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Therefore, the Simpsons rule is exact when approximating the integral of a cubic polynomial function.

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$\"\"$

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The definite integral is $\"\"$ , $\"\"$ .

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Consider the function $\"\"$ .

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The first derivative is $\"\"$ .

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The second derivative is $\"\"$ .

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The third derivative is $\"\"$ .

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The fourth derivative is $\"\"$ .

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The maximum value of $\"\"$ on the interval $\"\"$ is $\"\"$ .

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Approximate error in Simpsons rule $\"\"$ .

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Substitute $\"\"$ , $\"\"$ and $\"\"$ .

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$\"\"$ .

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The value of an error is zero.

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$\"\"$

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The Simpsons rule is exact when approximating the integral of a cubic polynomial function.

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Using Simpsons rule the error of $\"\"$ is zero.