$\"\"$

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(a)

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

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Let $\"\"$ .

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Error in Trapezoidal rule:

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If $\"image\"$ has a continuous second derivative on $\"\"$ , then the error approximating the integral $\"\"$ by Trapezoidal Rule is $\"\"$ , $\"\"$ .

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The function $\"\"$ .

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First derivative is $\"\"$ .

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Second derivative is $\"\"$ .

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The second derivative is continuous on the interval $\"\"$ .

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

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Approximate error in trapezoidal rule:

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$\"\"$

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

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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Error in Trapezoidal Rule is $\"\"$ .

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$\"\"$

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(b)

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

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Error in Simpsons Rule:

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If $\"image\"$ has a continuous fourth derivative on $\"\"$ , then the error approximating the integral $\"\"$ by Simpsons Rule is $\"\"$ , $\"\"$ .

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The function $\"\"$ .

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First derivative is $\"\"$ .

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Second derivative is $\"\"$ .

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Third derivative is $\"\"$ .

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Fourth derivative is $\"\"$ .

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The fourth derivative is continuous on the interval $\"\"$ .

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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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$\"\"$

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

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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Error in Simpsons rule is $\"\"$ .

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$\"\"$

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(a) Error in Trapezoidal rule is $\"\"$ .

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(b) Error in Simpsons rule is $\"\"$ .