Determine , the maximum of the absolute value of the second derivative of the integrand over the interval of integration.
Determine
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Context Panel: Assign Function
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From Figure 6.7.3(a), a graph of on , estimate .
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Figure 6.7.3(a) Graph of on
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Write and press the Enter key.
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Context Panel: Optimization≻Maximize (local)
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The numeric optimizer happily finds the "correct" maximum value and returns it as the first entry of a list. (The second member of the list is another list containing the equation that states where the maximum occurred.)
With , solve the inequality for .
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Context Panel: Solve≻Solve
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The appropriate choice of is the first positive integer greater than . Hence, guarantees that the Trapezoid rule will approximate with an error of no more than . From Example 6.7.1, take to be the number . To determine the actual value of for which the Trapezoid rule approximates with the desired accuracy, use the ApproximateInt command as per Table 6.7.3(a).
Initialize
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Tools≻Load Package: Student Calculus 1
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Loading Student:-Calculus1
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Define as the actual value of the integral.
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Use the ApproximateInt command and compare to
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Table 6.7.3(a) The smallest value of for which the Trapezoid rule approximates to within
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By experiment, it is determined that is the smallest value of for which the Trapezoid rule approximates with an error no worse than .