Chapter 6: Techniques of Integration
Section 6.7: Numeric Methods
Use the error bound in Table 6.7.1 to estimate the value of the partition n for which the Trapezoid rule makes an absolute error of no more than 10−4 when estimating λ, the value of the definite integral in Example 6.7.1.
What is the actual value of n for which the Trapezoid rule achieves this accuracy?
If fx=1+sinxlnx+1, write h212 maxx(|f″|) b−a, the error bound in Table 6.7.1, as b−a312 n2maxx(|f″|) and solve the inequality 4−1312 n2M≤10−4 for n, where M=maxx(|f″|)≐0.741. This gives n≐± 129.1, so the appropriate value is n=130. However, for n=105, the Trapezoid rule actually approximates λ with an error no worse than 10−4.
Determine M, the maximum of the absolute value of the second derivative of the integrand over the interval of integration.
Context Panel: Assign Function
fx=1+sinxlnx+1→assign as functionf
From Figure 6.7.3(a), a graph of f″x on 1,4, estimate M=maxx(|f″|)≐0.7.
Figure 6.7.3(a) Graph of f″x on 1,4
Write fx and press the Enter key.
Context Panel: Optimization≻Maximize (local)
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 a=1,b=4,M≐0.741, solve the inequality b−a312 n2maxx(|f″|)≤10−4 for n.
Write the inequality b−a312 n2maxx(|f″|).
Context Panel: Solve≻Solve
The appropriate choice of n is the first positive integer greater than 129.1. Hence, n=130 guarantees that the Trapezoid rule will approximate λ with an error of no more than 10−4. From Example 6.7.1, take λ to be the number 5.078061188. To determine the actual value of n for which the Trapezoid rule approximates λ with the desired accuracy, use the ApproximateInt command as per Table 6.7.3(a).
Tools≻Load Package: Student Calculus 1
Define L as the actual value of the integral.
Use the ApproximateInt command and compare to λ
L−ApproximateIntfx,x=1..4.0,partition=104,method=trapezoid = 0.000100629
L−ApproximateIntfx,x=1..4.0,partition=105,method=trapezoid = 0.000098721
Table 6.7.3(a) The smallest value of n for which the Trapezoid rule approximates λ to within 10−4
By experiment, it is determined that n=105 is the smallest value of n for which the Trapezoid rule approximates λ with an error no worse than 10−4.
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