Chapter 4: Integration
Section 4.6: Average Value and the Mean Value Theorem
Example 4.6.3
In the interval 0,π, find all values of c for which fc=favg.
Solution
Mathematical Solution
From Example 4.6.2, favg=2/π. Hence, the equation sinc=2/π must be solved for c. Figure 4.6.2(a) and Figure 4.6.2(b), and the appropriate calculations, show that c1=arcsin2/π≐0.69011, and c2=π−arcsin2/π≐2.4515.
Maple Solution
Tools≻Load Package: Student Calculus 1
Loading Student:-Calculus1
Find all solutions
Write the equation fc=favg; press the Enter key.
Context Panel: Student Calculus1≻Solve≻Find Roots See Figure 4.6.3(a) for Roots dialog. For exact roots, uncheck "calculate numerically" For numeric roots, check "calculate numerically"
Figure 4.6.3(a) Roots dialog
Exact Roots
Numeric Roots
sinc=2/π
sinc=2π
→roots
arcsin2π,π−arcsin2π
0.6901070914,2.451485562
Alternatively, apply the Roots command directly
Apply the Roots command in the Student Calculus1 package. In this form, exact solutions are returned, if possible.
Rootssinc=2/π,c=0..π
Apply the Roots command in the Student Calculus1 package. In this form, numeric solutions are returned.
Rootssinc=2/π,c=0..π,numeric
The Roots command, which attempts to return all real solutions (of an equation) in a bounded interval, is not presently available in the Context Panel's Solve options prior to Maple 18. The closest syntax-free access to the command in these earlier versions of Maple is via the task template shown in Table 4.6.3(a).
Tools≻Tasks≻Browse: Algebra≻Solve Analytically in Specified Interval
Solve Analytically in a Specified Interval
Enter an expression:
Find the roots in a specified interval:
StudentCalculus1Roots,0..2 π
Express the roots in floating-point form:
evalf
0.6901070912,2.451485563
Table 4.6.3(a) Roots command implemented via task template
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