Chapter 4: Integration
Section 4.6: Average Value and the Mean Value Theorem
Use fx=x3−x,x∈0,3, to illustrate the connection between the average value of f and the Mean Value theorem.
The average value of f is 13∫03x3−x ⅆx = 214.
The Mean Value theorem states that f will attain its average value at least once on the interval 0,3. This occurs at the solution of the equation fx=21/4, which is
Since 0,f0=0,0 and 3,f3=3,24, the slope of the "secant" line is 8. The solution of the equation f′x=3 x2−1=8 is x=3.
The point at which the function attains its average value is not the same point at which the tangent is parallel to the secant!
Tools≻Load Package: Student Calculus 1
Define the function f
Context Panel: Assign Function
fx=x3−x→assign as functionf
Figure 4.6.4(a) provides a screen-shot of the
tutor applied to fx. Figure 4.6.4(b) provides a screen-shot of the
tutor applied to fx.
Figure 4.6.4(a) Function Average tutor
Figure 4.6.4(b) Mean Value Theorem tutor
Table 4.6.4(a) details the calculations of the points where fx=favg and where the tangent line is parallel to the secant line.
Find where fx=favg
Write the appropriate equation.
Press the Enter key.
Context Panel: Solve≻Numerically Solve
Find where f′x=8
Context Panel: Solve≻Solve
Table 4.6.4(a) Further calculations for favg and the Mean Value theorem
The most direct way to obtain the exact solution of the equation fx=favg is shown below.
The default output for the MeanValueTheorem command is the graph shown in Figure 4.6.4(b). Alternate usages are shown in Table 4.6.4(b).
MeanValueTheoremfx,x=0..3,output=points = 3
MeanValueTheoremfx,x=0..3,output=points,numeric = 1.732050808
Table 4.6.4(b) Alternate usages for the MeanValueTheorem command
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