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
<< Previous Example Section 4.6
Next Chapter >>
© Maplesoft, a division of Waterloo Maple Inc., 2023. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document