Chapter 3: Applications of Differentiation
Section 3.4: Differentials and the Linear Approximation
Apply the Mean Value theorem to the function fx=x e−x,0≤x≤3.
Determine c from first principles.
Figure 3.4.2(a) contains an image of the
tutor applied to fx=x e−x on the interval 0,3.
The value of c=0.880 is returned for the point where the tangent line is parallel to the secant line drawn from 0,f0 to 3,f3.
Amongst the Student packages, only the LinearAlgebra subpackage provides control over the number of digits used by the tutors. In the Mean Value Theorem tutor, the number of digits used for c, namely, three, is hard-coded in the tutor.
Figure 3.4.2(a) Mean Value Theorem tutor
Solution from first principles
Type fx=…, being sure to use the exponential "e".
Context Panel: Assign Function
fx=x ⅇ−x→assign as functionf
Write the equation fb=fa+f′cb−a, being sure to put a space between f′c and b−a. Press the Enter key.
Context Panel: Solve≻Numerically Solve
Animation of Figure 3.4.2(b)
Figure 3.4.2(b) animates the graph in Figure 3.4.2(a). The slider controls the value of b, the right endpoint of the interval 0,b.
Click on the graph to access the animation toolbar, in the center of which is a slider that controls the value of b.
Figure 3.4.2(b) Animation of the Mean Value theorem
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