Mean Value Theorem and Inequalities

from below the graph until one of them first touches the graph. Alternatively, one may have to start
with a dotted line above the graph and move it down until it touches.
If the function isn’t differentiable, this approach goes wrong. For instance, it breaks down for
the function f(x) = |x|. The dotted line always touches the graph first at x = 0, no matter what its
slope is, and f?(0) is undefined (see Fig. 2).
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Lecture 14 18.01 Fall 2006
Figure 2: Graph of y = |x|, with secant line. (MVT goes wrong.)
Interpretation of the Mean Value Theorem
You travel from Boston to Chicago (which we’ll assume is a 1,000 mile trip) in exactly 3 hours. At
1000
some time in between the two cities, you must have been going at exactly mph.
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f(t) = position, measured as the distance from Boston.
f(3) = 1000, f(0) = 0, a = 0, and b = 3.
1000
= f(b) ? f(a)
= f?(c)
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where f?(c) is your speed at some time, c.
Versions of the Mean Value Theorem
There is a second way of writing the MVT:
f(b) ? f(a) = f?(c)(b ? a)
f(b) = f(a) + f?(c)(b ? a) (for some c, a < c < b)
There is also a third way of writing the MVT: change the name of b to x.
f(x) = f(a) + f?(c)(x ? a) for some c, a < c < x
The theorem does not say what c is. It depends on f, a, and x.
This version of the MVT should be compared with linear approximation (see Fig. 3).
f(x) ? f(a) + f?(a)(x ? a) x near a
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Lecture