文档介绍：Section
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Rolle’s Theorem and the
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The two theorems which are at the heart of this section draw connections between the
instantaneous rate of change and the average rate of change of a function. The Mean
Value Theorem, of which Rolle’s Theorem is a special case, says that if f is differentiable
on an interval, then there is some point in that interval at which the instantaneous rate
of change of the function is equal to the average rate of change of the function over the
entire interval. For example, if f gives the position of an object moving in a straight line,
the Mean Value Theorem says that if the average velocity over some interval of time is 60
miles per hour, then at some time during that interval the object was moving at exactly
60 miles per hour. This is not a surprising fact, but it does turn out to be the key to
understanding many useful applications.
Before we turn to a consideration of Rolle’s theorem, we need to establish another
fundamental result. Suppose an object is thrown vertically into the air so that its position
at time t is given by f(t) and its velocity by v(t) = f 0(t). Moreover, suppose it reaches its
maximum height at time t0. On its way up, the object is moving in the positive direction,
and so v(t) > 0 for t < t0; on the way down, the object is moving in the negative direction,
and so v(t) < 0 for t > t0. It follows, by the Intermediate Value Theorem and the fact that
v is a continuous function, that we must have v(t0) = 0. That is, at time t0, when f(t)
0
reaches its maximum value, we have f (t0) = 0. This is an extremely useful fact which
holds in general for differentiable functions, not only at maximum values but at minimum
values as well. Before providing a general demonstration, we first need a few definitions.
Definition A function f is said to have a local maximum at a point c if there exists