Section 4.6: The Mean Value Theorem
Why would this require continuity and differentiability?
Example: Determine whether Rolle’s theorem applies
to the function ( )
on the interval [-6, 0].
√
If so, find the point(s) that are guaranteed to exist by
that theorem.
Note: For ( )
secant from (
, what is the slope of the
√
( )) to ( ( ))?
As we’ve just seen, there’s a point (at x = -4) where the
slope of the tangent is equal to the slope of this secant.
In fact, for any points a and b that we pick in (-6, 0),
there must be some point c for which the slope of the
tangent at c is equal to the slope of the secant between
( )) and (
( )).
(
Example: A car travels 100 miles in two hours. At
some point(s) in that two-hour period, the speedometer
must have read 50 miles per hour. How do we know?
Example: Determine whether the Mean Value
Theorem applies to the function ( )
interval [1, 4]. If so, find the point(s) that are
guaranteed to exist by that theorem.
on the