
MATHEMATICS  THEORY



In this section, Rolle's theorem is introduced along with
three applications. 





Rolle's Theorem

Rolle's theorem hypotheses
Acceleration Equals 0


Rolles's theorem is used to find
a function's horizontal tangent line. It is a special case of the
mean
value theorem which is discussed in the next section. Rolle's
theorem states:
If a function satisfies the following conditions
 the function is continuous in a closed interval [a, b],
 the function is differentiable on this open interval (a, b),
 the value of the function at the two ends are equal,
then there is a point between a and b such that its derivative is 0.
This
theorem can be better understood by examining the following
cases:
 First, examine the case when the function f(x) is a constant.
In this case, df(x)/dx = 0 at all the points from a to b.
For example, when a car is driving
at an uniform velocity from time a to b, the derivative of the velocity,
acceleration, is 0 in this
period.




Velocity is 0 at the Point c 

 Next, take the case when the function value is larger than the
starting point value, f(x) > f(a).
In this case the function value is maximum,
when df(x)/dx equals 0. For example, when a ball is thrown directly upward,
velocity
is 0
when
the
displacement is maximum. In other words, the function maximum will be
at point c (see graphic) where the tangent is horizontal.




df(x)/dx = 0 at Point c 

 Another case is when the function value is smaller than the starting
value f(x) < f(a).
In this case, df(x)/dx = 0 happens at the minimum
value point. For example,
the derivative of a hanging cable at its lowest point is 0. At point
c (see graphic) the function f(x) is a horizontal tangent line.






Notice that when a function satisfy Rolle's
hypotheses, it may have multiple horizontal tangent line at various point
when the function local extreme values are reached. 



