Mean Value THeroijem

1)Mean Value Theorem states that when f(x) is continuous and differentiable at all points in a specific interval, there is a guaranteed point that will have the same slope as the average slope of that interval.
* To determine where the average slope is the same as point c, this equation can be used :f'(c)= f(b)-f(a)/b-a


An example in which this theorem works is found in f(x)=x^2, [0,2]
the average slope of the function could be found using f(b)-f(a)/b-a:
*4-0/2-0=2


to find c,plug in c into f'(x)and make it = to the average slope and
*2c=2 -> c=1
The slope at c=1 is equal to the average slope. As a result, the secant line(y=2x) is parallel to the tangent line(y=2x-1).


2)The mean value theorem is not guaranteed when the function isn't both continuous and differentiable, because the tangent line may not exist. If the tangent line doesn't exist.
Example:
Discontinuous:The Mean Value THeorem does not work on this graph because of the jump at x=0. It's discontinuity makes it impossible to find the tangent line at c.


Not DIfferentiable: y=x^(1/3)
For this graph, the mean value theorem works at all points except when x = 0 because there is a vertical slope which prevents it from being differentiable.

In order for the Mean Value Theorem to work, the slopes in the specific interval must actually exist. Also the graph must be "smooth" or without jumps, corners, or cusps in the specific interval.