revision!

1. Mean Value Theroem
f'(c)=f(b)-f(a)/b-a
What this means is that the slope of the tangent line is parallel to the slope of the secant line. Both of the lines will have the same slope and be equal to one another.

An example would be where:
f(x)=cos(x)+1the blue line would be the tangent line (y=2) and the green line would be the secant line (y=1) the interval would be (-2,2). they are parallel to each other and their slopes equal on another as well.

2. Why does the Mean Value Theorem only work for continuous and differentiable functions?
- A continuous function has the same limit from both the negative and positive sides.
- A diferentiable function has a slope at a certain point, If f is differentiable at a point c, then f is continuous at that point c.

Here are some examples:

Differentiable but not continuous:

In the function: x^(-1)+1, there is a hole at x=0, making it differentiable but not continuous because there is a hole discontinuity in the graph.

Continuous but not differentiable:The function:x^(1/2)+1 on the interval [-1,1] is continuous, but not differentiable because there is a cusp at x=0.

A very MEAN value theorem,

The mean value theorem shows how the slope of the tangent line and the slope of the secant line are parallel to each other. The equation that we would use would be x^2.
The blue line would be the tangent line (derivative) and the green line would be the secant line.
by using the mean value theorem, we can prove that the tangent and secant line are parallel to one another.

If we use the interval [0,2], then the equation of the tangent line is 2x-1.
The derivative would be 2x.
They would be parallel at x=1.


The mean value theorem fails when:

There is a point when the function is not differentiable.
such as: f(x)=abs(x)And where a function would be discontinuous
an example would be at f(x)=1/x

The function would be discontinuous at x=0.