Mean Value Theorem o.O (remix)

1. The graph below shows f(x)=x-sin(x)



There are two lines in this graph, the red and black line.

The red line is the slope at point c, f'(c)

The black line is connected by the points a and b.
It has a slope of f(b)-f(a) / b-a

These two lines are parallel to each other.



2. The graph below shows f(x)=1 / (x-5)^2



This function has an infinite discontinuity as x approaches 5.

This fails the Mean Value Theorem because there is not a tangent line at x=5 that would be parallel to the line joining points A and B




3. The graph below shows f(x)=sin(1/x)

This function is continuous at at all points but it is not differentiable at x=0

This is an oscillating discontinuity; it occurs at a value of x, (in this case, x→0), near to which a function refuses to settle down.

This fails the Mean Value Theorem because there is not a definite slope for point C.