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=0This 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.
huh?
ReplyDeletehahaha. random.
ReplyDeleteanyways, can you be a little more specific in terms of those lines and "a", "b", and "c"?? You seem to be interchanging them. For example, is "c" a point, a slope, or the name of the line?
Also, like I said to the rest of the class, for this week's post, add an example with actual equations.
W/ that being said, GREAT discontinuity example. My favorite one yet!! You even have an equation for it! =)