f'(c) = [f(b) - f(a)]/(b - a).
Graphically:
This means that in the interval [a,b] there will be a point c whose tangent line's slope will be the same as the slope from the secant line between points a and b, which will make both lines parallel to each other.
By breaking down the formula that defines the Mean Value Theorem (f'(c) = [f(b) - f(a)]/(b - a)) we have that:
Tangent line's slope at point c defined by f'(c) =
Secant line slope defined by [(b) - f(a)]/(b -a).
2.- However, the Mean Value Theorem fails if in the interval [a,b] the function is discontinuous or not differentiable.
DISCONTINUOUS:
Here the Mean Value Theorem Fails because in the interval [a,b] the function is discontinuous and there won't be any point c between [a,b] because it just doesn't exist.
NOT DIFFERENTIABLE:
The Mean Value Theorem fails in the interval [-4,4] because it has a corner on x=0 which makes the function not differentiable at that point. This means that a tangent line at x=0 doesn't exist. 
Once again, the second graph helped clear up where the Mean Value Theorem fails. =D
ReplyDeleteGREAT breakdown of what the crazy math symbols mean! Can you use equations as an example for f, f', and the equation of the tangent line?
ReplyDeleteFor your discontinuous example though, there does seem to be a c where the tangent is parallel to the secant. It's a little bit to the right of the discontinuity. I love this example. Can you explain what that means in your new post? Does it mean the Mean Value Theorem conditions are wrong?
hey i like how you explained why the theorem fails
ReplyDeleteGreat Graphssss and explanationss,
ReplyDeleteclear!
P.S. I miss 1st period :(