Mean+Value+Theorem!

Recall that we studied the Mean Value Theorem: If a function f(x) is **continuous** on the interval [a, b] and **differentiable** on the open interval (a, b), then there must be some value c within (a, b) such that f ' (c) = ( f(b) - f(a) )/ (b - a). In other words, there must be some point in the interval in which there is a tangent line that is parallel to the secant line!

You might also consider the fact that if the requirements for the mean value theorem hold, then there is a value of c within the interval (a, b) such that

f(b) - f(a) = f ' (c) (b - a).

Please feel free to contribute some thoughts! For example, why is the Mean Value Theorem even taught -- really, it is something to notice, but does it really have any far-reaching implications??

Mimi: The Mean Value Theorem is needed in order to prove the Fundamental Theorem of Calculus! Also, there are many consequences to the Mean Value Theorem that are worth noting, including: Also, I don't know if this is a far-reaching implication, but the Mean Value Theorem applies to the real world, and is important to understanding mechanics. For example, the Mean Value Theorem would state that a plane that traveled 1,000 miles in an hour (it's average speed was 1,000 miles/hour) must have, at some point, had an instantaneous velocity of precisely 1,000 miles/hour.
 * If f'(x) = 0 everywhere on an open interval, then f(x) must be constant on that interval.
 * If f'(x) > 0 for all x values on an open interval, then f(x) must be increasing on that interval.
 * If f'(x) < 0 for all x values on an open interval, then f(x) must be decreasing on that interval.

The Mean Value Theorem tells us that at point c (anywhere within a given interval on a continuous, differentiable function), the tangent line is equal to the secant line of the function. -Alex

I think you mean that the slope of the tangent line is equal to the slope of the secant line! Ms. Sweeney



The homework for Monday night is p. 285 # 7, 8, 9 - 15 odd, 16, 23, and 25. I am attaching my answers on this page!

Notes from today: