# Mean Value Theorem: Derivatives vs. Integrals

- Author:
- jeromeawhite

- The relationship between the Mean Value Theorem (MVT) for Derivatives and the MVT for Integrals.
- The Fundamental Theorem of Calculus (FTC) statement involving the derivative of a definite integral. (Whether this shall be referred to as the "first" FTC or the "second" FTC is not consistently agreed upon.)

## MVT for Integrals

Gradually drag point B to the right, counting the unit squares of ice that accumulate in the process. Stop at some point. Divide the ice area you've tallied by the width of the ice block. You have just calculated the _______ ________ of the function over your specified interval.

Drag the "ice" slider all the way to the right, causing it go through the "melting" process and convert into "water." Yes, we're pretending that ice and water have the same density in this metaphor. (Relax, chemistry nerds.) Continuing the metaphor, what does the MVT for Integrals guarantee will occur for at least one location an the specified interval between points A and B?

## MVT for Derivatives

Turning our attention to the F graph, turn on the "tangent & secant" checkbox. An unlabeled point appears on the F curve. Drag that point to a location on the F curve that satisfies the MVT for Derivatives. What does the Theorem guarantee at this particular point?

## Connect MVT for Derivatives to MVT for Integrals

**Tangent line: **At the same x-value(s) where you just dragged the unlabeled point on F(t) to satisfy the MVT for Derivatives, follow the vertical line that passes through that point back to the block of ice f(x). What of significance do you observe at that location on the ice?

**Secant line:** Compute the slope of the secant line for F(t) on the interval between points A and B. How does this slope connect to the block of ice f(x)?

## Fundamental Theorem of Calculus

With the "tangent & secant" checkbox still on, drag the unlabeled point back and forth along the differentiable F(t) curve. What is noteworthy about the slope of the line tangent to F(t) at the unlabeled point?