4.2 HW Problems

Table of Contents:

Rolle's Theorem Explained and Mean Value Theorem For Derivatives - Examples - Calculus

• We have already made clear that the derivative of constant functions are zero. But how about more complex functions that have a derivative that is ALWAYS zero?
• If two functions have identical derivatives over an interval, how are the two related?
• First let me introduce you to Rolle’s Theorem (which is used to prove the Mean Value Theorem).

Rolle’s Theorem

Rolle’s Theorem says that a differentiable curve has at least ONE horizontal tangent between two points where it crosses a horizontal line. It may have just one (a), or it may have more (b).

• Suppose that $y=f(x)$ is continuous over the closed interval [a,b] and differentiable at every point of its interior (a,b).
• If $f(a)=f(b)$, then thee is at least one number c in (a,b) at which $f’(c)=0$.

Proof

• The hypothesis of this Theorem is important. One failure (even at one point) will cause the graph to not even have a horizontal tangent.

There may be no horizontal tangent if the hypotheses of Rolle’s Theorem do not hold.

Rolle's Theorem with Examples

The Mean Value Theorem

• If you take Rolle’s Theorem and slant it, badabing badaboom you now have the Mean Value Theorem.