Learning Outcomes
 Understand the meaning of Rolle’s theorem.
 Understand the meaning of the Mean Value Theorem
 Verify that the Mean Value Theorem applies and find values $c$ guaranteed by the Mean Value Theorem.
 State three important consequences of the Mean Value Theorem.
Textbook
 Chapter 4.4 The Mean Value Theorem
Textbook Assignment
 p. 388: 161, 164, 168, 171, 174, 186, 188
WeBWork Assignment
 ApplicationMean Value Theorem
Exit problems of the session

Determine whether the Mean Value Theorem applies for the following functions over the given interval $[a, b]$. If yes, then find $c$ that satisfies the Mean Value Theorem.
(a). $y=x^3+2x+1$ over $[0, 2]$ (b). $y=\ln(2x+3)$ over $[1, 2]$
Key Concepts
 Rolle’s Theorem: If $f$ is continuous over $[a, b]$ and differentiable over $(a, b)$, and $f(a)=f(b)=0$, then there exist a point $c \in (a, b)$, such that $f'(c)=0$.
 The Mean Value Theorem: If $f$ is continuous over $[a, b]$ and differentiable over $(a, b)$, then there exist a point $c \in (a, b)$, such that
$f'(c)=\dfrac{f(b)f(a)}{ba}$
 Three important corollaries of the Mean Value Theorem:
 If $f'(x)=0$ over an interval I, then $f(x)$ is constant over I.
 If two differentiable functions $f$ and $g$ satisfy $f'(x)=g'(x)$ over I, then $f(x)=g(x)+C$ for some constant $C$.
 If $f'(x)>0$ over an interval I, then $f$ is increasing over I. If $f'(x)<0$ over an interval I, then $f$ is decreasing over I.
Videos and Practice Problems of Selected Topics
 $\rhd$ The Mean Value Theorem (6:36) The statement and what it means geometrically.
 $\rhd$ A polynomial example (4:49) Given $f(x) =x^26x+8$ and the interval $[2,5]$, find $c$ satisfying the Mean Value Theorem.
 $\rhd$ A square root function example (6:23) Given $f(x) =\sqrt{4x3}$ and the interval $[1,3]$, find $c$ satisfying the Mean Value Theorem.
 * Practice: Using the Mean Value Theorem. (4 problems)