Learning Outcomes

  1. Understand the meaning of Rolle’s theorem.
  2. Understand the meaning of the Mean Value Theorem
  3. Verify that the Mean Value Theorem applies and find values $c$ guaranteed by the Mean Value Theorem.
  4. 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

  • Application-Mean Value Theorem

Exit problems of the session 

  1. 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

$f'(c)=\dfrac{f(b)-f(a)}{b-a}$

  • Three important corollaries of the Mean Value Theorem: 
    1. If $f'(x)=0$ over an interval I, then $f(x)$ is constant over I.
    2. If two differentiable functions $f$ and $g$ satisfy $f'(x)=g'(x)$ over I, then $f(x)=g(x)+C$ for some constant $C$. 
    3. 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

  1. $\rhd$ The Mean Value Theorem  (6:36) The statement and what it means geometrically.
  2. $\rhd$ A polynomial example (4:49) Given $f(x) =x^2-6x+8$ and the interval $[2,5]$, find $c$ satisfying the Mean Value Theorem.
  3. $\rhd$ A square root function example (6:23) Given $f(x) =\sqrt{4x-3}$ and the interval $[1,3]$, find $c$ satisfying the Mean Value Theorem.
  4. * Practice: Using the Mean Value Theorem. (4 problems)