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)