#11: MEAN VALUE THEOREM

The Mean Value Theorem is used to prove many of the theorems and properties developed in Calculus. It is also used in Science when physical processes are studied and differential equations are developed to model the process. It is important in this course to know what the theorem says, its graphical interpretation, and when it applies. Rolle's Theorem is a simplified version of the Mean Value Theorem and is used to prove it.

ROLLE'S THEOREM: Let f be a continuous function defined on the closed interval [a,b], differentiable on the open interval (a,b), with f (a) = f ( b). Then there is a value of c in the interval (a,b) such that f ' (c) = 0. (Note: the theorem guarantees at least one value of c on the interval, but there may be more than one. )

1. f (x) = 1 / x on [-1,1]

  1. Evaluate f (1) and f (-1).

  2. Find f ' (x).

  3. Does Rolle's Theorem apply? If it does,

    1. find c

    2. graph f and the tangent line at x = c.

2. f (x) = sin x on [ 0, 2 ]

  1. Evaluate f (0) and f (2).

  2. Find f ' (x).

  3. Rolle's Theorem apply? If it does,

    1. find c

    2. graph f and the tangent line at x = c.

3. TRY THIS #1: f (x) = tan x on [ 0, ]

  1. Evaluate f (0) and f ().

  2. Find f ' (x).

  3. Does Rolle's Theorem apply? If it does,

    1. find c

    2. graph f and the tangent line at x = c.


4. f (x) = cos x on [ 0, ]

  1. Evaluate f (0) and f ().

  2. Find f ' (x).

  3. Does Rolle's Theorem apply? If it does,

    1. find c

    2. graph f and the tangent line at x = c.



5. TRY THIS #2: f (x) = x 2 - 4x + 5 on [0,4]

  1. Evaluate f (0) and f (4).

  2. Find f ' (x).

  3. Does Rolle's Theorem apply? If it does,

    1. find c

    2. graph f and the tangent line at x = c.


6. f (x) = | x | on [-1,1]

  1. Evaluate f (-1) and f (1).

  2. Find f ' (x).

  3. Does Rolle's Theorem apply? If it does,

    1. find c

    2. graph f and the tangent line at x = c.



7. TRY THIS #3: f (x) = x 1 / 3 on [-1,1]

  1. Evaluate f (-1) and f (1).

  2. Find f ' (x).

  3. Does Rolle's Theorem apply? If it does,

    1. find c

    2. graph f and the tangent line at x = c.


8. f (x) = x 2 + 5x + 1 on [-1,2]

  1. Evaluate f (-1) and f (2).  

  2. Find f ' (x).

  3. Does Rolle's Theorem apply? If it does,

    1. find c

    2. graph f and the tangent line at x = c.



9. TRY THIS #4: f (x) = x 3 + 1 on [1,2]

  1. Evaluate f (1) and f (2)

  2. Find f ' (x).

  3. Does Rolle's Theorem apply? If it does,

    1. find c

    2. graph f and the tangent line at x = c.


MEAN VALUE THEOREM: Let f be a continuous function defined on the closed interval [a,b] and differentiable on the open interval (a,b). Then there is a value of c in the interval (a,b) such that

(Note: the theorem guarantees at least one value of c on the interval, but there may be more than one. )

Geometric interpretation of the Mean Value Theorem: Where the theorem applies, there is a value of c on the interval (a,b) where the tangent line to the function at x = c is parallel to the secant line through the points on the curve at x = a and x = b.

10. f (x) = 1 / x on [-1,1]

  1. Evaluate f (-1) and f (1).       

  2. Find f ' (x).

  3. Does the Mean Value Theorem apply? If it does,

    1. find c

    2. graph f , the secant line and the tangent line at x = c.



11. f (x) = sin x on [ /2, 3/2 ]

  1. Evaluate f (/2) and f (3/2).

  2. Find f ' (x).

  3. Does the Mean Value Theorem apply? If it does

    1. find c

    2. graph f , the secant line and the tangent line at x = c.



12. TRY THIS #5: f (x) = cos x on [0, 3/2 ]

  1. Evaluate f (0) and f (3/2).

  2. Find f ' (x).

  3. Does the Mean Value Theorem apply? If it does,

    1. find c

    2. graph f , the secant line and the tangent line at x = c.


13. f (x) = | x | on [-1,1]

  1. a) Evaluate f (-1) and f (1).

  2. Find f ' (x).

  3. Does the Mean Value Theorem apply? If it does,

    1. find c

    2. graph f , the secant line and the tangent line at x = c.



14. TRY THIS #6: f (x) = x 1 / 3 on [-1,1]

  1. Evaluate f (-1) and f (1).

  2. Find f ' (x).

  3. Does the Mean Value Theorem apply? If it does,

    1. find c

    2. graph f , the secant line and the tangent line at x = c.


15. f (x) = x 2 + 5x + 1 on [-1,2]

  1. Evaluate f (-1) and f (2).

  2. Find f ' (x).

  3. Does the Mean Value Theorem apply? If it does,

    1. find c

    2. graph f , the secant line and the tangent line at x = c.



16. TRY THIS #7: f (x) = x 3 + 1 on [1,2]

  1. Evaluate f (1) and f (2).

  2. Find f ' (x).

  3. Does the Mean Value Theorem apply? If it does,

    1. find c

    2. graph f , the secant line and the tangent line at x = c.


ROXANNE BYRNE :UNIVERSITY OF COLORADO AT DENVER: ©:2002, Roxanne Byrne