Mean Value Theorem

4,277 views

Published on

Published in: Technology, Economy & Finance
0 Comments
2 Likes
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total views
4,277
On SlideShare
0
From Embeds
0
Number of Embeds
66
Actions
Shares
0
Downloads
38
Comments
0
Likes
2
Embeds 0
No embeds

No notes for slide

Mean Value Theorem

  1. 1. 4.2 Mean Value Theorem
  2. 2. Mean Value Theorem for Derivatives <ul><li>If y = f(x) is continuous at every point of the closed interval [a,b] and differentiable at every point of its interior (a,b) then there is at least one point c in (a,b) at which </li></ul>
  3. 3. Using Mean Value Theorem <ul><li>Show that f(x) = 2x 2 satisfies the mean value theorem on the interval [0,2]. Then find the solution to the equation on the interval. </li></ul><ul><li>Find f’(x) </li></ul><ul><li>f’(x) = 4x </li></ul><ul><li>4 = 4x </li></ul><ul><li>X= 1 </li></ul>
  4. 4. Using Mean Value Theorem <ul><li>f(x) = l x – 1 l on [0, 4] </li></ul><ul><li>f(a) = -1 </li></ul><ul><li>f(b) = 3 </li></ul><ul><li>1 = l x -1 l </li></ul>The function does not satisfy the mean value theorem because there is a cusp so the function is not continuous on [0,4]
  5. 5. Mean Value Theorem <ul><li>f(x) = -2x 3 + 6x – 2 , [-2 , 2] </li></ul><ul><li>f(-2) = -2(-2) 3 + 6(-2) - 2 = 2 f(2) = -2(2) 3 + 6(2) - 2 = - 6 </li></ul><ul><li>f(x) = -6 - 2 = -2 </li></ul><ul><li> 2 - (-2) </li></ul><ul><li>f '(x) = -6x 2 + 6 </li></ul><ul><li>-2 = -6x 2 + 6 </li></ul><ul><li>X = 2 -2 √3, √3 </li></ul>Mean value is satisfied because the function is continuous on [-2, 2]
  6. 6. Using Mean Value Theorem <ul><li>f(x) = x 3 + 3x – 1, [0,1] </li></ul><ul><li>f(b) = 3 </li></ul><ul><li>f(a) = -1 </li></ul><ul><li>f’(x) = 3x 2 + 3 </li></ul><ul><li>4 = 3x 2 +3 </li></ul><ul><li>X = </li></ul>
  7. 7. Mean Value with Trig. Functions
  8. 8. More mean value Solve.

×