Section 2.9
              What does f say about f ?

                           Math 1a


                      February 1...
Outline




   Cleanup



   Increasing and Decreasing functions



   Concavity and the second derivative
Last worksheet, problem 2

   Graphs of f , f , and f   are shown below. Which is which? How
   can you tell?
            ...
Solution
Again, look at the horizontal tangents. The short-dashed curve has
horizontal tangents where no other curve is ze...
Outline




   Cleanup



   Increasing and Decreasing functions



   Concavity and the second derivative
Definition
    Let f be a function defined on and interval I . f is called
    increasing if

                     f (x1 ) <...
Definition
    Let f be a function defined on and interval I . f is called
    increasing if

                     f (x1 ) <...
Examples: Increasing
Examples: Decreasing
Examples: Neither
Fact
       If f is increasing and differentiable on (a, b), then f (x) ≥ 0
       for all x in (a, b)
Fact
       If f is increasing and differentiable on (a, b), then f (x) ≥ 0
       for all x in (a, b)
       If f is decre...
Fact
       If f is increasing and differentiable on (a, b), then f (x) ≥ 0
       for all x in (a, b)
       If f is decre...
Example
Here is a graph of f . Sketch a graph of f .
Example
Here is a graph of f . Sketch a graph of f .
Fact
       If f (x) > 0 for all x in (a, b), then f is increasing on (a, b).
       If f (x) < 0 for all x in (a, b), the...
Fact
       If f (x) > 0 for all x in (a, b), then f is increasing on (a, b).
       If f (x) < 0 for all x in (a, b), the...
Outline




   Cleanup



   Increasing and Decreasing functions



   Concavity and the second derivative
Definition
    A function is called concave up on an interval if f is
    increasing on that interval.
Definition
    A function is called concave up on an interval if f is
    increasing on that interval.
    A function is ca...
Fact
       If f is concave up on (a, b), then f (x) ≥ 0 for all x in (a, b)
Fact
       If f is concave up on (a, b), then f (x) ≥ 0 for all x in (a, b)
       If f is concave down on (a, b), then f...
Fact
       If f (x) > 0 for all x in (a, b), then f is concave up on (a, b).
       If f (x) < 0 for all x in (a, b), the...
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Lesson 7: What does f' say about f?

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Given a function f, the derivative f' can be used to get important information about f. For instance, f is increasing when f'>0. The second derivative gives useful concavity information.

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Lesson 7: What does f' say about f?

  1. 1. Section 2.9 What does f say about f ? Math 1a February 15, 2008 Announcements no class Monday 2/18! No office hours 2/19. ALEKS due Wednesday 2/20 (10% of grade). Office hours Wednesday 2/20 2–4pm SC 323 Midterm I Friday 2/29 in class (up to §3.2)
  2. 2. Outline Cleanup Increasing and Decreasing functions Concavity and the second derivative
  3. 3. Last worksheet, problem 2 Graphs of f , f , and f are shown below. Which is which? How can you tell? y x
  4. 4. Solution Again, look at the horizontal tangents. The short-dashed curve has horizontal tangents where no other curve is zero. So its derivative is not represented, making it f . Now we see that where the bold curve has its horizontal tangents, the short-dashed curve is zero, so that’s f . The remaining function is f .
  5. 5. Outline Cleanup Increasing and Decreasing functions Concavity and the second derivative
  6. 6. Definition Let f be a function defined on and interval I . f is called increasing if f (x1 ) < f (x2 ) whenever x1 < x2 for all x1 and x2 in I .
  7. 7. Definition Let f be a function defined on and interval I . f is called increasing if f (x1 ) < f (x2 ) whenever x1 < x2 for all x1 and x2 in I . f is called decreasing if f (x1 ) > f (x2 ) whenever x1 < x2 for all x1 and x2 in I .
  8. 8. Examples: Increasing
  9. 9. Examples: Decreasing
  10. 10. Examples: Neither
  11. 11. Fact If f is increasing and differentiable on (a, b), then f (x) ≥ 0 for all x in (a, b)
  12. 12. Fact If f is increasing and differentiable on (a, b), then f (x) ≥ 0 for all x in (a, b) If f is decreasing and differentiable on (a, b), then f (x) ≤ 0 for all x in (a, b).
  13. 13. Fact If f is increasing and differentiable on (a, b), then f (x) ≥ 0 for all x in (a, b) If f is decreasing and differentiable on (a, b), then f (x) ≤ 0 for all x in (a, b). Proof. Suppose f is increasing on (a, b) and x is a point in (a, b). For h > 0 small enough so that x + h < b, we have f (x + h) − f (x) f (x + h) > f (x) =⇒ >0 h So f (x + h) − f (x) lim+ ≥0 h→0 h A similar argument works in the other direction (h < 0). So f (x) ≥ 0.
  14. 14. Example Here is a graph of f . Sketch a graph of f .
  15. 15. Example Here is a graph of f . Sketch a graph of f .
  16. 16. Fact If f (x) > 0 for all x in (a, b), then f is increasing on (a, b). If f (x) < 0 for all x in (a, b), then f is decreasing on (a, b).
  17. 17. Fact If f (x) > 0 for all x in (a, b), then f is increasing on (a, b). If f (x) < 0 for all x in (a, b), then f is decreasing on (a, b). The proof of this fact requires The Most Important Theorem in Calculus.
  18. 18. Outline Cleanup Increasing and Decreasing functions Concavity and the second derivative
  19. 19. Definition A function is called concave up on an interval if f is increasing on that interval.
  20. 20. Definition A function is called concave up on an interval if f is increasing on that interval. A function is called concave down on an interval if f is decreasing on that interval.
  21. 21. Fact If f is concave up on (a, b), then f (x) ≥ 0 for all x in (a, b)
  22. 22. Fact If f is concave up on (a, b), then f (x) ≥ 0 for all x in (a, b) If f is concave down on (a, b), then f (x) ≤ 0 for all x in (a, b).
  23. 23. Fact If f (x) > 0 for all x in (a, b), then f is concave up on (a, b). If f (x) < 0 for all x in (a, b), then f is concave down on (a, b).
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