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.

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. Outline
Cleanup
Increasing and Decreasing functions
Concavity and the second derivative

3. Last worksheet, problem 2
Graphs of f , f , and f are shown below. Which is which? How
can you tell?
y
x

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. Outline
Cleanup
Increasing and Decreasing functions
Concavity and the second derivative

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. 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. Examples: Increasing

9. Examples: Decreasing

10. Examples: Neither

11. Fact
If f is increasing and differentiable on (a, b), then f (x) ≥ 0
for all x in (a, b)

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. 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. Example
Here is a graph of f . Sketch a graph of f .

15. Example
Here is a graph of f . Sketch a graph of f .

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. 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. Outline
Cleanup
Increasing and Decreasing functions
Concavity and the second derivative

19. Definition
A function is called concave up on an interval if f is
increasing on that interval.

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. Fact
If f is concave up on (a, b), then f (x) ≥ 0 for all x in (a, b)

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. 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).