Lesson 7: What does f' say about f?

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 doesf
0
say aboutf?
Math 1a
February 15, 2008
Announcements
Ino class Monday 2/18! No oce hours 2/19.
IALEKS due Wednesday 2/20 (10% of grade).
IOce hours Wednesday 2/20 2{4pm SC 323
IMidterm I Friday 2/29 in class (up tox3.2)

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

3. Last worksheet, problem 2
Graphs off,f
0
, andf
00
are shown below. Which is which? How
can you tell?x
y

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
00
. Now we see that where the bold
curve has its horizontal tangents, the short-dashed curve is zero, so
that's f
0
. The remaining function is f .

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

6. Denition
ILetfbe a function dened on and intervalI.fis called
increasingif
f(x1)<f(x2) wheneverx1<x2
for allx1andx2inI.
Ifis calleddecreasingif
f(x1)>f(x2) wheneverx1<x2
for allx1andx2inI.

7. Denition
ILetfbe a function dened on and intervalI.fis called
increasingif
f(x1)<f(x2) wheneverx1<x2
for allx1andx2inI.
Ifis calleddecreasingif
f(x1)>f(x2) wheneverx1<x2
for allx1andx2inI.

8. Examples: Increasing

9. Examples: Decreasing

10. Examples: Neither

11. Fact
IIf f is increasing and dierentiable on(a;b), then f
0
(x)0
for all x in(a;b)
IIf f is decreasing and dierentiable on(a;b), then f
0
(x)0
for all x in(a;b).
Proof.
Supposefis increasing on (a;b) andxis a point in (a;b). For
h>0 small enough so thatx+h<b, we have
f(x+h)>f(x) =)
f(x+h)f(x)
h
>0
So
lim
h!0
+
f(x+h)f(x)
h
0
A similar argument works in the other direction (h<0). So
f
0
(x)0.

12. Fact
IIf f is increasing and dierentiable on(a;b), then f
0
(x)0
for all x in(a;b)
IIf f is decreasing and dierentiable on(a;b), then f
0
(x)0
for all x in(a;b).
Proof.
Supposefis increasing on (a;b) andxis a point in (a;b). For
h>0 small enough so thatx+h<b, we have
f(x+h)>f(x) =)
f(x+h)f(x)
h
>0
So
lim
h!0
+
f(x+h)f(x)
h
0
A similar argument works in the other direction (h<0). So
f
0
(x)0.

13. Fact
IIf f is increasing and dierentiable on(a;b), then f
0
(x)0
for all x in(a;b)
IIf f is decreasing and dierentiable on(a;b), then f
0
(x)0
for all x in(a;b).
Proof.
Supposefis increasing on (a;b) andxis a point in (a;b). For
h>0 small enough so thatx+h<b, we have
f(x+h)>f(x) =)
f(x+h)f(x)
h
>0
So
lim
h!0
+
f(x+h)f(x)
h
0
A similar argument works in the other direction (h<0). So
f
0
(x)0.

14. Example
Here is a graph off. Sketch a graph off
0
.

15. Example
Here is a graph off. Sketch a graph off
0
.

16. Fact
IIf f
0
(x)>0for all x in(a;b), then f is increasing on(a;b).
IIf f
0
(x)<0for all x in(a;b), then f is decreasing on(a;b).
The proof of this fact requires The Most Important Theorem in
Calculus.

17. Fact
IIf f
0
(x)>0for all x in(a;b), then f is increasing on(a;b).
IIf f
0
(x)<0for 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. Denition
IA function is calledconcave upon an interval iff
0
is
increasing on that interval.
IA function is calledconcave downon an interval iff
0
is
decreasing on that interval.

20. Denition
IA function is calledconcave upon an interval iff
0
is
increasing on that interval.
IA function is calledconcave downon an interval iff
0
is
decreasing on that interval.

21. Fact
IIf f is concave up on(a;b), then f
00
(x)0for all x in(a;b)
IIf f is concave down on(a;b), then f
00
(x)0for all x in
(a;b).

22. Fact
IIf f is concave up on(a;b), then f
00
(x)0for all x in(a;b)
IIf f is concave down on(a;b), then f
00
(x)0for all x in
(a;b).

23. Fact
IIf f
00
(x)>0for all x in(a;b), then f is concave up on(a;b).
IIf f
00
(x)<0for all x in(a;b), then f is concave down on
(a;b).