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MA103 Week 5 Lecture Notes
How Derivatives Affect the Shape of a Graph
(Text Reference: §4.3)
Monotonicity (Increasing/Decreasing)
S
Definition: A function
f
is called (strictly)
increasing on an interval
I
if
f
Ÿ
x
1
f
Ÿ
x
2
whenever
x
1
,
x
2
I
and
x
1
x
2
. (That is, for
all x
1
x
2
in
I
,
f
Ÿ
x
1
f
Ÿ
x
2
.)
Similarly,
f
is (strictly)
decreasing on
I
if
x
1
,
x
2
I
and
x
1
x
2
implies
f
Ÿ
x
1
f
Ÿ
x
2
.
S
f
is said to be
monotonic on
I
if
f
is either increasing on
I
or decreasing on
I
Ž
note that this term is not used in your text
S
Test for Monotonic Functions (Increasing/Decreasing Test):
1.
f
U
Ÿ
x
0 for all
x
Ÿ
a
,
b
implies
f
is increasing on
Ÿ
a
,
b
;
2.
f
U
Ÿ
x
0 for all
x
Ÿ
a
,
b
implies
f
is decreasing on
Ÿ
a
,
b
;
Ž
proof may be discussed in lecture
S
example: Examine the monotonicity (i.e., where the function is increasing/decreasing) of
f
Ÿ
x
1
4
x
4
"
x
3
x
2
1
sol’n:
f
U
Ÿ
x
x
3
"
3
x
2
2
x
x
Ÿ
x
"
1
Ÿ
x
"
2
The critical numbers are
x
0,1,2.
Analyze the sign of
f
U
Ÿ
x
using intervals determined by the critical numbers.
f
U
"
"
".
01
2
.
f
{

{
Thus
f
is decreasing on
Ÿ
".
,0
:
Ÿ
1,2
and increasing on
Ÿ
0,1
:
Ÿ
2,
.
.
S
First Derivative Test: Suppose
c
is a critical number of a continuous function
f
.
1.
If the sign of
f
U
Ÿ
x
is positive to the left of
c
and negative to the right of
c
, then
f
has a
local maximum at
c
.
2.
If the sign of
f
U
Ÿ
x
is negative to the left of
c
and positive to the right of
c
, then
f
has a
local minimum at
c
.
3.
If the sign of
f
U
Ÿ
x
does not change at
c
, then
f
has no local extremum at
c
.
S
example: In the above example,
f
would have a local maximum value at
x
1 and local
minimum values at
x
0 and
x
2 by the First Derivative Test.
S
Note that if a function
f
has just one critical number
c
in an interval
I
, then if
f
has a local
minimum at
c
,
f
Ÿ
c
is the minimum value of
f
on
I
, and if
f
has a local maximum at
c
,
f
Ÿ
c
is the maximum value of
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This note was uploaded on 04/30/2011 for the course MATH 205 taught by Professor Tseng during the Spring '08 term at American College of Computer & Information Sciences.
 Spring '08
 tseng
 Calculus, Derivative

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