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Derivative and Graphs
OBJECTIVES
1)
Find the Intervals on which a function increasing, or decreasing.
2)
Find relative extrema of a continuous function using the FirstDerivative Test.
3)
Sketch graphs of continuous functions
Definition:
A function
f
is
increasing
over an interval
I
if, for every
a
and
b
in
I
,
if
a
<
b
,
then
f
(
a
) <
f
(
b
).
(If the input
a
is less than the input
b
, then the output for
a
is less than the output for
b
.)
A function
f
is
decreasing
over an interval
I
if, for every
a
and
b
in
I
,
if
a
<
b
,
then
f
(
a
) >
f
(
b
).
(If the input
a
is less than the input
b
, then the output for
a
is greater than the output for
b
.)
Increasing and Decreasing Functions.
Suppose a function
f
has a derivative at each point in an open Interval
I,
then
1. If
f
′
(
x
) > 0
for all
x
in an interval
I
, then
f
is increasing over
I
.
2. If
f
′
(
x
) < 0
for all
x
in an interval
I
, then
f
is decreasing over
I
.
3. If
f
′
(
x
) = 0
for all
x
in an interval
I
, then
f
is constant over
I
.
DEFINITION:
A
critical value
of a function
f
is any number
c
in the domain of
f
for which the
tangent line at (
c
,
f
(
c
)) is horizontal or for which the derivative does not exist.
That is,
c
is a critical value if
f
(
c
)
exists and
f
′
(
c
) = 0
or
f
′
(
c
) does not exist.
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 Spring '10
 Cox
 Marketing

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