Section 2.8: Derivatives
Instructor: Ms. Hoa Nguyen
([email protected])
Derivatives
The
derivative of a function
f
at a number
a
is:
f
(
a
) = lim
h
→
0
f
(
a
+
h
)

f
(
a
)
h
if this limit exists.
Let
x
=
a
+
h
⇔
h
=
x

a
, then:
f
(
a
) = lim
x
→
a
f
(
x
)

f
(
a
)
x

a
if this limit exists.
The function
f
is called the derivative of
f
because it has been “derived” from
f
by the
limiting operation in the above equations.
Refer to Section 2.7 to understand the interpretation of the
derivatives
as
a slope of a tangent
or
an instantaneous rate of change
.
Example 1 of Section 2.8
(textbook):
The graphs of a function
f
(
x
)
and its derivative
f
(
x
):
f
(
x
)
>
0 on (
a, b
)
⇔
f
increases on (
a, b
).
f
(
x
)
<
0 on (
a, b
)
⇔
f
decreases on (
a, b
).
f
(
x
) = 0 at
x
=
c
⇔
the tangent at the point (
c, f
(
c
)) is horizontal.
Questions
:
•
Is the derivative a function?
Recall that a function is a map which associates one
number to another.
•
What does the sign of the derivative (positive or negative) tell about the behavior of
the function as you go forward?
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 Fall '08
 Noohi
 Calculus, Derivative, Continuous function, Ms. Hoa Nguyen

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