Section 4.2: The Mean Value Theorem (continued)
Mean Value Theorem
: Let
f
be a function on the interval
[
a
,
b
] (with
a
<
b
) that satisfies
1.
f
is continuous on the closed interval [
a
,
b
]
2.
f
is differentiable on the open interval (
a
,
b
).
(We assume
a
<
b
.)
Then there is a number
c
in (
a
,
b
) such
that
f
′
(
c
) = (
f
(
b
)–
f
(
a
))/(
b
–
a
).
To see why the Mean Value Theorem is intuitively
reasonable, note that (
f
(
b
)–
f
(
a
))/(
b
–
a
) is the slope of the
secant joining (
a
,
f
(
a
)) and (
b
,
f
(
b
)).
This secant line touches
the curve
y
=
f
(
x
) in at least two points.
If we imagine
sliding the secant line up or down, there ought to be some
position where it becomes an actual tangent line.
If we
write the intersection of the tangent line with the curve as
(
c
,
f
(
c
)), then this is exactly the point
c
that we need.
A special case of the Mean Value Theorem is
Rolle’s Theorem: Let
f
be a function that satisfies
1.
f
is continuous on the closed interval [
a
,
b
],
2.
f
is differentiable on the open interval (
a
,
b
),
3.
f
(
a
)=
f
(
b
).
Then there is a number
c
in (
a
,
b
) such that
f
′
(
c
) = 0.
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Rolle’s Theorem can be seen as a special case of the Mean
Value Theorem.
But you can also derive the Mean Value Theorem from
Rolle’s Theorem, as in Stewart.
Consequences of Mean Value Theorem:
Theorem 5: If
f
is continuous on [
a
,
b
] and differentiable on
(
a
,
b
) and the derivative of
f
on (
a
,
b
) is constantly 0, the
function
f
is constant on [
a
,
b
].
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 Fall '11
 Staff
 Calculus, Derivative, Mean Value Theorem, Mathematical analysis

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