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Calculus Section 4.1
Page5
4.2
Mean Value Theorem
ROLLE'S THEOREM
Let
f
be differentiable on (a,b) and continuous on [a,b].
If
f
(a) =
f
(b) = 0, then there is at least one point c in (a,b) where
′
f
c
( ) =
0 .
(geometrically obvious)
eg:
(Set III #57)
If c is the number defined by Rolle's Theorem, then for
f x
(
)
=
2x
3

6x
on the
interval
0
≤
x
≤
3
, then c is
A. 1
B. 1
C.
2
D. 0
E.
3
eg: (Set III #56)
If
f
(a) =
f
(b) = 0 and f(x) is continuous on [a,b], then
A.
f(x) must be identically zero
B.
′
f
x
( )
may be different from zero for all x on [a,b]
C.
there exists at least one number c, a < c < b, such that
′
f
c
( ) =
0
D.
′
f
x
( )
must exist for every x on (a,b)
E.
none of the preceding is true
MEAN VALUE THEOREM
Let f be differentiable on (a,b) and continuous on [a,b].
There there
is at least one point c in (a,b) where
′
f
c
( )
=
f b
(
) 
f a
( )
b

a
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Page 6
(geometrically obvious)
eg:
Let
f x
(
) =
x
3
+
1.
Show that f(x) satisfies the hypotheses of the
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This note was uploaded on 02/04/2011 for the course MATH 116 taught by Professor John during the Spring '10 term at St. Michael.
 Spring '10
 John
 Calculus, Mean Value Theorem

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