section 4.2 rolle’s theorem & mean value theorem calculus winter, 2010
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Section 4.2Rolle’s Theorem & Mean Value Theorem
Calculus
Winter, 2010
Calculus, Section 4.1 2
Rolle’s Theorem (or “What goes up must come down”)
IF (condition) f is continuous on [a,b] f is differentiable on (a,b) f(a)=f(b)
THEN (conclusion)There exist a number c in (a,b) such that f’(c)=0
Calculus, Section 4.1 3
Rolle’s Theorem (or “What goes up must come down”)
IF f is continuous on [a,b] f is differentiable on
(a,b) f(a)=f(b)
THEN There exist a number c in (a,b) such that f’(c)=0
a bc
Calculus, Section 4.1 4
Rolle’s Theorem (or “What goes up must come down”)
Since we know such a c exists, we now can solve from c with confidence.
a bc2( ) 2 8
( ) 2 2
0 2 2
2 2
2 2
2 21
f x x x
f x x
x
x
x
x
Calculus, Section 4.1 5
Using Rolle’s Theorem
Prove that the equation x3+x-1=0 has exactly one real root.
Let f(x)=x3+x-1 continuous and differentiable everywhere
Since f(-10) is a big negative number and f(10) is a big positive number, the Intermediate Value Theorem says that somewhere on (-10,10)
f(x) = 0. Therefore there exists at least one root.
Calculus, Section 4.1 6
Using Rolle’s Theorem
Prove that the equation x3+x-1=0 has exactly one real root.
Suppose there are two roots a and b If there are two roots, then f(a)=f(b)=0. Rolle’s Theorem says that somewhere there is c
where f’(c) = 0, but we see the f’(x)=3x2+1 which is ALWAYS POSITIVE.
Therefore our supposition must be false. Therefore there is exactly one root.
Calculus, Section 4.1 7
Mean Value Theorem(or “someone’s got to be average”)
IF
is continuous on [ , ]
is differentiable on ( , )
THEN
There exist a number in ( , )
such that
( ) ( ) ( )=
or
( ) ( )= ( )( )
f a b
f a b
c a b
f b f af c
b a
f b f a f c b a
Translation:
On the interval (a,b) there is at least one place where the average slope is the instantaneous slope.
Calculus, Section 4.1 8
Mean Value Theorem(or “someone’s got to be average”)IF
is continuous on [ , ]
is differentiable on ( , )
THEN
There exist a number in ( , )
such that
( ) ( ) ( )=
or
( ) ( )= ( )( )
f a b
f a b
c a b
f b f af c
b a
f b f a f c b a
2( ) 2 8
1
4
5 0 51
1 4 5ab
f x x x
a
b
m
Calculus, Section 4.1 9
Mean Value Theorem(or “someone’s got to be average”) There must be a
place on (a,b) where f’(x) = -1
2( ) 2 8
1
4
5 0 51
1 4 5ab
f x x x
a
b
m
2( ) 2 8
( ) 2 2
1 2 2
3 2
3 2
2 23
2
f x x x
f x x
x
x
x
x
Calculus, Section 4.1 10
Warnings!
Don’t apply Rolle’s Theorem or The Mean Value Theorem unless the conditions are metContinuous on [a,b]Differentiable on (a,b)
Calculus, Section 4.1 11
Assignment
Section 4.2, # 1, 4, 6, 9, 11, 15, 17, 19, 21, 26, 29, 31, 40, 43