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CalcBC
Chp 4
2
Rolle’s Theorem
and Mean Value Theorem
NOTES
1
ROLLE’S TH
EOREM
–
A set of conditions used to determine if
there is a turning point within the given interval.
Let
f
be a function that satisfies the following three hypotheses:
(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)
(the
endpoints
have the
same yvalue
)
Then, there is at least one number
c
in the
(open)
interval
(a, b)
such that
f’(c) = 0
Visual example of how Rolle's involves extreme value points:
f(a)
f(a)
f(a)
f(b)
f(b)
f(b)
f(c)
f(c)
f(c)
Why
deriv.= 0?
Because the segment
connecting the endpoints is
horizontal
and
at x = c
there may be an EVP.
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Ex(Apply Rolle’s Thm)—
Verify
the function satisfies the three hypotheses of Rolle’s Thm on the given interval.
If
the conditions are met, find all numbers c that satisfy the conclusions
of Rolle’s Thm.
f(x) =
[
6, 0]
<Vocabulary assistance>
hypotheses are the three conditions:
continuous,
differentiable,
f(a) = f(b)
conclusion:
f
'(c) = 0
<Answer>
continuous
(no undefined yvalues)
function does not negative value under the squareroot for the given interval
it is all continuous on the interval
[
6, 0]
(check)
differentiable
(no undefined slopes)
could be a problem where the derivative has variable in denominator
get derivative to see what it looks like
(product and chain rules used)
f
'(x) =
+
(fast add)
f'(x) =
f
'(x) = 0
(use numerator=0)
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 Spring '10
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 Mean Value Theorem

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