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Rolle's Theorem
- f(x) satisfies 3 conditions:
- 1.) f is cont on [a,b]
- 2.) f is dx-able on (a,b)
- 3.) f(a) = f(b)then there is a # c in (a,b) where f'(c) = 0
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MVT
- if: f(x) satisfies 2 conditions:
- 1.) f is cont on [a,b]
- 2.) f is dx-able on (a,b)
then: there is a #c in (a,b) where:
1.) f'(c) =  or
2.) f(b) - f(a) = f'(c)(b-a)
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What makes Rolle's Theorem plausible?
- there is at least 1 point ( c, f(c) ) where the tan line is horizontal to the graph,
- & therefore where f'(c) = 0
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PROOF c 3 cases of Rolle's Theorem
in all 3 cases, there is at least one point ( c, f(c) ) on the graph of f(x) where the tangent is horizontal, making f'(c) equal to 0
- I. f(x) = k
- y = a constant f'(x) = 0, so c can be any # in (a,b)
- II. f(x) > f(a) for some x in (a,b) * because of condition 1 that f(x) is cont on [a,b], EVT can be used/applied & f(x) has a max val in [a,b]
- * since f(a) = f(b), f(x) has to reach this max val @ a #c in (a,b)
- * then f(x) has a local max @ #c &, by condition 2 that f(x) is dx-able on (a,b), f(x) is dx-able at c * therefore, f'(c) = 0 by Fermat's theorem
- III. f(x) < f(a) for some x in (a, b)
- * by EVT, f(x) has a min val in [a,b] since f(a) = f(b), f(x) gets to this min val @ #c in (a,b)
- * f'(c) = 0 by Fermat's theorem
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main use of Rolle's Theorem
prove MVT
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What makes MVT Theorem plausible/reasonable?
How does MVT make sense?
- 1. imagine 2 points a, f(a) & b, f(b) on the graph of a dx-able fxn
- 2. the slope of the secant line will be
- 3. f'(c), which is the slope of the tangent line @ ( c, f(c) ), is equal to #2,
4. given #3, there is at least one point P ( c, f(c) ) where the slope of the tan line is equal to the slope of the secant line connecting points A and B
- 5. the tan line is parallel to the secant line @ point P
- visually, imagine a line parallel to the secant line connecting points A and B, whose position is initally off the graph, but inches closer and closer toward the graph/curve until the point it reaches the graph/just touches it
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MVT basically says
- there is a # where the instantaneous rate of change is equal to the average rate of change over an interval
- i.e., if you traveled 100 miles in 2 hours, the average speed was 50 miles per hour.
- at at least 1 point in time, the speedometer read 50 mph
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Why is MVT important?
What is its main significance?
MVT allows you to obtain information about a function from information about its derivative
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if f '(x) = 0 for all x in an interval (a, b), then f(x) is:
constant on (a, b)
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