4.2 MVT
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Rolle's Theorem
 f(x) satisfies 3 conditions:
 1.) f is cont on [a,b]
 2.) f is dxable on (a,b)
 3.) f(a) = f(b)then there is a # c in (a,b) where f'(c) = 0

MVT
 if: f(x) satisfies 2 conditions:
 1.) f is cont on [a,b]
 2.) f is dxable on (a,b)
then: there is a #c in (a,b) where:
1.) f'(c) =
or
2.) f(b)  f(a) = f'(c)(ba)

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

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 dxable on (a,b), f(x) is dxable 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

main use of Rolle's Theorem
prove MVT

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 dxable 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

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

Why is MVT important?
What is its main significance?
MVT allows you to obtain information about a function from information about its derivative

if f '(x) = 0 for all x in an interval (a, b), then f(x) is:
constant on (a, b)