4.2 MVT

Card Set Information

Author:
jojobean0203
ID:
215007
Filename:
4.2 MVT
Updated:
2013-04-22 02:20:24
Tags:
calc
Folders:

Description:
MVT
Show Answers:

Home > Flashcards > Print Preview

The flashcards below were created by user jojobean0203 on FreezingBlue Flashcards. What would you like to do?


  1. 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
  2. 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)
  3. 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
  4. 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
  5. main use of Rolle's Theorem
    prove MVT
  6. 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
  7. 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
  8. Why is MVT important? 
    What is its main significance?
    MVT allows you to obtain information about a function from information about its derivative
  9. if f '(x) = 0 for all x in an interval (a, b), then f(x) is:
    constant on (a, b)

What would you like to do?

Home > Flashcards > Print Preview