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Cauchy Mean Value theorem

  • 02-01-2011 12:13pm
    #1
    acb
    Registered Users, Registered Users 2 Posts: 120 ✭✭


    Hi guys,
    This is my 3rd Q this week alone and everyone has been great answering me so I have another one!!

    I wasnt in when we covered the Cauchy MVT in college and havent a clue what its about- in english let alone all the notation.

    I know its like a specialised case of Mean Value theorem- which I do know and understand, yeh for once!! But I just dont know what the theorem is saying at a very basic and fundamental level?

    Is it saying something like if we have two functions and we take a point on them each that we will be able to find two tangent lines to the points that are parallel to eachother???:o (this is a complete guess -that was inspired from a diagram I seen and am embarrased Im on a maths course and cant read any of the notation in the proof!)
    I know I need to understand it as its used the the next theorem I need to understand!

    Thank you all so so much for your help- boards.ie is fab!!:D


Comments

  • #2
    Thomas_S_Hunterson
    Closed Accounts Posts: 6,151 ✭✭✭Thomas_S_Hunterson


    If you imagine any two points a and b with some continuous differentiable curve between them then there will be at least one point on the curve where the slope of a tangent is equal to the slope of a straight line between a and b.

    This image might help you visualise it:
    300px-Mvt2.svg.png


  • #3
    acb
    Registered Users, Registered Users 2 Posts: 120 ✭✭acb


    Sean_K wrote: »
    If you imagine any two points a and b with some continuous differentiable curve between them then there will be at least one point on the curve where the slope of a tangent is equal to the slope of a straight line between a and b.

    This image might help you visualise it:
    300px-Mvt2.svg.png


    I understand Mean value thm above- I think well, but is this the same as Cauchys??
    Mean value is essentially saying there is AT LEAST one point c in the interval [a,b] where the slope at the tangent to c is parallel (same) as slope of secant line AB.

    But I dont see what Cauchy is saying??? Thanks for replying though;)


  • #4
    acb
    Registered Users, Registered Users 2 Posts: 120 ✭✭acb


    306px-Cauchy.png
    What in the name of god is all this about???


  • #5
    sponsoredwalk
    Registered Users, Registered Users 2 Posts: 3,038 ✭✭✭sponsoredwalk


    https://www.youtube.com/watch?v=KQNeFUTgB-s


    The mean value theorem is the special case of Cauchy's mean value theorem when g(t) = t

    http://en.wikipedia.org/wiki/Mean_value_theorem#Cauchy.27s_mean_value_theorem

    Basically the only difference is that instead of

    { [f(x₂) - f(x₁) ] / (t₂ - t₁) } it's { [f(x₂) - f(x₁) ] / [ g(t₂) - g(t₁) ] }

    If you're reading the proof on the wiki page & you already understand the
    proof of the normal MVT it's probably best to read it backwards, it makes
    more sense to me that way.


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