Cauchy Mean Value theorem

acb

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

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:

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:

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;)

acb

What in the name of god is all this about???

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.