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Topic: 1.39 - What exactly is the Mean value Theorem?
Replies: 29   Last Post: Jun 20, 2014 4:00 PM

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 Dirk Van de moortel Posts: 157 Registered: 12/6/11
Re: 1.39 - What exactly is the Mean value Theorem?
Posted: Jun 19, 2014 9:51 AM

John Gabriel <thenewcalculus@gmail.com> wrote:
> On Thursday, 19 June 2014 13:31:35 UTC+2, Dirk Van de moortel wrote:
>

>>>> If f is a differentiable function over (a,b), then
>>>> {f(b)-f(a)}/(b-a) is the (natural) average of all the ordinates of
>>>> f' over (a,b).

>
>>> OMG ... no. Really? That is simply hilarious. You're a total
>>> fucking moron.

>
>> It is true.
>> Try it out with a simple example such as
>> f(x) = x^2
>> a = 4
>> b = 7
>> Dirk Vdm

>
> ( 7^2 - 4^2 )/ 3 = (49-16)/ 3 = 33/3 = 11
>
> The ordinates of f'(x) are given by 2x. The average of the ordinates
> of f'(x) on the interval (4,7) is 11, as expected.
>
> So who is the fucking moron? YOU!

You snipped the part where I said it is true.

Wiz,
the average of a function F over an interval (a,b) is defined as
A(F(a,b)) = 1/(b-a) Int_a^b F(x) dx
Take F = f' and verify:
A(f'(a,b)) = 1/(b-a) Int_a^b f'(x) dx
= 1/(b-a) ( f(b) - f(a) )
= ( f(b) - f(a) ) / (b-a)
I assume you didn't notice the apostophe in f'.

Dirk Vdm

Date Subject Author
6/19/14 Virgil
6/19/14 johngabriel2009@gmail.com
6/19/14 Virgil
6/19/14 Wizard-Of-Oz
6/19/14 Dirk Van de moortel
6/19/14 Dirk Van de moortel
6/19/14 Virgil
6/19/14 Dirk Van de moortel
6/19/14 Dirk Van de moortel
6/19/14 Port563
6/19/14 Dirk Van de moortel
6/19/14 johngabriel2009@gmail.com
6/19/14 Virgil
6/20/14 Virgil
6/19/14 Virgil
6/19/14 Virgil
6/19/14 Virgil
6/19/14 Jeff Hebert
6/19/14 Jeff Hebert
6/19/14 Virgil
6/20/14 hoffman@spectre.com
6/20/14 Virgil