Differentiation and Integration

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March 23rd 2006, 03:30 AM
Maitham

Differentiation and Integration

IS it possible to get an answer?

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March 23rd 2006, 05:37 AM
CaptainBlack

Quote:

Originally Posted by Maitham

IS it possible to get an answer?

url=http://www.eyelash.ps/up]http://www.eyelash.ps/up/uploads/448c4941bb.jpg[/url]

Lets do the first derivative problem:

Differentiate:

$ f(t)=\exp(\mu t + \sigma^2t^2/2) $

Here we need the chain rule:

$ \frac{d}{dt}g(h(t))=\frac{dh}{dt}(t) \times \frac{dg}{dt}(h(t)) $ ,

so put $h(t)=\mu t + \sigma^2t^2/2$ and $g(t)=exp(t)$ .

Then:

$ \frac{d}{dt}f(t)=[\mu+\sigma^2 t][\exp(\mu t + \sigma^2t^2/2)] $

RonL
March 23rd 2006, 05:43 AM
CaptainBlack

Now lets do the first of the integrals:

$ \int_0^{\infty} \frac{1}{\beta}e^{-x/\beta}dx $

The integrand is the derivative of $-e^{-x/\beta}$ so we know
(probably from the fundamentaly theorem of calculus) that:

$ \int_0^{\infty} \frac{1}{\beta}e^{-x/\beta}dx=\left[-e^{-x/\beta}\right]_0^{\infty}=-0+1=1 $

RonL
March 27th 2006, 09:51 PM
Maitham

Thanks For the answer and i`ll try to do the rest and post the answer to check them for me if they are right, thats if it possible with you.