Modeling the HEart with Calculus

[Jrb599]

I'm receiving a lecture on Modeling the Heart for about three weeks. We were assigned a problem which required us to find the mean Psa value.. Where

Psa(t) = Psa(0)*exp(-t/(Csa*Rs))

and Psa(0) is the initial value, and Csa and Rs are constants. Now What i did is intergrate this like

1/T * Int(Psa(t),t=0..T). And I got the right answer. The second part of the problem wants us to approach it another way.

They want us to evaluate the average of N equal spaced samples of Psa(T). and then take the lim N -> infinity. As a hint they remind us that Psa(T) forms a finite geometric series.

I can't remember any of this from Calculus II. Can someone show me how to do it? Thanks.

 

[matt grime]

http://en.wikipedia.org/wiki/Geometric_series

first hit in google - contains all the information you need (i.e.the formula for the sum of a geometric series.

 

[Jrb599]

Yeah I tried that. I couldn't get it to work for me. Like I said it's been a few years. I also tried maple using that definition and it wouldn't give me an answer

 

[matt grime]

What do you mean when you say you can't make it work? You've written down a geometric series (we presume, sicne you've not posted your work), now look at the formula. Identify the terms: what is the ratio, what is the starting value. Put that into the formula. Now, if you want, take the limit as N tends to infinity. If you post what you're attempting then it helps people to help you.

There is also no need to use the terms given to you, like Csa or Ra. Csa*Ra is just a constant, call it C. e^{-1/C} is just another constant, so let's call it K. Psa(0) is just a constant, so let's call it S. Psa is ugly, so let's write P(t).

So, really we have:

P(t)=S*K^t

and you want to evaluate

(P(0)+P(t/n)+P(2t/n)+...+P(n*t/n))/n

the thing inside is just a geometric series, and the link tells you its value.

 

[Jrb599]

anyone else have some advice..

Matt, I'm sorry I didn't post my work. But that's what I've been doing. However, when I simplify I do not come out with the correct result. I come out with Zero. and that's just not righ

 

[cristo]

any other help?

Have you read matt's post? If so, and you still don't understand, then you need to show some work-- we're not mind-readers!

 

[Jrb599]

and why is it T divide by n?

 

[Jrb599]

The geometric series I get is

Pso*(1-exp(-t/(N*Csa*Rs))))/(1-exp(-1/(Rs*Csa)))/N

and then evaluate that as N -> Infinity you get 0.

 

[matt grime]

You really should make it easier on us and write in a different notation (they're just letters, they don't mean anything).It is divided by N cos the question asks you to average the value over N equally spaced intervals between 0 and t.

It isn't zero: you've not added up the intervals properly.

What is the common ratio, r? What is the intial term, a? What is the sum of a geometric series with n+1 terms common ratio r and initial term a?

You appear to have r wrong.

 

[Jrb599]

a = 1

r = e^-h/K where h = T/N and k= Csa*Rs(1- e^-ht/k)/(1-e^-h/k)

 

[matt grime]

a = 1

The first term is Psa(0), which is not necessarily 1

r = e^-h/K where h = T/N and k= Csa*Rs

we want to remember the N and T in this, so let's not obscure that

(1- e^-ht/k)/(1-e^-h/k)

And this is what? (People are not psychics - you have to explain yourself in words, sentences and paragraphs.)

We have a geometric series:

P(0)(1+ c^{t/n} + c^{2t/n}+..+c^t)

so the sum of that is

S(n,t):=\frac{P(0)(1-c^{t(n+1)/n})}{ 1- c^{t/n}}

c is just exp{-1/Csa*Rs}

so you want the limit of S(n,t)/n as n tends to infinity