Exponential modeling of G-force

[cgi093]

Homework Statement

Derive an equation of the form y=ax^b to model the given data: (35, 0.01) (28, 0.03) (20, 0.1) (15, 0.3) (11, 1) (9,3) (6, 10) (4.5, 30)

Homework Equations

Well, I know the answer is y = (7790 +/- 1246)x^(-3.698+/-0.1036) because that's what LoggerPro spits out, but I don't know how to derive it correctly without a program.

The Attempt at a Solution

Among the many, many attempts:

y = Ax^b
b = log(y)-log(A)
****** log(x)

Then I inserted numbers from two different data points then set the equations equal to each other, resulting in:

log(0.1)-log(A) = log(1)-log(A)
*** log(20) ****** *** log(11)

Which, after a step or two, became:

log(20) = log(11)log(0.1) +log(11)
************* log(A)

So, log(A) = log(11)log(0.1)
*********** log (20) - log(11)

Yielding an answer of A ~ -4.01

However, that isn't right, so I didn't even try to solve for b.

Could anyone please help? This is a big assignment, so sorry if I bump this thread a bit until I get help. And please take me through the steps, because I don't want to copy, I want to understand. I just need some help getting there. Thanks.

P.S. Ignore the asterisks. They're there only to get the denominators in the right place.

 

[zachzach]

I would use the method of least squares.

 

[lanedance]

so starting with
$$y = ax^b$$

taking logs
$$log(y) = log(ax^b) = log(x^b) + log(a)= b.log(x) + log(a)$$

so plotting up log(y) vs log(x) should be a graph of a straight line... if you can work out the line of best fit for that straight line, you should be able to cacaluate a & b from them...

 

[lanedance]

if its really experimental data, the data is not perfect, so you can't use a single pair of data points - you should use the whole data set to find the line of best fit...

as zachzach points of least squares is a good idea

 

[cgi093]

@ zach: Thanks, but no idea how to do that. And I posted this in the wrong subforum, because I'm in advanced pre-calc. So no calculus to help me please.

@lane: it's not experimental. That's an interesting idea though. I'll try it. Thanks a lot.

 

[lanedance]

then just plot the points on a log-log graph & draw a straight line of best fit on graph paer & work out the gradient & intercept

 

[zachzach]

http://www.efunda.com/math/leastsquares/lstsqr1dcurve.cfm

I do not know to not use calculus.

 

[cgi093]

http://www.efunda.com/math/leastsquares/lstsqr1dcurve.cfm

I do not know to not use calculus.

Thanks again, but that looks like gibberish to me. I've never used sigma outside of physics, for example. Maybe it wouldn't be that hard to learn, but I did google least squares and none of it really made sense to me.

 

[lanedance]

least squares is just a statistical method to give you the line of best fit through a set of data points

its derived by calculus, but doesn't require any to use it...

 

[zachzach]

least squares is just a statistical method to give you the line of best fit through a set of data points

its derived by calculus, but doesn't require any to use it...

Good point.

 

[lanedance]

so after some quick googling, you can do it very quickly in excel http://phoenix.phys.clemson.edu/tutorials/excel/regression.html

and the equations to do it by hand are there as well if you are so inclined..