In this screencast we're going to look at the linearization of non-linear equations.
Some examples that we're going to look at are the sine of Y equals X times AX plus B
to the minus 2, here we want to find A and B, why?
Because if we can determine those parameters we can put them in a form: Y equals MX plus
B, where B is the intercept, M is the slope, and once we have those numbers we can use
the model for any X, Y data that we have.
So the first thing to remember is that our Y and our X don't always have to be only a
function of Y or X, so we can have any combination of X and Y.
As long as our Y is equal to some function of X, Y and/or our X is equal to some function
of X and/or Y we can plot this.
And we'll go back to our problem, let's say we're given all sorts of X data and all sorts
of Y data, and what we want to do is put this in a form that we can plot our X versus Y
and solve for this A and B. Let's see how we do that.
So the first thing that we're going to do is divide both sides by X, why?
This way we leave B by itself and not attached to either X or Y.
So we have here the sine of Y divided by X equals AX plus B to the minus 2.
Now let's take the inverse of both sides, generally when you see something raised to
a negative power it's easier to invert both sides first, so now we're getting something
that looks closer to a linear equation and we're going to take the square root of both
sides, and now what we can do is we can plot the square root of X over sine of Y versus
X, our slope is going to be our A and our intercept is going to be our B. So how would
we set this up on a spreadsheet?
The first thing you would do is take all your X data and all your Y data and then you would
make a column here that is just our X divided by sine of Y and now you can plot this column,
versus this column to get our linear equation.
So let's look at another one.
Here we have 1 over Y equals A, X minus 4 to the negative B.
The first thing we're going to do is divide both sides by A, so get rid of this A right
here, and again because we have a negative here I'm going to invert this.
Now whenever you see something raised to a parameter you should be thinking "natural
log" so let's take the natural log of both sides, this still isn't quite what we want
because of this A times Y, so we use the natural log such that ln of A times Y is the same
thing as ln of A plus ln of Y, and this equals B ln of X minus 4, and hopefully you can see
where we're going with this.
We go to ln of Y equals B ln of X minus 4 minus ln of A, so now we can plot ln of Y
versus ln of X minus 4, our slope is B, our intercept is minus ln of A, however remember
what we're looking for is that A, so we're going to have to take this A equals e to the
minus ln of A or our intercept to get A. And finally let's look at Y equals AX plus B over
X to the minus 1, let's invert this, now we have both parameters as a function of X and
we can only have one of them because that's what would be our slope.
What do we do here?
Well we multiply everything through by X, we plot X over Y versus X squared, our slope
is A and our intercept is B.