Quadratic Approximation of Potential Function using Taylor Expansion Method

[Crazy Gnome]

Homework Statement

What is the quadratic approximation to the potential function?

Homework Equations

U(x) = U₀((a/x)+(x/a))
U₀= 20
a=4

The Attempt at a Solution

This is just the last part of a question on my engineering homework, I never learned Taylor expansions before even though I have taken all the class prerequisites. So if you could just walk me through it that would be much appreciated.

 

[Dick]

Read this http://en.wikipedia.org/wiki/Taylor_series and try it out yourself. It's just a cookbook formula. You have to add various derivatives of U(x) at x=4 times powers of (x-4) up to quadratic. It's not that mysterious. We'll be glad to look at your efforts.

 

[Crazy Gnome]

Read this http://en.wikipedia.org/wiki/Taylor_series and try it out yourself. It's just a cookbook formula. You have to add various derivatives of U(x) at x=4 times powers of (x-4) up to quadratic. It's not that mysterious. We'll be glad to look at your efforts.

So for my first try I got U=(1.25)x²-10x+60. I am pretty sure I remember something about only going out to the second order place to make it "quadratic" so I got.

40 for the first place, 0 zero for the first order, and 1.25x²-10x+20 for the second order.

Does this look right?

 

[Dick]

Yes, that's right. For most purposes you probably want to leave that in the form 40+5*(x-4)^2/4, since you are thinking of x's near 4.