Taylor Expansion Problem, Fractional Change

[Ziyonex]

I hate for this to be my first post here, as this is the kind of community I'd like to contribute to. Bad timing I guess.

 

Basically, I am dealing with a "simple" Taylor expansion problem. I think the goal is to expand f(m) and f(∆m), and divide them part by part, but I do not know how to make that expansion.

 

Rather than type out the problem and deal with formatting, I just took a screenshot of the problem.

 

Can someone confirm that this is what I need to do, and maybe give me a hint as to how to expand this? It seems like, since m is in the denominator of a square root, taking n derivatives would accumulate a ton of junk. I guess it is only asking for an answer accurate to one significant figure though.

 

Thank you.

[timo]

"Your life consists of sliding on a horizontal, frictionless surface" is quite an uncommon phrasing of a math question. Anyways, the goal here is to calculate [math]\Delta f[/math] as a result of the mass increasing by 2%.

 

There is no [math]f(\Delta m)[/math] that you could expand. My guess is that you have not fully understood Taylor expansions. The Taylor expansion is a method to estimate the value of a function f at a point x from the function's properties at another point x0. Hint: [math]\Delta f = f(m + \Delta m) - f(m)[/math].

 

 

I don't quite see how to calculate the result to one significant figure, given that no numbers are given.

Edited January 23, 2012 by timo

[Ziyonex]

I agree that the first line is a bit odd.

 

I understand that in Taylor series you expand about a point. I know the point I want to expand about is m so that I can obtain a good approximation for m + ∆m.

I am uncertain what to expand, though.

 

This is what I was thinking, but I'm lost. I just really want to be able to understand what to do when I see a problem like this.

 

[DrRocket]

I agree that the first line is a bit odd.

 

I understand that in Taylor series you expand about a point. I know the point I want to expand about is m so that I can obtain a good approximation for m + ∆m.

I am uncertain what to expand, though.

 

This is what I was thinking, but I'm lost. I just really want to be able to understand what to do when I see a problem like this.

 

 

Expand the inverse of the square root as a (truncated) Taylor series, around the fixed point [math]m[/math] that is your mass. That gives you [math] f(m)[/math], once you have accounted for the constants in your equation. Then look at [math] f(m+\Delta m) - f(m)[/math] for small values of [math] \Delta m[/math]. The number of significant figures will be determined by the value of [math]m[/math] and the number of terms in the truncation of the Taylor series.

Edited January 24, 2012 by DrRocket

[timo]

I am uncertain what to expand, though.

I would expand f(m+dm), since that physically makes the most sense. But it is easy to see that the (final) result is the same as if you try to expand f(m+dm)-f(m) or (f(m+dm)-f(m))/f(m). If you don't see that, you may want to try out the other two ways to get a better understanding of the Taylor expansion.

[DrRocket]

I would expand f(m+dm), since that physically makes the most sense.

 

But it doesn't make much sense mathematically.

 

A Taylor series is normally spoken of as "expanded around a point" . In this case you want to expand the series arount the point m.

 

It makes little sense to expand f(m+dm) around 0, when in fact the radius of convergence when expanded near zero may not even include m. Since we are talking about, in essence the reciprocal of the square root, it will not even be expandable around 0. It blows up near 0.

[timo]

But [what you just said] doesn't make much sense mathematically. A Taylor series is normally spoken of as "expanded around a point" . In this case you want to expand the series arount the point m. It makes little sense to expand f(m+dm) around 0 …

Given that Ziyonex said "I know the point I want to expand about is m so that I can obtain a good approximation for m + ∆m" I think that much was already clear.

Edited January 24, 2012 by timo

[DrRocket]

Given that Ziyonex said "I know the point I want to expand about is m so that I can obtain a good approximation for m + ∆m" I think that much was already clear.

 

The key to effective application of mathematics in any field, physics included, is precision.

 

In fact the approach as you stated it makes very little sense, and if taken as you state it, would result in nonsense-- trying to base a power series at (m+dm), whatever that is supposed to mean. So while the OP may know what he intends to do, your suggestion only muddies the waters and would confuse any innocent lurker.

 

I is generally better to say what you mean and mean what you say than to speak in riddles and hope the listener can puzzle out something that makes sense.

Edited January 25, 2012 by DrRocket