Deriving the frequency of a rubber band

Question

When a rubber band is stretched to have a tension T, it produces a frequency f. You change the tension by a very small amount ∆T . Show that the new frequency of the rubber band is f(1+∆T/2T)

Answer 1

I'm not sure how this was intended to be solved, but this is what I'd do. Assume the frequency depends on the tension and some other parameters (such as the properties of the rubber band, length, linear density, etc. that do not have dimensions involving time). Frequency is expressed in units of time$^{-1}$, and the units of force (tension) are mass $\times$ length $\times$ time$^{-2}$, so the frequency must be proportional to $T^{1/2}$, hence $f(T)=k\sqrt{T}$ for some constant $k$ that depends on the rubber band.

To approximate $f(T+\Delta T)$ you can just use a Taylor series.