The frequency of vibrations of a vibrating violin string is given by $\displaystyle f=\frac{1}{2L} \sqrt{\frac{T}{\rho}}$, where $L$ is the length of the string, $T$ is the tension, and $\rho$ is its linear density.

Find the rate of change of the frequency with respect to:
(a) the length (when $T$ and $\rho$ are constant)
(b) the tension (when $L$ and $\rho$ are constant)
(c) the linear density (when $L$ and $T$ are constant)

The pitch of a note is determined by the frequency $f$. (The higher the frequency, the higher the pitch.) Use the signs of the derivatives in (a) through (c) to determine what happens to the pitch of a note:

(d) when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates.
(e) when the tension is increased by turning a tuning peg.
(f) when the linear density is increased by switching to another string.

*For parts (a) through (c), use "p" for "$\rho$".
*For parts (d) through (f), enter "h" for higher note, or "l" for lower note.

(a)
(b)
(c)
(d)
(e)
(f)