The frequency of vibrations of a vibrating violin string is given by [math], where [math] is the length of the string, [math] is the tension, and [math] is its linear density.

Find the rate of change of the frequency with respect to:
(a) the length (when [math] and [math] are constant)
(b) the tension (when [math] and [math] are constant)
(c) the linear density (when [math] and [math] are constant)

The pitch of a note is determined by the frequency [math]. (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 "[math]".
*For parts (d) through (f), enter "h" for higher note, or "l" for lower note.

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