We're learning related rates/rates of change currently in my AP calc class. Just have a few questions, wondering if I'm understanding these problems correctly. The " " is my questions, etc.
f = 1/2L * sq. root ( T / P )
The frequency of vibrations of a violin string is given by the above eq. where L is the length of the string, T is its tension, and P is linear density.
#1. Find rate of change of f with respect to: a) the length (when T & P are constant), b) the tension (when T & P are constant), c) the linear density (when T & P are density)
" Do I take the first three derivatives of f to find these? First deriv being length, second being tension, third being linear density? "
#2 The pitch of a note (how high or low the note sounds) is determined by the frequency F. Use the signs of the derivatives in #1 to determine what happens to the pitch of a note:
" I don't understand this at all "
a) when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates
b) when the tension is increased by turning a tuning peg
c) when the linear density is increased by switching to another string
When you shorten the string the frequency becomes:
where denotes the derivative of with respect to evaluated at .
Part a) asks: Is this an increase or decrease in frequency?
The others are similar except that the variables in question are increased rather than decreased.
CB