Single Variable Calculus, Chapter 3, 3.7, Section 3.7, Problem 26

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

a.) Find the rate of change of the frequency with respect to
$(i)$ the length (when $T$ and $\rho$ are constant)
$(ii)$ the tension (when $L$ and $\rho$ are constant), and
$(iii)$ the linear density (when $L$ and $T$ are constant)
b.)The pitch of a note is determined by the frequency $f$. Use the signs of the derivative in part(a) to determine what happens to the pitch of a note.
$(i)$ when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates.
$(ii)$ when the tension is increased by turning a tuning peg.
$(iii)$ when the linear density is increased by switching another string.


$
\begin{equation}
\begin{aligned}
\text{a.) } & (i) & \frac{df}{dL} &= \frac{1}{2} \sqrt{\frac{T}{\rho}} \frac{df}{dL} \left( \frac{1}{L} \right)\\
\\
& \phantom{x} & \frac{df}{dL} &= \frac{1}{2} \sqrt{\frac{T}{\rho}} \left( \frac{-1}{L^2} \right)\\
\\
& \phantom{x}& \frac{df}{dL} &= \frac{-1}{2L^2} \sqrt{\frac{T}{\rho}}\\
\\
\\
\\
& (ii) & \frac{df}{dT} &= \frac{1}{2L(\rho)^{\frac{1}{2}}} \cdot \frac{d}{dT} (T)^{\frac{1}{2}} \\
\\
& \phantom{x} & \frac{df}{dT} &= \frac{1}{2L(\rho)^{\frac{1}{2}}} \cdot \frac{1}{2} T^{\frac{-1}{2}}\\
\\
& \phantom{x} & \frac{df}{dT} &= \frac{1}{4L\sqrt{\rho T}} \\
\\
\\
& (iii) & \frac{df}{d\rho} & = \frac{\sqrt{T}}{2L} \cdot \frac{d}{d\rho} (\rho)^{\frac{-1}{2}}\\
\\
& \phantom{x} & \frac{df}{d\rho} &= \frac{T}{2L} \cdot -\frac{1}{2}(\rho)^{\frac{-3}{2}}\\
\\
& \phantom{x} & \frac{df}{d\rho} &= \frac{-1}{4L} \sqrt{\frac{T}{\rho^3}}

\end{aligned}
\end{equation}
$


b.) $(i)$ A decrease in length $L$ is associated with the increase in the frequency since $\displaystyle \frac{df}{dL}$ from part(a) is negative.
$(ii)$ An increase in the tension $T$ is associated with an increase in frequency since $\displaystyle \frac{df}{dT}$ from part (a) is positive.
$(iii)$ An increase in the linear density $\rho$ is associated with a decrease in frequency since $\displaystyle \frac{df}{d\rho}$ from part (a) is negative.