Chapter 4.6, Problem 40E

Sound If a string with a fundamental frequency of $110$ hertz is plucked in the middle, it will vibrate at the odd harmonics of $110,330,550,\dots$ hertz but not at the even harmonics of $220,440,660,\dots$ hertz. The resulting $P$ pressure on the ear-drum caused by the string can be approximated using the equation

$P\left(t\right)=0.003\sin \left(220\pi t\right)+\frac{0.003}{3}\sin \left(660\pi t\right)+\frac{0.003}{5}\sin \left(1100\pi t\right)+\frac{0.003}{7}\sin \left(1540\pi t\right)$,

where $P$ is in pounds per square foot at a time of $t$ seconds after the string is plucked. Source: Fundamentals of Musical Acoustics and The Physics and Psychophysics of Music: An Introduction.

a. Graph $P\left(t\right)$ on $\left[0,0.01\right]$.

b. Find $P^{\text{'}}\left(0.002\right)$ and interpret.