Exercise 27

In the following questions, we investigate two different applied settings using the differential.

1. Let $f$ represent the vertical displacement in centimeters from the rest position of a string (like a guitar string) as a function of the distance $x$ in centimeters from the fixed left end of the string and $y$ the time in seconds after the string has been plucked. (An interesting video of this can be seen at https://www.youtube.com/watch?v=TKF6nFzpHBUA.) A simple model for $f$ could be $$\begin{gathered} f(x,y) = \cos(x)\sin(2y). \end{gathered}$$ Use the differential to approximate how much more this vibrating string is vertically displaced from its position at $(a,b) = \left(\frac{\pi}{4}, \frac{\pi}{3} \right)$ if we decrease $a$ by $0.01$ cm and increase the time by $0.1$ seconds. Compare to the value of $f$ at the point $\left(\frac{\pi}{4}-0.01, \frac{\pi}{3}+0.1\right)$.

2. Resistors used in electrical circuits have colored bands painted on them to indicate the amount of resistance and the possible error in the resistance. When three resistors, whose resistances are $R_1$, $R_2$, and $R_3$, are connected in parallel, the total resistance $R$ is given by $$\begin{gathered} \frac1R = \frac1{R_1} + \frac1{R_2} + \frac1{R_3}. \end{gathered}$$ Suppose that the resistances are $R_1=25\Omega$, $R_2=40\Omega$, and $R_3=50\Omega$. Find the total resistance $R$. If you know each of $R_1$, $R_2$, and $R_3$ with a possible error of $0.5$%, estimate the maximum error in your calculation of $R$.