In the following questions, we investigate two different applied settings using the differential.
a. 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.
A simple model for $f$ could be
$$f(x, y)=\cos (x) \sin (2 y)$$
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 $0.01 \mathrm{~cm}$ and increase the time by 0.1 seconds. Compare to the value of $f$ at the point $\left(\frac{\pi}{4}-0.01, \frac{\pi}{3}+0.1\right)$.
b. 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
$$\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}$$
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 $R$.
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