This small group activity is designed to provide practice with the multivariable chain rule.
Students determine a particular rate of change using given information involving other rates of change.
The discussion emphasizes the equivalence of a variety of approaches, including the use of differentials.
Good “review” problem; can also be used as a homework problem.
Techniques to help make the transition from single- to multi-variable derivatives.
How to utilize infinitesimal reasoning using differentials.
The voltage \(V\) (in volts) across a circuit is given by \(V=IR\) (Ohm's Law), where \(I\) is the current (in amps) flowing through the circuit and \(R\) is the resistance (in ohms). If we place two circuits, with resistance \(R_1\) and \(R_2\), in parallel, then their combined resistance \(R\) is given by
\[{1\over R} = {1\over R_1} + {1\over R_2}\]
Suppose the current is 2 amps and increasing at \(10^{-2}\) amp/sec and \(R_1\) is 3 ohms and increasing at 0.5 ohm/sec, while \(R_2\) is 5 ohms and decreasing at 0.1 ohm/sec. Calculate the rate at which the voltage is changing.