Activity 11.6.2.
In the following questions, we apply the Chain Rule in several different contexts.
(a)
Suppose that we have a function \(z\) defined by \(z(x,y) = x^2+xy^3\text{.}\) In addition, suppose that \(x\) and \(y\) are restricted to points that move around the plane by following a circle of radius \(2\) centered at the origin that is parameterized by
\begin{equation*}
x(t) = 2\cos(t), \ \mbox{ and } \ y(t) = 2\sin(t)
\end{equation*}
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Use the Chain Rule to find the resulting instantaneous rate of change \(\frac{dz}{dt}\text{.}\)
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Substitute \(x(t)\) for \(x\) and \(y(t)\) for \(y\) in the rule for \(z\) to write \(z\) in terms of \(t\) and calculate \(\frac{dz}{dt}\) directly.
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Write a couple of sentences comparing your answers to parts i. and ii.
(b)
Suppose that the temperature on a metal plate is given by the function \(T\) with
\begin{equation*}
T(x,y) = 100-(x^2 + 4y^2)
\end{equation*}
where the temperature is measured in degrees Fahrenheit and \(x\) and \(y\) are each measured in feet.
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Suppose an ant is walking along the \(x\)-axis at the rate of 2 feet per minute toward the origin. When the ant is at the point \((2,0)\text{,}\) what is the instantaneous rate of change in the temperature \(dT/dt\) that the ant experiences. Include units on your response.
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Suppose instead that the ant walks along an ellipse with \(x = 6\cos(t)\) and \(y = 3\sin(t)\text{,}\) where \(t\) is measured in minutes. Find \(\frac{dT}{dt}\) at \(t = \pi/6\text{,}\) \(t=\pi/4\text{,}\) and \(t = \pi/3\text{.}\) What does this seem to tell you about the path along which the ant is walking?
(c)
Suppose that you are walking along a surface whose elevation is given by a function \(f\text{.}\) Furthermore, suppose that if you consider how your location corresponds to points in the \(xy\)-plane, you know that when you pass the point \((2,1)\text{,}\) your velocity vector is \(\vv=\langle -1,2\rangle\text{.}\) If some contours of \(f\) are as shown in Figure 11.6.2, estimate the rate of change \(df/dt\) when you pass through \((2,1)\text{.}\)
