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Activity 11.3.2.

In this activity, we will be examining the function \(f\) defined by

\begin{equation*} f(x,y) = \frac{x y^2}{x+1} \end{equation*}

at the point \((1,2)\text{.}\) Specifically, we will be looking how the traces of this function will be related to the values of the partial derivatives.

(a)

Give the trace \(f(x,2)\) at the fixed value \(y=2\) as a function of \(x\text{.}\) On Figure 11.3.7, draw the graph of the trace with \(y=2\) around the point where \(x=1\text{,}\) indicating the scale and labels on the axes. On the graph of the trace with \(y=2\text{,}\) sketch the tangent line at the point \(x=1\text{.}\)

Figure 11.3.7. A plot of \(f(x,2)\)
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(b)

Find the partial derivative \(f_x(1,2)\) and relate its value to the sketch you just made.

(c)

Give the trace \(f(1,y)\) as a function of \(y\) at the fixed value \(x=1\text{.}\) On Figure 11.3.8, draw the graph of the trace with \(x=1\) indicating the scale and labels on the axes. On your graph of the trace corresponding to \(x=1\text{,}\) sketch the tangent line at the point \(y=2\text{.}\)

Figure 11.3.8. A plot of \(f(x,2)\)
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(d)

Find the partial derivative \(f_y(1,2)\) and relate its value to the sketch you just made.