Activity 11.4.3.
(a)
In Figure 11.4.8, we see the trace of \(f(x,y) = \sin(x) e^{-y}\) with \(x\) held constant with \(x = 1.75\) plotted in blue. You can use the slider in Figure 11.4.8 to look at how the slope of the tangent line changes as you vary the \(y\)-coordinate along this trace. Write a couple of sentences that describe whether the slope of the tangent lines to this curve increase or decrease as \(y\) increases along the trace \(x=1.75\text{.}\) Be sure to pay attention to which direction corresponds to each coordinate increasing.
(b)
Compute \(f_{yy}(x,y)\) algebraically and explain how your observations in the previous task are related to the value of \(f_{yy}(1.75,y)\text{.}\) Be sure to address the notion of concavity in your response and be careful to note the directions in which \(y\) is increasing.
(c)
We want to explore what is described by the mixed partial derivative, \(f_{xy}\text{,}\) and we will focus on the point given by \((x,y)=(1.75,-1.5)\text{.}\) In this task, we will work through the definition for \(f_{xy} (1.75,-1.5)\) carefully.
The partial derivative \(f_x(1.75,-1.5)\) measures the slope of the line tangent to the trace given by \(y=-1.5\text{.}\) The tangent line we are examining is changing in the \(x\)-direction (parallel to the \(x\)-axis). When we look at \(f_{xy}\text{,}\) we are taking the partial derivative of \(f_x\) with respect to \(y\text{,}\) \(\frac{\partial}{\partial y} \Bigl[ f_x \Bigr]\text{.}\) So we are looking at how the slope of the tangent line in the \(x\)-direction will change when we vary the \(y\)-coordinate a little bit. We would approximate \(f_{xy}(1.75,-1.5)\) with the following difference quotient:
\begin{equation*}
\frac{f_x(1.75,-1.5+h)-f_x(1.75,-1.5)}{h}
\end{equation*}
You might think of sliding your pencil down the trace with \(x=1.5\) so that the pencil is in the \(x\)-direction and tangent to the surface. In Figure 11.4.9, the pencil would be the black tangent line drawn. You can use the slider at the top of Figure 11.4.9 to change the \(y\)-coordinate of the point where the black tangent line is drawn. You should look at what happens to the slope of the black tangent line as you increase the \(y\)-coordinate with the slider. The tangent line in the \(x\)-direction at the point \((1.75,-1.5)\) is drawn in gray for reference purposes.
Based on your exploration with Figure 11.4.9, write a few sentences about whether \(f_{xy}(1.75, -1.5)\) is positive or negative and justify your reasoning.
(d)
Compute \(f_{xy}(x,y)\) algebraically and evaluate \(f_{xy}(1.75, -1.5)\text{.}\) Write a couple of sentences about how this value compares with your observations in the previous task.
(e)
We know that \(f_{xx}(1.75, -1.5)\) measures the concavity of the \(y = -1.5\) trace, and that \(f_{yy}(1.75, -1.5)\) measures the concavity of the \(x = 1.75\) trace. What do you think the quantity \(f_{xy}(1.75, -1.5)\) measures?
(f)
On Figure 11.4.10, the trace with \(y = -1.5\) is highlighted with the point \((1.75,-1.5,f(1.75,-1.5))\) drawn in black. Sketch three tangent lines whose slopes correspond to the value of \(f_{yx}(x,-1.5)\) for three different values of \(x\) (near \(x=1.75\)). Use your tangent lines to state whether \(f_{yx}(1.75, -1.5)\) is positive or negative. Justify your reasoning and describe what you think \(f_{yx}(1.75, -1.5)\) measures.
