Preview Activity 11.4.1.
Once again, we consider the function \(f\) defined by \(f(x,y) = \frac{x^2\sin(2y)}{32}\) that measures a projectile’s range as a function of its initial speed \(x\) and launch angle \(y\text{.}\) The graph of this function, including traces with \(x=150\) and \(y=0.6\text{,}\) is shown in the interactive graph below.
(a)
(b)
Notice that \(f_x\) itself is a new function of \(x\) and \(y\text{,}\) so we may now compute the partial derivatives of \(f_x\text{.}\) Find the partial derivative \(\frac{\partial}{\partial x} \Bigl[ f_x \Bigr]\text{,}\) which we will denote by \(f_{xx} \text{.}\) Verify that \(f_{xx}(150,0.6) \approx 0.058\text{.}\)
(c)
The graph below shows the trace of \(f\) with \(y=0.6\) with three tangent lines included. Write a few sentences to explain how \(f_{xx}(150,0.6) \approx 0.058\) is reflected in this figure.
(d)
Find the partial derivative \(\frac{\partial}{\partial y} \Bigl[ f_y \Bigr]\text{,}\) which we will denote by \(f_{yy}\) and compute the value of \(f_{yy}(150, 0.6)\text{.}\)
(e)
The graph below shows the trace \(f(150, y)\) and includes three tangent lines. Write a couple of sentences to explain how the value of \(f_{yy}(150,0.6)\) is reflected in this figure.
(f)
Because \(f_x\) and \(f_y\) are each functions of both \(x\) and \(y\text{,}\) they each have two partial derivatives. You have already computed two of them. Now compute
\begin{equation*}
\frac{\partial}{\partial y} \Bigl[ f_x \Bigr]\qquad\text{and}\qquad\frac{\partial}{\partial x} \Bigl[ f_y \Bigr]
\end{equation*}
for the range function \(f(x,y) = \frac{x^2\sin(2y)}{32}\) by using your earlier computations of \(f_x\) and \(f_y\text{.}\) Write a sentence to explain how you calculated these “mixed” partial derivatives.
