Preview Activity 11.4.1.
Once again, let’s 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 Figure 11.4.1.
(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 \(f_{xx} = (f_x)_x= \frac{\partial}{\partial x} \Bigl[ f_x \Bigr]\) and show that \(f_{xx}(150,0.6) \approx 0.058\text{.}\)
(c)
Figure 11.4.2 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 \(f_{yy}=(f_y)_y=\frac{\partial}{\partial y} \Bigl[ f_y \Bigr]\) and compute the value of \(f_{yy}(150, 0.6)\text{.}\)
(e)
Figure 11.4.3 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. Not only can we compute \(f_{xx} = (f_x)_x\text{,}\) but also \(f_{xy} = (f_x)_y =\frac{\partial}{\partial y} \Bigl[ f_x \Bigr]\text{;}\) likewise, in addition to \(f_{yy} = (f_y)_y\text{,}\) but also \(f_{yx} = (f_y)_x =\frac{\partial}{\partial x} \Bigl[ f_y \Bigr]\text{.}\) For the range function \(f(x,y) = \frac{x^2\sin(2y)}{32}\text{,}\) use your earlier computations of \(f_x\) and \(f_y\) to now determine \(f_{xy}\) and \(f_{yx}\) (as functions of \(x\) and \(y\)). Write one sentence to explain how you calculated these “mixed” partial derivatives.
