Problem 47
Show that the function defined by $$ f(x, y, z)=\frac{x y z}{x^{3}+y^{3}+z^{3}} \quad \text { for }(x, y, z) \neq(0,0,0) $$ and \(f(0,0,0)=0\) is not continuous at \((0,0,0)\).
Problem 48
Show that the function defined by $$ f(x, y, z)=(y+1) \frac{x^{2}-z^{2}}{x^{2}+z^{2}} \quad \text { for }(x, y, z) \neq(0,0,0) $$ and \(f(0,0,0)=0\) is not continuous at \((0,0,0)\).
Problem 48
A CAS can be used to calculate and graph partial derivatives. Draw the graphs of each of the following: (a) \(\sin \left(x+y^{2}\right)\) (b) \(D_{x} \sin \left(x+y^{2}\right)\) (c) \(D_{y} \sin \left(x+y^{2}\right)\) (d) \(D_{x}\left(D_{y} \sin \left(x+y^{2}\right)\right)\)
Problem 49
Give definitions in terms of limits for the following partial derivatives: (a) \(f_{y}(x, y, z)\) (b) \(f_{z}(x, y, z)\) (c) \(G_{x}(w, x, y, z)\) (d) \(\frac{\partial}{\partial z} \lambda(x, y, z, t)\) (e) \(\frac{\partial}{\partial b_{2}} S\left(b_{0}, b_{1}, b_{2}, \ldots, b_{n}\right)\)
Problem 50
Find each partial derivative. (a) \(\frac{\partial}{\partial w}(\sin w \sin x \cos y \cos z)\) (b) \(\frac{\partial}{\partial x}[x \ln (w x y z)]\) (c) \(\lambda_{t}(x, y, z, t)\), where \(\lambda(x, y, z, t)=\frac{t \cos x}{1+x y z t}\)
