91Ó°ÊÓ

\(f(x, y)=-x /\left(x^{2}+y^{2}\right), f(0,0)=0 ;-1 \leq x \leq 1\), \(-1 \leq y \leq 1\); global maximum and global minimum. Be careful.

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 \((0,0,0)\).

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) \(D_{x}\left(D_{y} \sin \left(x+y^{2}\right)\right)\)

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) \(\frac{\partial}{\partial b_{2}} S\left(b_{0}, b_{1}, b_{2}, \ldots, b_{n}\right)\)

\(f(x, y)=8 \cos (x y+2 x)+x^{2} y^{2},-3 \leq x \leq 3\), \(-3 \leq y \leq 3\); global maximum and global minimum.

\(f(x, y)=(\sin x) /(6+x+|y|) ;-3 \leq x \leq 3\), \(-3 \leq y \leq 3\); global maximum and global minimum.

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 \(\lambda(x, y, z, t)=\frac{t \cos x}{1+x y z t}\)

\(f(x, y)=\cos \left(|x|+y^{2}\right)+10 x \exp \left(-x^{2}-y^{2}\right)\); \(-2 \leq x \leq 2,-2 \leq y \leq 2\); global maximum and global minimum.

\(f(x, y)=2 \sin x+\sin y-\sin (x+y)\); \(0 \leq x \leq 2 \pi, 0 \leq y \leq 2 \pi\); global maximum and global minimum.