Chapter 12

If \(f(x, y, z)=(x y / z)^{1 / 2}\), find \(f_{x}(-2,-1,8)\).

Identify the graph of \(f(x, y)=x^{2}-x+3 y^{2}+\) \(12 y-13\), state where it attains its minimum value, and find this minimum value.

Draw the graph and the corresponding contour plot. \(f(x, y)=\sin \sqrt{2 x^{2}+y^{2}} ;-2 \leq x \leq 2,-2 \leq y \leq 2\)

Plot the graphs of each of the following functions on \(-2 \leq x \leq 2,-2 \leq y \leq 2\), and determine where on this set they are discontinuous. (a) \(f(x, y)=x^{2} /\left(x^{2}+y^{2}\right), f(0,0)=0\) (b) \(f(x, y)=\tan \left(x^{2}+y^{2}\right) /\left(x^{2}+y^{2}\right), f(0,0)=0\)

Let \(A(x, y)\) be the area of a nondegenerate rectangle of dimensions \(x\) and \(y\), the rectangle being inside a circle of radius 10. Determine the domain and range for this function.

Draw the graph and the corresponding contour plot. \(f(x, y)=\sin \left(x^{2}+y^{2}\right) /\left(x^{2}+y^{2}\right), f(0,0)=1 ;\) \(-2 \leq x \leq 2,-2 \leq y \leq 2\)

Draw the graph and the corresponding contour plot. \(f(x, y)=\left(2 x-y^{2}\right) \exp \left(-x^{2}-y^{2}\right) ;-2 \leq x \leq 2\) \(-2 \leq y \leq 2\)

Give definitions of continuity at a point and continuity on a set for a function of three variables.

The wave equation \(c^{2} \partial^{2} u / \partial x^{2}=\partial^{2} u / \partial t^{2}\) and the heat equation \(c \partial^{2} u / \partial x^{2}=\partial u / \partial t\) are two of the most important equations in physics ( \(c\) is a constant). These are called partial differential equations. Show each of the following: (a) \(u=\cos x \cos c t\) and \(u=e^{x}\) cosh \(c t\) satisfy the wave equation. (b) \(u=e^{-c t} \sin x\) and \(u=t^{-1 / 2} e^{-x^{2} /(4 c t)}\) satisfy the heat equation.

Draw the graph and the corresponding contour plot. \(f(x, y)=(\sin x \sin y) /\left(1+x^{2}+y^{2}\right) ;-2 \leq x \leq 2\) \(-2 \leq y \leq 2\)

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

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

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