91Ó°ÊÓ

Identify and briefly describe the surfaces defined by the following equations. $$x^{2}+y^{2}+4 z^{2}+2 x=0$$

Use the gradient rules of Exercise 81 to find the gradient of the following functions. $$f(x, y)=\ln \left(1+x^{2}+y^{2}\right)$$

Identify and briefly describe the surfaces defined by the following equations. $$9 x^{2}+y^{2}-4 z^{2}+2 y=0$$

Use the formal definition of a limit to \(\lim _{(x, y) \rightarrow(a, b)} c f(x, y)=c \lim _{(x, y) \rightarrow(a, b)} f(x, y)\)

Use the gradient rules of Exercise 81 to find the gradient of the following functions. $$f(x, y, z)=\sqrt{25-x^{2}-y^{2}-z^{2}}$$

Identify and briefly describe the surfaces defined by the following equations. $$x^{2}+4 y^{2}=1$$

Use the gradient rules of Exercise 81 to find the gradient of the following functions. $$f(x, y, z)=(x+y+z) e^{x y z}$$

The flow of heat along a thin conducting bar is governed by the one- dimensional heat equation (with analogs for thin plates in two dimensions and for solids in three dimensions) $$\frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}},$$ where \(u\) is a measure of the temperature at a location \(x\) on the bar at time t and the positive constant \(k\) is related to the conductivity of the material. Show that the following functions satisfy the heat equation with \(k=1\). $$u(x, t)=e^{-t}(2 \sin x+3 \cos x)$$

The flow of heat along a thin conducting bar is governed by the one- dimensional heat equation (with analogs for thin plates in two dimensions and for solids in three dimensions) $$\frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}},$$ where \(u\) is a measure of the temperature at a location \(x\) on the bar at time t and the positive constant \(k\) is related to the conductivity of the material. Show that the following functions satisfy the heat equation with \(k=1\). \(u(x, t)=A e^{-\alpha^{2} t} \cos a x,\) for any real numbers \(a\) and \(A\)

Identify and briefly describe the surfaces defined by the following equations. $$y^{2}-z^{2}=2$$