다변수 미분 전체 문제

Find the level surface for the functions of three variables and describe it.

$w(x,y,z)={x}^{2}+{y}^{2}-{z}^{2},c=4$

Find the level surface for the functions of three variables and describe it.

$w(x,y,z)=9{x}^{2}-4{y}^{2}+36{z}^{2},c=0$

For the following exercises, find an equation of the level curve of $f$ that contains the point $P.$

$f(x,y)=1-4{x}^{2}-{y}^{2},P(0,1)$

For the following exercises, find an equation of the level curve of $f$ that contains the point $P.$

$g(x,y)={y}^{2}\arctan \,x,P(1,2)$

For the following exercises, find an equation of the level curve of $f$ that contains the point $P.$

$g(x,y)={e}^{xy}({x}^{2}+{y}^{2}),P(1,0)$

The strength $E$ of an electric field at point $(x,y,z)$ resulting from an infinitely long charged wire lying along the $z\text{-axis}$ is given by $E(x,y,z)=k/\sqrt{{x}^{2}+{y}^{2}},$ where $k$ is a positive constant. For simplicity, let $k=1$ and find the equations of the level surfaces for $E=10\,\text{and}\,E=100.$

Refer to the preceding problem. Find the level curves for $T=40^\circ \text{C and}\,T=100^\circ \text{C},$ and describe what the level curves represent.

For the following exercises, find the limit of the function.

$\lim_{(x,y)\to (1,2)}x$

For the following exercises, find the limit of the function.

$\lim_{(x,y)\to (1,2)}\frac{5{x}^{2}y}{{x}^{2}+{y}^{2}}$

For the following exercises, evaluate the limits at the indicated values of $x\,\text{and}\,y.$ If the limit does not exist, state this and explain why the limit does not exist.

$\lim_{(x,y)\to (0,0)}\frac{4{x}^{2}+10{y}^{2}+4}{4{x}^{2}-10{y}^{2}+6}$

For the following exercises, evaluate the limits at the indicated values of $x\,\text{and}\,y.$ If the limit does not exist, state this and explain why the limit does not exist.

$\lim_{(x,y)\to (11,13)}\sqrt{\frac{1}{xy}}$

For the following exercises, evaluate the limits at the indicated values of $x\,\text{and}\,y.$ If the limit does not exist, state this and explain why the limit does not exist.

$\lim_{(x,y)\to (0,1)}\frac{{y}^{2}\sin \,x}{x}$

For the following exercises, evaluate the limits at the indicated values of $x\,\text{and}\,y.$ If the limit does not exist, state this and explain why the limit does not exist.

$\lim_{(x,y)\to (0,0)}\sin \left(\frac{{x}^{8}+{y}^{7}}{x-y+10}\right)$

For the following exercises, evaluate the limits at the indicated values of $x\,\text{and}\,y.$ If the limit does not exist, state this and explain why the limit does not exist.

$\lim_{(x,y)\to (\pi /4,1)}\frac{y\,\tan \,x}{y+1}$

For the following exercises, evaluate the limits at the indicated values of $x\,\text{and}\,y.$ If the limit does not exist, state this and explain why the limit does not exist.

$\lim_{(x,y)\to (0,\pi /4)}\frac{\sec \,x+2}{3x-\tan \,y}$

For the following exercises, evaluate the limits at the indicated values of $x\,\text{and}\,y.$ If the limit does not exist, state this and explain why the limit does not exist.

$\lim_{(x,y)\to (2,5)}\left(\frac{1}{x}-\frac{5}{y}\right)$

For the following exercises, evaluate the limits at the indicated values of $x\,\text{and}\,y.$ If the limit does not exist, state this and explain why the limit does not exist.

$\lim_{(x,y)\to (4,4)}x\,\ln \,y$

For the following exercises, evaluate the limits at the indicated values of $x\,\text{and}\,y.$ If the limit does not exist, state this and explain why the limit does not exist.

$\lim_{(x,y)\to (4,4)}{e}^{-{x}^{2}-{y}^{2}}$

For the following exercises, evaluate the limits at the indicated values of $x\,\text{and}\,y.$ If the limit does not exist, state this and explain why the limit does not exist.

$\lim_{(x,y)\to (0,0)}\sqrt{9-{x}^{2}-{y}^{2}}$

For the following exercises, evaluate the limits at the indicated values of $x\,\text{and}\,y.$ If the limit does not exist, state this and explain why the limit does not exist.

$\lim_{(x,y)\to (1,2)}({x}^{2}{y}^{3}-{x}^{3}{y}^{2}+3x+2y)$