91影视

Use at least two methods to prove that $\frac{dr}{dt}=0$when $r=\sqrt{x^2+y^2}$$x=伪\cos t,\text{and}y=伪\sin t$if $伪$ is constant.

Express the volume, V, and surface area, S, of a right circular cone with radius r and height h as functions of two variables. What is the domain of each function?

$f(x,y)=\ln 鈦�\left(x{y}^2\right),P=(1,鈭�3)$

The arithmetic mean of the real numbers $a_1,a_2,....,a_n$is $\frac{1}{n}\left(a_1+a_2+....+a_n\right)$. If $a_i>0$for 1 鈮� i 鈮� n, then the geometric mean of $a_1,a_2,....,a_{n}is{\left(a_1{a}_2...a_n\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}$. In Exercises 64鈥�66 we ask you to prove that the geometric mean is always less than the arithmetic mean for a set of positive numbers.

Use the method of Lagrange multipliers to show that $\sqrt{xy}猢�\frac{1}{2}\left(x+y\right)$when x and y are both positive.

Solve the exact differential equations in Exercises 63鈥�66.$y\cos \left(xy\right)鈭�3+(x\cos (xy)+2)\frac{dy}{dx}=0.$

Prove Theorem 12.33. That is, show that if $z=f(x,y),x=u(s,t)$, and $y=v(s,t)$, then, for all values of $s$and $t$at which $u$and $v$are differentiable, and if $f$is differentiable at $u(s,t),v(s,t))$, it follows that

$\frac{鈭�z}{鈭�s}=\frac{鈭�z}{鈭�x}\frac{鈭�x}{鈭�s}+\frac{鈭�z}{鈭�y}\frac{鈭�y}{鈭�s}$and$\frac{鈭�z}{鈭�t}=\frac{鈭�z}{鈭�x}\frac{鈭�x}{鈭�t}+\frac{鈭�z}{鈭�y}\frac{鈭�y}{鈭�t}$

Let S be a subset of $R^2$or $R^3$. Prove that a set S is open if and only if $鈭�S鈭�S=鈭�$

Use the first-order partial derivatives of the functions in Exercises$65$and $66$to find the equation of the hyperplane tangent to the graph of the function at the indicated point P. Note that these are the same functions as in Exercises $53$and $54.$

$f(x,y,z)=x^2+y^2鈭�z^3,P=(1,鈭�5,3)$

Solve the exact differential equations in Exercises 63鈥�66.$e^x\ln y+x^3+\frac{e^x}{y}\frac{dy}{dx}=0$

The arithmetic mean of the real numbers $a_1,a_2,....,a_n$is $\frac{1}{n}\left(a_1+a_2+....+a_n\right)$. If $a_i>0$for 1 鈮� i 鈮� n, then the geometric mean of $a_1,a_2,....,a_{n}is{\left(a_1{a}_2...a_n\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}$. In Exercises 64鈥�66 we ask you to prove that the geometric mean is always less than the arithmetic mean for a set of positive numbers.

Use the method of Lagrange multipliers to show that $\sqrt[3]{xyz}猢�\frac{1}{3}\left(x+y+z\right)$when x, y and z are both positive.