Chapter 13

Evaluate the limit. $$ \lim _{x, y) \rightarrow(-1,1)} \frac{x^{2}+2 x y^{2}+y^{4}}{1+y^{2}} $$

Find the domain of the function. \(g(x, y)=\sqrt{25-x^{2}-y^{2}}\)

Find the minimum value of \(f\) subject to the given constraint. In each case assume that the minimum value exists. $$ f(x, y)=4 x^{2}+y^{3}+3 y+7 ; 2 x^{2}+\frac{3}{2} y^{2}=\frac{3}{2} $$

Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ f(x, y)=x^{2} y-2 x y+2 y^{2}-15 y $$

Approximate the value of \(f\) at the given point. $$ f(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}} ;(3.01,4.02,11.98) $$

Find the gradient of the function. $$ g(x, y, z)=\frac{-x+y}{-x+z} $$

Compute \(\partial z / \partial u\) and \(\partial z / \partial v\). $$ z=\frac{4}{x y}-\frac{x}{y} ; x=u^{2}, y=u v $$

Find the directional derivative of \(f\) at the point \(P\) in the direction of a. $$ f(x, y)=\tan (x+2 y) ; P=(0, \pi / 6) ; \mathbf{a}=-4 \mathbf{i}+5 \mathbf{j} $$

Find the first partial derivatives of the function. $$ f(x, y)=\sqrt{4-x^{2}-9 y^{2}} $$

Evaluate the limit. $$ \lim _{(x, y) \rightarrow(2,1)} \frac{x^{3}+2 x^{2} y-x y-2 y^{2}}{x+2 y} $$

Find the domain of the function. \(f(x, y)=\frac{1}{x+y}\)

Find the minimum value of \(f\) subject to the given constraint. In each case assume that the minimum value exists. $$ f(x, y, z)=x^{2}+2 y^{2}+z^{2} ; x+y+z=4 $$

Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ f(x, y)=x^{3}-6 x^{2}-3 y^{2} $$

Approximate the value of \(f\) at the given point. $$ f(x, y, z)=x y z^{2} ;(-2.1,1.01,0.989) $$

Find the gradient of the function. $$ g(x, y, z)=-x^{2} y^{3} e^{\left(z^{2}\right)} $$

Compute \(\partial z / \partial u\) and \(\partial z / \partial v\). $$ z=16-4 x^{2}-y^{2} ; x=u \sin v, y=v \cos u $$

Find the directional derivative of \(f\) at the point \(P\) in the direction of a. $$ f(x, y, z)=3 x-2 y+4 z ; P=(1,-1,2) ; \mathbf{a}=\mathbf{i}+\mathbf{j}+\mathbf{k} $$

Find the first partial derivatives of the function. $$ z=\sqrt{\frac{1}{4} x^{2}-y^{2}} $$

Evaluate the limit. $$ \lim _{(x, y) \rightarrow(\ln 2,0)} e^{2 x+y^{2}} $$

Find the domain of the function. \(f(u, v)=\ln \frac{u^{2}+v^{2}}{\left(u^{2}-v^{2}\right)^{2}}\)

Find the minimum value of \(f\) subject to the given constraint. In each case assume that the minimum value exists. $$ f(x, y, z)=x^{4}+8 y^{4}+27 z^{4} ; x+y+z=\frac{11}{12} $$

Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ f(x, y)=3 x^{2}-3 x y^{2}+y^{3}+3 y^{2} $$

Approximate the number. $$ \sqrt[4]{(1.9)^{3}+(2.1)^{3}} $$

Find the gradient of the function at the given point. $$ f(x, y)=\frac{x+3 y}{5 x+2 y} ;\left(-1, \frac{3}{2}\right) $$

Compute \(\partial z / \partial u\) and \(\partial z / \partial v\). $$ z=\ln \left(x^{2}-y^{2}\right) ; x=u-v, y=u^{2}+v^{2} $$

Find the directional derivative of \(f\) at the point \(P\) in the direction of a. $$ f(x, y, z)=x^{3} y^{2} z ; P=(2,-1,2) ; \mathbf{a}=2 \mathbf{i}-\mathbf{j}-2 \mathbf{k} $$

Find the first partial derivatives of the function. $$ z=\sqrt{\left(1-x^{2 / 3}\right)^{3}-y^{2}} $$

Evaluate the limit. $$ \lim _{(x, y, z) \rightarrow(\pi / 2,-\pi / 2,0)} \cos (x+y+z) $$

Find the domain of the function. \(f(x, y, z)=\sqrt{1-x^{2}-y^{2}-z^{2}}\)

Find the minimum value of \(f\) subject to the given constraint. In each case assume that the minimum value exists. $$ f(x, y, z)=3 z-x-2 y ; z=x^{2}+4 y^{2} $$

Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ f(u, v)=u^{3}+v^{3}-6 u v $$

Approximate the number. $$ (16.05)^{1 / 4}(7.95)^{2 / 3} $$

Find the gradient of the function at the given point. $$ f(x, y)=x \cos x y ;(1,-\pi) $$

Compute \(\partial z / \partial u\) and \(\partial z / \partial v\). $$ z=2 e^{x^{2} y} ; x=\sqrt{u v}, y=1 / u $$

Find the directional derivative of \(f\) at the point \(P\) in the direction of a. $$ \begin{aligned} &f(x, y, z)=x y-y z+3 x z ; P=(1,-1,3) ; \\ &\mathbf{a}=-\mathbf{i}+3 \mathbf{j}+2 \mathbf{k} \end{aligned} $$

Find the first partial derivatives of the function. $$ w=\cos \frac{u}{v} $$

Evaluate the limit. $$ \lim _{(x, y) \rightarrow(0,1)} \frac{\sin x y}{y} $$

Find the domain of the function. \(f(x, y, z)=\frac{1}{x y z}\)

Find the extreme values of \(f\) in the region described by the given inequalities. In each case assume that the extreme values exist. $$ f(x, y)=2 x^{2}+y^{2}+2 y-3 ; x^{2}+y^{2} \leq 4 $$

Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ f(x, y)=4 x y+2 x^{2} y-x y^{2} $$

Compute \(\partial z / \partial r\) and \(\partial z / \partial s\). $$ z=\sin 2 u \cos 3 v ; u=(r+s)^{2}, v=(r-s)^{2} $$

Find the directional derivative of \(f\) at the point \(P\) in the direction of a. $$ f(x, y, z)=\frac{x-y-z}{x+y+z} ; P=(2,1,-1) ; \mathbf{a}=-2 \mathbf{i}-\mathbf{j}-\mathbf{k} $$

Find the first partial derivatives of the function. $$ z=\left(\sin x^{2} y\right)^{3} $$

Evaluate the limit. $$ \lim _{(x, y) \rightarrow(0,0)} \frac{\sin \left(x^{2}+y^{2}\right)}{x^{2}+y^{2}} $$

Find the domain of the function. \(g(x, y, z)=\frac{x}{y}-\frac{y}{z}+\frac{z}{x}\)

Find the extreme values of \(f\) in the region described by the given inequalities. In each case assume that the extreme values exist. $$ f(x, y)=x^{3}+x^{2}+\frac{y^{2}}{3} ; x^{2}+y^{2} \leq 36 $$

Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ f(x, y)=\frac{1}{x}+\frac{1}{y}+x y $$

Approximate the number. $$ \sin \frac{9 \pi}{20} \cos \frac{9 \pi}{30} $$

Compute \(\partial z / \partial r\) and \(\partial z / \partial s\). $$ z=\ln u+\ln v ; u=4^{r s}, v=4^{r / s} $$

Find the gradient of the function at the given point. $$ f(x, y, z)=z-\sqrt{x^{2}+y^{2}} ;(3,-4,7) $$