Chapter 13

Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ f(x, y)=x^{2}+2 y^{2}-6 x+8 y-1 $$

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

Approximate the value of \(f\) at the given point. $$ f(x, y)=\sqrt{x^{2}+y^{2}} ;(3.01,4.03) $$

Find the gradient of the function. $$ f(x, y)=3 x-5 y $$

Find the directional derivative of \(f\) at the point \(P\) in the direction of a. $$ \begin{aligned} &f(x, y)=2 x^{2}-3 x y+y^{2}+15 ; P=(1,1) \\ &\mathbf{a}=\frac{1}{\sqrt{2}} \mathbf{i}+\frac{1}{\sqrt{2}} \mathbf{j} \end{aligned} $$

Compute \(d z / d t\). \(z=2 x^{2}-3 y^{3} ; x=\sqrt{t}, y=e^{2 t}\)

Find the first partial derivatives of the function. $$ f(x, y)=\frac{2}{3} x^{3 / 2} $$

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

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

Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ f(x, y)=x^{2}-2 y^{2}-6 x+8 y+3 $$

Find the extreme values of \(f\) subject to the given constraint. In each case assume that the extreme values exist. $$ f(x, y)=x y ;(x+1)^{2}+y^{2}=1 $$

Approximate the value of \(f\) at the given point. $$ f(x, y)=\sqrt{x^{2}+y} ;(3.02,-4.98) $$

Find the gradient of the function. $$ f(x, y)=y^{2}+x \sin x^{2} y $$

Compute \(d z / d t\). \(z=\ln \left(3 x^{2}+y^{3}\right) ; x=e^{2 t}, y=t^{1 / 3}\)

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

Evaluate the limit. $$ \lim _{(x, y) \rightarrow(1,-2)}\left(2 x^{3}-4 x y+5 y^{2}\right) $$

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

Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ f(x, y)=x^{2}+6 x y+2 y^{2}-6 x+10 y-2 $$

Find the extreme values of \(f\) subject to the given constraint. In each case assume that the extreme values exist. $$ f(x, y)=x^{3}+2 y^{3} ; x^{2}+y^{2}=1 $$

Approximate the value of \(f\) at the given point. $$ f(x, y)=\ln \left(x^{2}+y^{2}\right) ;(-0.03,0.98) $$

Find the gradient of the function. $$ g(x, y)=e^{-2 x} \ln (y-4) $$

Compute \(d z / d t\). $$ z=\sin x+\cos x y ; x=t^{2}, y=1 $$

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

Find the first partial derivatives of the function. $$ f(x, y)=2 x+3 x^{2} y^{4} $$

Evaluate the limit. $$ \lim _{(x, y) \rightarrow(1,0)} \frac{x^{2}-x y+1}{x^{2}+y^{2}} $$

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

Find the extreme values of \(f\) subject to the given constraint. In each case assume that the extreme values exist. $$ f(x, y, z)=y^{3}+x z^{2} ; x^{2}+y^{2}+z^{2}=1 $$

Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ g(x, y)=x^{2}-x y-2 y^{2}+7 x-8 y+3 $$

Approximate the value of \(f\) at the given point. $$ f(x, y)=\sin \pi x y ;(-1.97,2.005) $$

Find the gradient of the function. $$ f(x, y)=\frac{x y-1}{x^{2}+y^{2}} $$

Compute \(d z / d t\). $$ z=\tan ^{-1}\left(y^{2}-x^{2}\right) ; x=\sin t, y=\cos t $$

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

Find the first partial derivatives of the function. $$ g(x, y)=x^{3} e^{2 y} $$

Evaluate the limit. $$ \lim _{(x, y, z) \rightarrow(-1,2,0)}\left(x^{2}+3 y-4 z^{2}+2\right) $$

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

Find the extreme values of \(f\) subject to the given constraint. In each case assume that the extreme values exist. $$ f(x, y, z)=x y z ; x^{2}+y^{2}+4 z^{2}=6 $$

Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ k(x, y)=-x^{2}-2 x y-2 y^{2}+6 x-10 y+5 $$

Approximate the value of \(f\) at the given point. $$ f(x, y)=\tan x y ;(0.99 \pi, 0.24) $$

Find the gradient of the function. $$ f(x, y, z)=2 x^{2}-y^{2}-4 z^{2} $$

Compute \(d z / d t\). $$ z=\sqrt{2 x-4 y} ; x=\ln t, y=1-3 t^{3} $$

Find the first partial derivatives of the function. $$ g(u, v)=\frac{u^{3}+v^{3}}{u^{2}+v^{2}} $$

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

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

Find the extreme values of \(f\) subject to the given constraint. In each case assume that the extreme values exist. $$ f(x, y, z)=x y+y z ; x^{2}+y^{2}+z^{2}=8 $$

Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ f(x, y)=-x^{2}+4 x y+y^{2}-2 x+9 $$

Approximate the value of \(f\) at the given point. $$ f(x, y)=\sqrt{6-x^{2}-y^{2}} ;(0.987,1.013) $$

Find the gradient of the function. $$ f(x, y, z)=\left(2 x+y^{2}+z^{3}\right)^{5 / 2} $$

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

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

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