Chapter 10

Find the largest possible domain and the corresponding range of each function. Determine the equation of the level curves \(f(x, y)=c\), together with the possible values of \(c .\) \(f(x, y)=\exp \left[-\left(x^{2}+y^{2}\right)\right]\)

In Problems 1-16, find \(\partial f / \partial x\) and \(\partial f / \partial y\) for the given functions. $$ f(x, y)=\log _{5}(3 x y) $$

The functions are defined on the rectangular domain $$ D=\\{(x, y):-1 \leq x \leq 1,-1 \leq y \leq 1\\} $$ Find the absolute maxima and minima of \(f\) on \(D\). \(f(x, y)=x^{2}+y^{2}\)

Show that $$ \lim _{(x, y) \rightarrow(0,0)} \frac{3 x^{2}-y^{2}}{x^{2}+y^{2}} $$ does not exist by computing the limit along the positive \(x\) -axis and the positive \(y\) -axis.

Find the gradient of each function. $$ f(x, y)=x^{3} y^{2} $$

Show that 0 0 is an equilibrium of $$ \left[\begin{array}{l} x_{1}(t+1) \\ x_{2}(t+1) \end{array}\right]=\left[\begin{array}{ll} -0.7 & 0 \\ -0.3 & 0.2 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ and determine its stability.

Find the linearization of \(f(x, y)\) at the indicated point \(\left(x_{0}, y_{0}\right).\) $$ f(x, y)=x-3 y ;(3,1) $$

Find the largest possible domain and the corresponding range of each function. Determine the equation of the level curves \(f(x, y)=c\), together with the possible values of \(c .\) \(f(x, y)=\frac{x-y}{x+y}\)

In Problems 17-24, find the indicated partial derivatives. $$ f(x, y)=3 x^{2}-y-2 y^{2} ; f_{x}(1,0) $$

Find the absolute maxima and minima of $$ f(x, y)=x^{2}+y^{2}-x+2 y $$ on the set $$ D=\\{(x, y)=0 \leq x \leq 1,-2 \leq y \leq 0\\} $$

Compute $$\lim _{(x, y) \rightarrow(0,0)} \frac{4 x y}{x^{2}+y^{2}} $$ along the \(x\) -axis, the \(y\) -axis, and the line \(y=x\). What can you conclude?

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

Show that \(\left[\begin{array}{l}0 \\ 0\end{array}\right]\) is an equilibrium of $$ \left[\begin{array}{l} x_{1}(t+1) \\ x_{2}(t+1) \end{array}\right]=\left[\begin{array}{lr} 0.4 & 0.2 \\ 0 & -0.9 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ and determine its stability.

Find the linearization of \(f(x, y)\) at the indicated point \(\left(x_{0}, y_{0}\right).\) $$ f(x, y)=2 x y ;(1,-1) $$

Find the largest possible domain and the corresponding range of each function. Determine the equation of the level curves \(f(x, y)=c\), together with the possible values of \(c .\) \(f(x, y)=\frac{x+y}{x-y}\)

In Problems 17-24, find the indicated partial derivatives. $$ f(x, y)=x^{1 / 3} y-x y^{1 / 3} ; f_{y}(1,1) $$

Find the absolute maxima and minima of $$ f(x, y)=x^{2}-y^{2}+4 x+y $$ on the set $$ D=\\{(x, y)=-4 \leq x \leq 0,0 \leq y \leq 1\\} $$

Compute $$\lim _{(x, y) \rightarrow(0,0)} \frac{3 x y}{x^{2}+y^{3}} $$ along lines of the form \(y=m x\), for \(m \neq 0 .\) What can you conclude?

Find the gradient of each function. $$ f(x, y)=\sqrt{x^{3}-3 x y} $$

Show that \(\begin{array}{ll}0 & \text { is an equilibrium of }\end{array}\) $$ \left[\begin{array}{l} x_{1}(t+1) \\ x_{2}(t+1) \end{array}\right]=\left[\begin{array}{ll} -1.4 & 0 \\ -0.5 & 0.1 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ and determine its stability.

Find the linearization of \(f(x, y)\) at the indicated point \(\left(x_{0}, y_{0}\right).\) $$ f(x, y)=\sqrt{x}+2 y ;(1,0) $$

In Problems 17-24, find the indicated partial derivatives. $$ g(x, y)=e^{x+3 y} ; g_{y}(2,1) $$

Maximize the function $$ f(x, y)=2 x y-x^{2} y-x y^{2} $$ on the triangle bounded by the line \(x+y=2\), the \(x\) -axis, and the \(y\) -axis.

Compute $$ \lim _{(x, y) \rightarrow(0,0)} \frac{2 x y}{x^{3}+y x} $$ along lines of the form \(y=m x\), for \(m \neq 0\), and along the parabola \(y=x^{2} .\) What can you conclude?

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

Show that \(\left[\begin{array}{l}0 \\ 0\end{array}\right]\) is an equilibrium of $$ \left[\begin{array}{l} x_{1}(t+1) \\ x_{2}(t+1) \end{array}\right]=\left[\begin{array}{rr} 0.1 & 0.4 \\ 0.1 & -0.2 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ and determine its stability.

Find the linearization of \(f(x, y)\) at the indicated point \(\left(x_{0}, y_{0}\right).\) $$ f(x, y)=\cos \left(x^{2} y\right) ;\left(\frac{\pi}{2}, 0\right) $$

In Problems 17-24, find the indicated partial derivatives. $$ h(u, v)=e^{l \prime} \sin (u+v) ; h_{u}(1,-1) $$

Maximize the function $$ f(x, y)=x y(15-5 y-3 x) $$ on the triangle bounded by the line \(5 y+3 x=15\), the \(x\) -axis, and the \(y\) -axis.

Compute $$ \lim _{(x, y) \rightarrow(0,0)} \frac{3 x^{2} y^{2}}{x^{3}+y^{6}} $$ along lines of the form \(y=m x\), for \(m \neq 0\), and along the parabola \(x=y^{2} .\) What can you conclude?

Find the gradient of each function. $$ f(x, y)=\exp \left[\sqrt{x^{2}+y^{2}}\right] $$

Show that \(\left[\begin{array}{l}0 \\ 0\end{array}\right]\) is an equilibrium of $$ \left[\begin{array}{l} x_{1}(t+1) \\ x_{2}(t+1) \end{array}\right]=\left[\begin{array}{ll} 1 & 2 \\ 3 & 2 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ and determine its stability.

Find the linearization of \(f(x, y)\) at the indicated point \(\left(x_{0}, y_{0}\right).\) $$ f(x, y)=\tan (x+y) ;(0,0) $$

In Problems 17-24, find the indicated partial derivatives. $$ f(x, z)=\ln (x z) ; f_{z}(e, 1) $$

Find the absolute maxima and minima of $$ f(x, y)=x^{2}+y^{2}+4 x-1 $$ on the disk $$ D=\left\\{(x, y): x^{2}+y^{2} \leq 9\right\\} $$

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

Find the linearization of \(f(x, y)\) at the indicated point \(\left(x_{0}, y_{0}\right).\) $$ f(x, y)=e^{3 x+2 y} ;(1,2) $$

In Problems 17-24, find the indicated partial derivatives. $$ g(v, w)=\frac{w^{2}}{v+w} ; g_{v}(1,1) $$

Find the absolute maxima and minima of $$ f(x, y)=x^{2}+y^{2}-6 y+3 $$ on the disk $$ D=\left\\{(x, y): x^{2}+y^{2} \leq 16\right\\} $$

Use the definition of continuity to show that $$ f(x, y)=\sqrt{9+x^{2}+y^{2}} $$ is continuous at \((0,0)\).

Find the gradient of each function. $$ f(x, y)=\ln \left(\frac{x}{y}+\frac{y}{x}\right) $$

Show that the equilibrium \(\left[\begin{array}{l}0 \\ 0\end{array}\right]\) $$ \left[\begin{array}{c} x_{1}(t+1) \\ x_{2}(t+1) \end{array}\right]=\left[\begin{array}{rr} -0.2 & -0.4 \\ 0.6 & 0.1 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ is stable.

Find the linearization of \(f(x, y)\) at the indicated point \(\left(x_{0}, y_{0}\right).\) $$ f(x, y)=\ln \left(x^{2}+y\right) ;(1,1) $$

In Problems 17-24, find the indicated partial derivatives. $$ f(x, y)=\frac{x y}{x^{2}+2} ; f_{x}(-1,2) $$

Find the absolute maxima and minima of $$ f(x, y)=x^{2}+y^{2}+x-y $$ on the disk $$ D=\left\\{(x, y): x^{2}+y^{2} \leq 1\right\\} $$

Find the gradient of each function. $$ f(x, y)=\cos \left(3 x^{2}-2 y^{2}\right) $$

Find the linearization of \(f(x, y)\) at the indicated point \(\left(x_{0}, y_{0}\right).\) $$ f(x, y)=x^{2} e^{y} ;(1,0) $$

In Problems 17-24, find the indicated partial derivatives. $$ f(u, v)=e^{u^{2} / 2} \ln (u+v) ; f_{u}(2,1) $$

Find the absolute maxima and minima of $$ f(x, y)=x^{2}+y^{2}+x+2 y $$ on the disk $$ D=\left\\{(x, y): x^{2}+y^{2} \leq 4\right\\} $$

Compute the directional derivative of \(f(x, y)\) at the given point in the indicated direction. $$ f(x, y)=\sqrt{2 x^{2}+y^{2}} \text { at }(1,2) \text { in the direction }\left[\begin{array}{l} 1 \\ 1 \end{array}\right] $$