Chapter 10

Show that the equilibrium \(\left[\begin{array}{l}0 \\ 0\end{array}\right]\) of $$ \left[\begin{array}{l} x_{1}(t+1) \\ x_{2}(t+1) \end{array}\right]=\left[\begin{array}{ll} 4.2 & -3.4 \\ 2.4 & -1.1 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ is unstable

Find the linear approximation of $$f(x, y)=e^{x+y}$$ at \((0,0)\), and use it to approximate \(f(0.1,0.05)\). Using a calculator, compare the approximation with the exact value of \(f(0.1,0.05)\).

Let $$ f(x, y)=4-x^{2}-y^{2} $$ Compute \(f_{x}(1,1)\) and \(f_{y}(1,1)\), and interpret these partial derivatives geometricallv.

Can a continuous function of two variables have two maxima and no minima? Describe in words what the properties of such a function would be, and contrast this behavior with a function of one variable.

Compute the directional derivative of \(f(x, y)\) at the given point in the indicated direction. $$ f(x, y)=x^{2} \sin y \text { at }(-1,0) \text { in the direction }\left[\begin{array}{r} 2 \\ -1 \end{array}\right] $$

Show that the equilibrium \(\left[\begin{array}{l}0 \\ 0\end{array}\right]\) of $$ \left[\begin{array}{c} x_{1}(t+1) \\ x_{2}(t+1) \end{array}\right]=\left[\begin{array}{ll} 2 & -4 \\ 5 & -6 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ is stable.

Find the linear approximation of $$f(x, y)=\sin (x+2 y)$$ at \((0,0)\), and use it to approximate \(f(-0.1,0.2)\). Using a calculator, compare the approximation with the exact value of \(f(-0.1,0.2)\).

Let $$ f(x, y)=\sqrt{4-x^{2}-y^{2}} $$ Compute \(f_{x}(1,1)\) and \(f_{y}(1,1)\), and interpret these partial derivatives geometrically.

Suppose \(f(x, y)\) has a horizontal tangent plane at \((0,0)\). Can you conclude that \(f\) has a local extremum at \((0,0) ?\)

Compute the directional derivative of \(f(x, y)\) at the given point in the indicated direction. $$ f(x, y)=e^{x+y} \text { at }(0,0) \text { in the direction }\left[\begin{array}{l} -1 \\ -1 \end{array}\right] $$

Show that the equilibrium \(\left[\begin{array}{l}0 \\ 0\end{array}\right]\) of $$ \begin{array}{l} x_{1}(t+1)=\frac{x_{2}(t)}{4\left(1+x_{1}^{2}(t)\right)} \\ x_{2}(t+1)=\frac{2 x_{1}(t)}{1+x_{2}^{2}(t)} \end{array} $$ is locally stable.

Find the linear approximation of $$f(x, y)=\ln \left(x^{2}-3 y\right)$$ at \((1,0)\), and use it to approximate \(f(1.1,0.1)\). Using a calculator, compare the approximation with the exact value of \(f(1.1,0.1)\).

Let $$ f(x, y)=1+x^{2} y $$ Compute \(f_{x}(-2,1)\) and \(f_{y}(-2,1)\), and interpret these partial derivatives geometrically.

Suppose crop yield \(Y\) depends on nitrogen \((\mathrm{N})\) and phosphorus (P) concentrations as $$ Y(N, P)=N P e^{-(N+P)} $$ Find the value of \((N, P)\) that maximizes crop yield.

(a) Write $$ h(x, y)=\sin \left(x^{2}+y^{2}\right) $$ as a composition of two functions. (b) For which values of \((x, y)\) is \(h(x, y)\) continuous?

Compute the directional derivative of \(f(x, y)\) at the given point in the indicated direction. $$ f(x, y)=x^{3} y^{2} \text { at }(2,3) \text { in the direction }\left[\begin{array}{r} -2 \\ 1 \end{array}\right] $$

Show that the equilibrium \(\left[\begin{array}{l}0 \\ 0\end{array}\right]\) of $$ \begin{array}{l} x_{1}(t+1)=\frac{3 x_{2}(t)}{1+x_{1}^{2}(t)} \\ x_{2}(t+1)=\frac{2 x_{1}(t)}{1+x_{2}^{2}(t)} \end{array} $$ is unstable.

Find the linear approximation of $$f(x, y)=\tan \left(2 x-3 y^{2}\right)$$ at \((0,0)\), and use it to approximate \(f(0.03,0.05) .\) Using a calculator, compare the approximation with the exact value of \(f(0.03,0.05).\)

Let $$ f(x, y)=2 x^{3}-3 y x $$ Compute \(f_{x}(1,2)\) and \(f_{y}(1,2)\), and interpret these partial derivatives geometrically.

Choose three numbers \(x, y\), and \(z\) so that their sum is equal to 60 and their product is maximal.

(a) Write $$ h(x, y)=\sqrt{x+y} $$ as a composition of two functions.

Compute the directional derivative of \(f(x, y)\) at the given point in the indicated direction. $$ f(x, y)=2 x y^{3}-3 x^{2} y \text { at }(1,-1) \text { in the direction }\left[\begin{array}{l} 3 \\ 1 \end{array}\right] $$

Show that the equilibrium \(\left[\begin{array}{l}0 \\ 0\end{array}\right]\) of $$ \begin{array}{l} x_{1}(t+1)=x_{2}(t) \\ x_{2}(t+1)=\frac{2 x_{2}(t)-x_{1}(t)}{2+x_{1}(t)} \end{array} $$ is locally stable.

Find the Jacobi matrix for each given function. $$ \mathbf{f}(x, y)=\left[\begin{array}{c} x+y \\ x^{2}-y^{2} \end{array}\right] $$

Find the maximum volume of a rectangular closed (top, bottom, and four sides) box with surface area \(48 \mathrm{~m}^{2}\).

(a) Write $$ h(x, y)=e^{x y} $$ as a composition of two functions. (b) For which values of \((x, y)\) is \(h(x, y)\) continuous?

Compute the directional derivative of \(f(x, y)\) at the given point in the indicated direction. $$ f(x, y)=y e^{x^{2}} \text { at }(0,2) \text { in the direction }\left[\begin{array}{r} 4 \\ -1 \end{array}\right] $$

Show that, for any \(a>1\), the equilibrium \(\left[\begin{array}{l}0 \\\ 0\end{array}\right]\) of $$ \begin{aligned} x_{1}(t+1) &=x_{2}(t) \\ x_{2}(t+1) &=\frac{a x_{2}(t)-(a-1) x_{1}(t)}{a+x_{1}(t)} \end{aligned} $$ is locally stable.

Find the Jacobi matrix for each given function. $$ \mathbf{f}(x, y)=\left[\begin{array}{c} 2 x-3 y \\ 4 x^{2} \end{array}\right] $$

Find the maximum volume of a rectangular open (bottom and four sides, no top) box with surface area \(75 \mathrm{~m}^{2}\).

a) Write $$ h(x, y)=\cos (y-x) $$ as a composition of two functions. (b) For which values of \((x, y)\) is \(h(x, y)\) continuous?

Compute the directional derivative of \(f(x, y)\) at the point \(P\) in the direction of the point \(Q\). $$ f(x, y)=2 x^{2} y-3 x, P=(2,1), Q=(3,2) $$

Show that \(\left[\begin{array}{l}0 \\ 0\end{array}\right]\) is an equilibrium point of $$ \begin{array}{l} x_{1}(t+1)=a x_{2}(t) \\ x_{2}(t+1)=2 x_{1}(t)-\cos \left(x_{2}(t)\right)+1 \end{array} $$ Assume that \(a>0\). For which values of \(a\) is \(\left[\begin{array}{c}0 \\\ 0\end{array}\right]\) locally stable?

Find the Jacobi matrix for each given function. $$ \mathbf{f}(x, y)=\left[\begin{array}{l} e^{x-y} \\ e^{x+y} \end{array}\right] $$

In Problems \(31-38\), find \(\partial f / \partial x, \partial f / \partial y\), and \(\partial f / \partial z\) for the given functions. $$ f(x, y, z)=x^{2} z+y z^{2}-x y $$

Find the minimum surface area of a rectangular closed (top, bottom, and four sides) box with volume \(216 \mathrm{~m}^{3}\).

Draw an open disk with radius 2 centered at \((1,-1)\) in the \(x-y\) plane, and give a mathematical description of this set.

Compute the directional derivative of \(f(x, y)\) at the point \(P\) in the direction of the point \(Q\). $$ f(x, y)=4 x y+y^{2}, P=(-1,1), Q=(3,2) $$

Show that \(\left[\begin{array}{c}0 \\ 0\end{array}\right]\) and \(\left[\begin{array}{r}-\pi \\ \pi\end{array}\right]\) are equilibria of $$ \begin{array}{l} x_{1}(t+1)=-x_{2}(t) \\ x_{2}(t+1)=\sin \left(x_{2}(t)\right)-x_{1}(t) \end{array} $$ and analyze their stability.

Find the Jacobi matrix for each given function. $$ \mathbf{f}(x, y)=\left[\begin{array}{c} (x-y)^{2} \\ \sin (x-y) \end{array}\right] $$

In Problems \(31-38\), find \(\partial f / \partial x, \partial f / \partial y\), and \(\partial f / \partial z\) for the given functions. $$ f(x, y, z)=x y z $$

Find the minimum surface area of a rectangular open (bottom and four sides, no top) box with volume \(256 \mathrm{~m}^{3}\).

Draw a closed disk with radius 3 centered at \((2,0)\) in the \(x-y\) plane, and give a mathematical description of this set.

Compute the directional derivative of \(f(x, y)\) at the point \(P\) in the direction of the point \(Q\). $$ f(x, y)=\sqrt{x y-2 x^{2}}, P=(1,6), Q=(3,1) $$

Find all nonnegative equilibria of $$ \begin{array}{l} x_{1}(t+1)=x_{2}(t) \\ x_{2}(t+1)=\frac{1}{2} x_{1}(t)+\frac{2}{3} x_{2}(t)-x_{2}^{2}(t) \end{array} $$ and analyze their stability.

Find the Jacobi matrix for each given function. $$ \mathbf{f}(x, y)=\left[\begin{array}{l} \cos (x-y) \\ \cos (x+y) \end{array}\right] $$

In Problems \(31-38\), find \(\partial f / \partial x, \partial f / \partial y\), and \(\partial f / \partial z\) for the given functions. $$ f(x, y, z)=x^{3} y^{2} z+\frac{x}{y z} $$

The distance between the origin \((0,0,0)\) and the point \((x, y, z)\) is $$ \sqrt{x^{2}+y^{2}+z^{2}} $$ Find the minimum distance between the origin and the plane \(x+y+z=1 .\) (Hint: Minimize the squared distance between the origin and the plane.)

Give a geometric interpretation of the set $$ A=\left\\{(x, y) \in \mathbf{R}^{2}: \sqrt{x^{2}+y^{2}-4 y+4}<3\right\\} $$

Compute the directional derivative of \(f(x, y)\) at the point \(P\) in the direction of the point \(Q\). $$ f(x, y)=e^{x-y}, P=(2,2), Q=(1,-1) $$