Chapter 10

In what direction does \(f(x, y)=e^{x} \cos y\) increase most rapidly at \((0, \pi / 2) ?\)

20\. Earthquake Frequency The time between two earthquakes is sometimes modeled using a Poisson process model. According to this model the probability that a second earthquake follows within a time \(t\) of the first earthquake is: $$ P(t)=1-e^{-\lambda t} $$ where \(\lambda\) is a positive constant. We define the average time between earthquakes to be a time \(T\) as follows: The probability that a second earthquake follows the first earthquake within time \(T\) is exactly \(1 / 2\), i.e.: $$ 1 / 2=1-e^{-\lambda T} $$ \(T\) is a function of \(\lambda\). Show that, if the coefficient \(\lambda\) is increased in the model, then the average time between earthquakes will decrease.

Show that \(\left[\begin{array}{l}0 \\ 0\end{array}\right]\) is an equilibrium of $$ \left[\begin{array}{l} x_{1}(t+1) \\ x_{2}(t+1) \end{array}\right]=\left[\begin{array}{rr} 0.1 & 0.3 \\ 0.1 & -1.8 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ and determine its stability.

Find the indicated partial derivatives. \(f(x, z)=\ln (x z) ; f_{z}(e, 1)\)

Use the definition of continuity to show that $$f(x, y)=2 x^{2}+y^{2}+1$$ is continuous at \((0,0)\).

Find the range of each function \(f(x, y)\), when defined on the specified domain \(D\). \(f(x, y)=x-y^{2} ; D=\\{(x, y):-1 \leq x \leq 1,1 \leq y \leq 2\\}\)

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

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

In what direction does \(f(x, y)=\sqrt{x^{2}-y^{2}}\) increase most rapidly at \((5,3)\) ?

Find the indicated partial derivatives. \(g(v, w)=\frac{w^{2}}{v+w} ; g_{v}(1,1)\)

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

Find the range of each function \(f(x, y)\), when defined on the specified domain \(D\). \(f(x, y)=\frac{x}{y} ; D=\\{(x, y): 0 \leq x \leq 1,1 \leq y \leq 2\\}\)

Find the linearization of \(f(x, y)\) at the indicated point \(\left(x_{0}, y_{0}\right) .\) \(f(x, y)=e^{9 x+2 y} ;(1,2)\)

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

In what direction does \(f(x, y)=\ln \left(x^{2}+y^{2}\right)\) increase most rapidly at \((1,1) ?\)

Show that \(\left[\begin{array}{l}0 \\ 0\end{array}\right]\) is an equilibrium of $$ \left[\begin{array}{l} x_{1}(t+1) \\ x_{2}(t+1) \end{array}\right]=\left[\begin{array}{ll} 1.5 & 0.2 \\ 0.08 & 0 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ and determine its stability.

Find the indicated partial derivatives. \(f(x, y)=\frac{x y}{x^{2}+2} ; f_{x}(-1,2)\)

Show that $$f(x, y)=\left\\{\begin{array}{cc}\frac{4 x y}{x^{2}+y^{2}} & \text { for }(x, y) \neq(0,0) \\\0 & \text { for }(x, y)=(0,0)\end{array}\right.$$ is discontinuous at \((0,0)\).

Find the range of each function \(f(x, y)\), when defined on the specified domain \(D\). \(f(x, y)=\frac{x}{y} ; D=\\{(x, y): 0 \leq x \leq 1,0 \leq y \leq 1, y>x]\)

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

Find the global 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 a unit vector that is normal to the level curve of the function $$ f(x, y)=3 x+4 y $$ at the point \((-1,1)\).

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

Find the indicated partial derivatives. \(f(u, v)=e^{\mu+3 v^{2}} ; f_{u}(2,1)\)

Show that $$f(x, y)=\left\\{\begin{array}{cl}\frac{3 r(y+x)}{x^{2}+y^{3}} & \text { for }(x, y) \neq(0,0) \\\0 & \text { for }(x, y)=(0,0)\end{array}\right.$$ is discontinuous at \((0,0)\).

Find the range of each function \(f(x, y)\), when defined on the specified domain \(D\). \(\left.f(x, y)=x^{2} y ; D=(x, y):-2 \leq x \leq 1,0 \leq y \leq 1, y

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

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

Find a unit vector that is normal to the level curve of the function $$ f(x, y)=x^{2}+\frac{y^{2}}{9} $$ at the point \((1,3)\).

Show that the equilibrium \(\begin{array}{ll}0 & \text { of } \\ 0\end{array}\) $$ \left[\begin{array}{l} x_{1}(t+1) \\ x_{2}(t+1) \end{array}\right]=\left[\begin{array}{rr} 0.2 & 0.3 \\ -0.5 & -0.4 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ is stable.

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

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 \(f(0.1,0.05)\)

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.

Find a unit vector that is normal to the level curve of the function $$ f(x, y)=x^{2}-y^{3} $$ at the point \((1,3)\).

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

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.

Show that $$f(x, y)=\left\\{\begin{array}{cc} \frac{3 r^{2} y}{\left(2 x^{4}+y^{2}\right)} & \text { for }(x, y) \neq(0,0) \\\ 0 & \text { for }(x, y)=(0,0)\end{array}\right.$$ is discontinuous at \((0,0) .\)

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 \(f(-0.1,0.2)\)

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

Find a unit vector that is normal to the level curve of the function $$ f(x, y)=x y $$ at the point \((2,3)\).

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

(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?

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

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

Chemotaxis Chemotaxis is the chemically directed movement of organisms up a concentration gradient \(-\) that is, in the direction in which the concentration increases most rapidly. The slime mold Dictyostelium discoideum exhibits this phenomenon. Single-celled amoebas of this species move up the concentration gradient of a chemical called cyclic adenosine monophosphate (AMP). Suppose the concentration of cyclic AMP at the point \((x, y)\) in the \(x-y\) plane is given by $$ f(x, y)=\frac{4}{\sqrt{x^{2}+y^{2}+1}} $$ If you place an amoeba at the point \((3,1)\) in the \(x-y\) plane, determine in which direction the amoeba will move if its movement is directed by chemotaxis.

Show that the equilibrium \(\left[\begin{array}{l}0 \\ 0\end{array}\right]\) of $$ \begin{array}{l} x_{1}(t+1)=\frac{x_{2}(t)}{4\left(1+\left(x_{1}(t)\right)^{2}\right)} \\ x_{2}(t+1)=\frac{3 x_{1}(t)}{1+\left(x_{2}(t)\right)^{2}} \end{array} $$ is locally stable.

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.

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

Find the linear approximation of $$ f(x, y)=\left(x^{2}+y^{2}\right) e^{-\left(x^{2}+y^{2}\right)} $$ at \((0,0)\), and use it to approximate \(f(0.01,0.05) .\) Using a calculator, compare the approximation with the exact value of \(f(0.01,0.05)\)

Choose three numbers \(x, y\), and \(z\) so that their sum is equal to 75 and their product is maximal.