Chapter 10

Find \(\partial f / \partial x\) and \(\partial f / \partial y\) for the given functions. \(f(x, y)=\ln \left(3 x^{2}-x y\right)\)

Use the properties of limits to calculate the following limits: \(\lim _{(x, y) \rightarrow(1,-2)} \frac{2 x^{2}+y}{2 x y+3}\)

Evaluate each function at the given point. \(g(n, p)=n p(1-p)^{n-1}\) at \((5,0.1)\)

Show that \(f(x, y)\) is differentiable at the indicated point. \(f(x, y)=\sin (x-y) ;(1,0)\)

The functions are defined on the rectangular domain $$D=\\{(x, y):-1 \leq x \leq 1,-1 \leq y \leq 1\\}$$ Find the global maxima and minima of \(f\) on \(D .\) $$ f(x, y)=3-x+2 y $$

In Problems 9-14, compute the directional derivative of \(f(x, y)\) at the given point in the indicated direction. $$ f(x, y)=\frac{1}{\sqrt{x^{2}+y^{2}}} \text { at }(0,1) \text { in the direction }\left[\begin{array}{r} 4 \\ -1 \end{array}\right] $$

Find \(\frac{d y}{d x}\) if \(y=\arctan x\).

In the negative binomial model, the fraction of hosts escap ing parasitism is given by $$ f(P)=\left(1+\frac{a P}{k}\right)^{-k} $$ (a) Graph \(f(P)\) as a function of \(P\) for \(a=0.1\) and \(a=0.01\) when \(k=0.75\) (b) For \(k=0.75\) and a given value of \(P\), how are the chances of escaping parasitism affected by increasing \(a\) ?

Find \(\partial f / \partial x\) and \(\partial f / \partial y\) for the given functions. \(f(x, y)=\ln \left(\frac{x^{2}+y}{y}\right)\)

Show that $$\lim _{(x, y) \rightarrow(0.0)} \frac{2 x^{2}-y^{2}}{x^{2}+y^{2}}$$ does not exist by computing the limit along the positive \(x\) -axis and the positive \(y\) -axis.

Evaluate each function at the given point. \(h\left(x_{1}, x_{2}\right)=x_{2} e^{-x_{1}}\) at \((2,-1)\)

Show that \(f(x, y)\) is differentiable at the indicated point. \(f(x, y)=x e^{-y} ;(0,1)\)

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

In Problems 15-18, compute the directional derivative of \(f(x, y)\) at the point \(P\) in the direction of the point \(Q .\) $$ f(x, y)=2 x^{2} y-3 x, P=(2,1), Q=(3,2) $$

15\. The growth rate \(r\) of a particular organism is affected by both the availability of food and the number of competitors for the food source. Denote the amount of food available at time \(t\) by \(F(t)\) and the number of competitors at time \(t\) by \(N(t)\). The growth rate \(r\) can then be thought of as a function of the two timedependent variables \(F(t)\) and \(N(t) .\) Assume that the growth rate is an increasing function of the availability of food and a decreasing function of the number of competitors. How is the growth rate \(r\) affected if the availability of food decreases over time while the number of competitors increases?

In the negative binomial model, the fraction of hosts escaping parasitism is given by $$ f(P)=\left(1+\frac{a P}{k}\right)^{-k} $$ (a) Graph \(f(P)\) as a function of \(P\) for \(k=0.75\) and \(k=0.5\) when \(a=0.02\) (b) For \(a=0.02\) and a given value of \(P\), how are the chances of escaping parasitism affected by increasing \(k ?\)

Find \(\partial f / \partial x\) and \(\partial f / \partial y\) for the given functions. \(f(x, y)=\ln \left(\frac{x y}{x^{2}+x y}\right)\)

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

Evaluate each function at the given point. \(g\left(x_{1}, x_{2}, x_{3}\right)=x_{1} \sqrt{x_{2} x_{3}}\) at \((1,2,1)\)

Show that \(f(x, y)\) is differentiable at the indicated point. \(f(x, y)=\left(x^{2}+y^{2}\right) e^{-x^{2}-y^{2}} ;(1,1)\)

The functions are defined on the rectangular domain $$D=\\{(x, y):-1 \leq x \leq 1,-1 \leq y \leq 1\\}$$ Find the global maxima and minima of \(f\) on \(D .\) $$ f(x, y)=x^{2}+y^{2} / 4 $$

In Problems 15-18, compute the directional derivative of \(f(x, y)\) at the point \(P\) in the direction of the point \(Q .\) $$ f(x, y)=4 x y+y^{2}, P=(-1,1), Q=(3,2) $$

Suppose that you travel along an environmental gradient, along which both temperature and precipitation increase. If the abundance of a particular plant species increases with both temperature and precipitation, would you expect to encounter this species more often or less often during your journey? (Use calculus to answer this question.)

The negative binomial model can be reduced to the Nicholson-Bailey model by letting the parameter \(k\) in the negative binomial model go to infinity. Show that $$ \lim _{k \rightarrow \infty}\left(1+\frac{a P}{k}\right)^{-k}=e^{-a P} $$

Find the indicated partial derivatives. \(f(x, y)=3 x^{2}-y+2 y^{2} ; f_{x}(1,0)\)

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

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

In Problems 15-18, compute the directional derivative of \(f(x, y)\) at the point \(P\) in the direction of the point \(Q .\) $$ f(x, y)=\sqrt{x y-2 x^{2}}, P=(1,6), Q=(3,1) $$

17\. Air Density The density of air changes with height. Under some conditions density \(\rho\), depends on height \(z\), and temperature \(T\) according to the equation: $$ \rho(z, T)=\rho_{0} e^{-\lambda z / T} $$ where \(\rho_{0}\) and \(\lambda\) are both constants. A meteorological balloon ascends (i.e., starts at \(z=1\) and gains height) over the course of several hours. (a) Assuming that the balloon ascends at a speed \(v\) (i.e., \(d z / d t=\) \(v\) ) and that the temperature changes over time (i.e., that \(T\) is given by a function \(T(t))\), derive, using the chain rule, an expression for the rate of change of air density, as measured by the weather balloon. (b) Assume that \(v=1, \rho_{0}=1\), and \(\lambda=1\) and that when \(t=0, T=1\). Are there any conditions under which the density, as measured by the balloon will not change in time? That is, find a differential equation that \(T\) must satisfy, if \(d \rho / d t=0\), and solve this differential equation.

Show that \(\left.\right|_{0} ^{0}\) is an equilibrium of $$ \left[\begin{array}{l} x_{1}(t+1) \\ x_{2}(t+1) \end{array}\right]=\left[\begin{array}{ll} -0.6 & 0 \\ -0.3 & 0.3 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ and determine its stability.

Find the indicated partial derivatives. \(f(x, y)=x^{1 / 3} y-x y^{-1 / 3} ; f_{y}(1,1)\)

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

Find the range of each function \(f(x, y)\), when defined on the specified domain \(D\). \(f(x, y)=\sqrt{9-x^{2}-y^{2}} ; D=\left\\{(x, y): x^{2}+y^{2} \leq 9\right\\}\)

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

In Problems 15-18, compute the directional derivative of \(f(x, y)\) at the point \(P\) in the direction of the point \(Q .\) $$ f(x, y)=e^{x-y}, P=(2,2), Q=(1,-1) $$

Show that \(\underset{0}^{0}\) is an equilibrium of $$ \left[\begin{array}{l} x_{1}(t+1) \\ x_{2}(t+1) \end{array}\right]=\left[\begin{array}{cr} -0.4 & 0.3 \\ 0 & -0.8 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ and determine its stability.

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

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

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.

In what direction does \(f(x, y)=3 x y-x^{2}\) increase most rapidly at \((-1,1) ?\)

19\. Maturity Time for Fish In Chapter 8 we met the von Bertalanffy equation as a model for the growth of a fish. The length \(L(t)\) of the fish is modeled by a function of age \(t\) by the function: $$ L(t)=L_{\infty}+\left(L_{0}-L_{\infty}\right) e^{-k t} $$ where \(L_{0}, L_{\infty}\), and \(k\) are all positive coefficients. We define the maturity age of the fish \(m\) to be the age \(x\), at which the fish reaches \(90 \%\) of its maximum length \(L_{\infty}\); that is $$ 0.9 L_{\infty}=L_{\infty}+\left(L_{0}-L_{\infty}\right) e^{-k m} $$ Show that, if \(L_{\infty}\) and \(L_{0}\) are constants, then the maturity age decreases if \(k\) increases. [Hint: show that \(d m / d k<0 .]\)

Show that \(\left.\right|_{0} ^{0} \mid\) is an equilibrium of $$ \left[\begin{array}{l} x_{1}(t+1) \\ x_{2}(t+1) \end{array}\right]=\left[\begin{array}{ll} -1.6 & 0 \\ -0.5 & 0.1 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ and determine its stability.

Compute $$\lim _{(x, y) \rightarrow(0.0)} \frac{3 x^{2} y}{\left(2 x^{4}+y^{2}\right)}$$ 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 range of each function \(f(x, y)\), when defined on the specified domain \(D\). \(f(x, y)=\exp \left[-\left(x^{2}+y^{2}\right)\right] ; D=\\{(x, y):-1 \leq x \leq 1,0 \leq\) \(y \leq 2\\}\)

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

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.