Chapter 13

Find the first partial derivatives of \(f\) at the given point. $$ f(x, y)=\left\\{\begin{array}{ll} \frac{x^{2} y^{3}}{x^{2}+4 y^{3}} & \text { for }(x, y) \neq(0,0) \\ 0 & \text { for }(x, y)=(0,0) \end{array} ;(0,0)\right. $$

Sketch the graph of the equation. \(x=-3\)

Find the extreme values of \(f\) on \(R\). $$ \begin{aligned} &f(x, y)=y e^{-x} ; R \text { is the rectangular region with vertices }\\\ &(0,0),(\ln 2,0),(\ln 2,3),(0,3) \end{aligned} $$

Find \(d y / d x\) by implicit differentiation. $$ x^{2 / 3}+y^{2 / 3}=2 $$

Find a vector that is normal to the graph of the equation at the given point. Assume that each curve is smooth. $$ e^{x^{2} y}=2 ;(1, \ln 2) $$

Let \(f(x, y)=\int_{1}^{x} P(t) d t+\int_{1}^{y} Q(t) d t\), where \(P\) and \(Q\) are continuous. Find \(f_{x}\) and \(f_{y}\).

Sketch the graph of the equation. \(z=y^{2}\)

Find the extreme values of \(f\) on \(R\). $$ \begin{aligned} &f(x, y)=2 \sin x+3 \cos y ; R \text { is the square region with }\\\ &\text { vertices }(0,-\pi / 2),(\pi,-\pi / 2),(\pi, \pi / 2),(0, \pi / 2) \text { . } \end{aligned} $$

Find \(d y / d x\) by implicit differentiation. $$ x^{2}+y^{2}+\sin x y^{2}=0 $$

Let \(f(x, y)=\int_{\pi}^{x^{2}+y^{2}} \sin t^{2} d t .\) Find \(f_{x}\) and \(f_{y} .\)

Sketch the graph of the equation. \(z=x^{3}+1\)

A rectangular box, open at the top, is to have a volume of 1728 cubic inches. Find the dimensions that will minimize the cost of the box if a. the material for the bottom costs 16 times as much per unit area as the material for the sides. b. the material for the bottom costs twice as much per unit area as the material for the sides.

Find the extreme values of \(f\) on \(R\). $$ \begin{aligned} &f(x, y)=e^{x^{2}-y^{2}} ; R \text { is the ring bounded by the circles }\\\ &x^{2}+y^{2}=\frac{1}{2} \text { and } x^{2}+y^{2}=2 \text { . } \end{aligned} $$

Find \(d y / d x\) by implicit differentiation. $$ e^{x / y}+\ln y / x+15=0 $$

Explain why \(f\) is continuous. $$ f(x, y)=\left\\{\begin{array}{ll} \frac{x^{2} y^{2}}{x^{2}+y^{2}} & \text { for }(x, y) \neq(0,0) \\ 0 & \text { for }(x, y)=(0,0) \end{array}\right. $$

Find the three positive numbers whose sum is 48 and whose product is as large as possible. Calculate the product.

Find \(d y / d x\) by implicit differentiation. $$ x^{2}=\frac{y^{2}}{y^{2}-1} $$

Find a vector that is normal to the graph of \(f\) at the given point. $$ f(x, y)=1-x^{2} ;(0,2,1) $$

Find \(f_{x y}\) and \(f_{y x}\) $$ f(x, y)=3 x^{2}-\sqrt{2} x y^{2}+y^{5}-2 $$

Let $$ f(x, y)=\left\\{\begin{array}{ll} \frac{x y^{2}}{x^{4}+y^{4}} & \text { for }(x, y) \neq(0,0) \\ 0 & \text { for }(x, y)=(0,0) \end{array}\right. $$ a. Show that \(f\) is continuous in each variable separately at \((0,0)\), that is, \(f(x, 0)\) is a continuous function of \(x\) at 0, and \(f(0, y)\) is a continuous function of \(y\) at \(0 .\) b. Show that \(f\) is not continuous at \((0,0)\).

Sketch the graph of the equation. \(x=\sqrt{1-y^{2}}\)

Suppose that on your vacation you plan to spend \(x\) days in San Francisco, \(y\) days in your home town, and \(z\) days in New York. You calculate that your total enjoyment \(f(x, y, z)\) will be given by $$ f(x, y, z)=2 x+y+2 z $$ If plans and financial limitations dictate that $$ x^{2}+y^{2}+z^{2}=225 $$ how long should each stay be to maximize your enjoyment?

Find the three positive numbers whose product is 48 and whose sum is as small as possible. Calculate the sum.

Assume that the equation defines \(z\) implicitly as a function of \(x\) and \(y\), and use "implicit partial differentiation" to find \(\partial z / \partial x\) and \(\partial z / \partial y\). $$ x^{2} z^{2}-2 x y z+z^{3} y^{2}=3 $$

Find a vector that is normal to the graph of \(f\) at the given point. $$ f(x, y)=y^{2} e^{x} ;(0,-3,9) $$

Find \(f_{x y}\) and \(f_{y x}\) $$ f(x, y)=\frac{x^{2}-y^{2}}{x^{2}+y^{2}} $$

Let $$ f(x, y)=\left\\{\begin{array}{ll} \frac{\sin x y}{x^{2}+y^{2}} & \text { for }(x, y) \neq(0,0) \\ 0 & \text { for }(x, y)=(0,0) \end{array}\right. $$ Show that \(f\) is not continuous at \((0,0)\).

Sketch the graph of the equation. \(x=\sqrt{4-y^{2}-z^{2}}\)

The ground state energy \(E(x, y, z)\) of a particle of mass \(m\) in a rectangular box with dimensions \(x, y\), and \(z\) is given by $$ E(x, y, z)=\frac{h^{2}}{8 m}\left(\frac{1}{x^{2}}+\frac{1}{y^{2}}+\frac{1}{z^{2}}\right) $$ where \(h\) is a constant. Assuming that the volume \(V\) of the box is fixed, find the values of \(x, y\), and \(z\) that minimize the value of \(E\).

Show that the box in the shape of a rectangular parallelepiped whose volume is the largest of any inscribed in a given sphere is a cube.

Assume that the equation defines \(z\) implicitly as a function of \(x\) and \(y\), and use "implicit partial differentiation" to find \(\partial z / \partial x\) and \(\partial z / \partial y\). $$ x-y z+\cos x y z=2 $$

Find an equation of the plane tangent to the graph of the given function at the indicated point(s). $$ f(x, y)=x y-x+y+5 ;(0,2,7) $$

Find \(f_{x y}\) and \(f_{y x}\) $$ f(x, y)=\sqrt{x^{2}+y^{2}} $$

Describe the boundaries of the following regions. a. The disk with center \((-3,2)\) and radius 6 b. The rectangular region with vertices \((0,0),(2,0)\), \((0,-3)\), and \((2,-3)\) c. The triangular region with vertices \((-1,1),(1,1)\), and \((0,-5)\) d. The upper half of the \(x y\) plane, consisting of all \((x, y)\) such that \(y \geq 0\) e. The graph of the parabola \(y=4 x^{2}\) f. The entire plane except the origin

The object distance \(p\), image distance \(q\), and focal length fof a simple lens satisfy the equation $$ \frac{1}{p}+\frac{1}{q}=\frac{1}{f} $$ Determine the minimum distance \(p+q\) between the object and the image for a given focal length.

A rectangular box without top is to have a volume of 32 cubic meters. Find the dimensions of such a box having the smallest possible surface area.

Assume that the equation defines \(z\) implicitly as a function of \(x\) and \(y\), and use "implicit partial differentiation" to find \(\partial z / \partial x\) and \(\partial z / \partial y\). $$ \frac{1}{z}+\frac{1}{y+z}+\frac{1}{x+y+z}=\frac{1}{2} $$

Find an equation of the plane tangent to the graph of the given function at the indicated point(s). $$ f(x, y)=\frac{x+2}{y+1} ;(2,3,1) $$

Find \(f_{x y}\) and \(f_{y x}\) $$ f(x, y, z)=x^{4}-2 x^{2} y \sqrt{z}+3 y z^{4}+2 $$

Find the point in space the sum of whose coordinates is 48 and whose distance from the origin is minimum.

$$ \text { Let } z=f(x-y) . \text { Show that } \frac{\partial z}{\partial x}=-\frac{\partial z}{\partial y} \text { . } $$

Find an equation of the plane tangent to the graph of the given function at the indicated point(s). $$ g(x, y)=\sin \pi x y ;(-\sqrt{2}, \sqrt{2}, 0) \text { and }\left(-\frac{1}{2}, \frac{1}{3},-\frac{1}{2}\right) $$

Find \(f_{x y}\) and \(f_{y x}\) $$ f(x, y, z)=z \cos x y $$

Determine whether \(f\) is continuous on the given region \(R\). \(f(x, y)=\left\\{\begin{array}{ll}1 & \text { for } x^{2}+y^{2} \leq 9 \\ 0 & \text { for } x^{2}+y^{2}>9\end{array}\right.\) \(R\) is the disk \(x^{2}+y^{2} \leq 9\)

Sketch the level surface \(f(x, y, z)=c\). \(f(x, y, z)=2 x-4 y+z ; c=-1\)

Let \(w=f(x-y, y-z, z-x) .\) Show that $$ \frac{\partial w}{\partial x}+\frac{\partial w}{\partial y}+\frac{\partial w}{\partial z}=0 $$

Find an equation of the plane tangent to the graph of the given function at the indicated point(s). $$ f(x, y)=e^{x+y^{2}} ;\left(-1,0, e^{-1}\right) \text { and }(0,1, e) $$

Find \(f_{x x}, f_{y y}\), and \(f_{z z}\) (where applicable). $$ f(x, y)=\sqrt{16-9 x^{2}-4 y^{2}} $$

Sketch the level surface \(f(x, y, z)=c\). \(f(x, y, z)=x^{2}+y^{2}+z^{2} ; c=2\)

Let \(x\) represent capital and \(y\) labor in the manufacture of \(f(x, y)\) units of a given product. Assume that capital costs \(a\) dollars per unit and labor costs \(b\) dollars per unit and that there are \(c\) dollars available, so that \(a x+b y=c\). a. Using Lagrange multipliers, show that production is maximum at the point \(\left(x_{0}, y_{0}\right)\) such that $$ \frac{f_{x}\left(x_{0}, y_{0}\right)}{a}=\frac{f_{y}\left(x_{0} y_{0}\right)}{b}=\lambda $$ where \(\lambda\) is the Lagrange multiplier, called the equimarginal productivity of the production function \(f\). b. Let \(f\) be the Cobb-Douglas production function given by $$ f(x, y)=x^{\alpha} y^{\beta} \quad \text { for } x>0 \text { and } y>0 $$ where \(\alpha\) and \(\beta\) are positive constants less than 1 . Using (a), show that at maximum production \(f\left(x_{0}, y_{0}\right)\) we have $$ \frac{y_{0}}{x_{0}}=\frac{\beta a}{\alpha b} $$ which is independent of the money available.