Chapter 13

If \(z=e^{-a y} \cos a x\), show that $$ \frac{\partial^{2} z}{\partial x^{2}}=a \frac{\partial z}{\partial y} $$

Sketch the quadric surface. \(x=z^{2}+3\)

Show that the surfaces \(z=\sqrt{x^{2}+y^{2}}\) and \(10 z=\) \(25+x^{2}+y^{2}\) have the same tangent plane at \((3,4,5)\).

Let \(z=x^{c} e^{-y / x}\), where \(c\) is a constant. Find the value of \(c\) such that $$ \frac{\partial z}{\partial x}=y \frac{\partial^{2} z}{\partial y^{2}}+\frac{\partial z}{\partial y} $$

Sketch the quadric surface. \(z=y^{2}-4 x^{2}\)

Show that the surfaces \(z=x y-2\) and \(x^{2}+y^{2}+z^{2}=3\) have the same tangent plane at \((1,1,-1)\).

Show that the functions \(u\) and \(v\) satisfy the Cauchy-Riemann equations \(u_{x}=v_{y}\) and \(u_{y}=-v_{x}\). $$ u=x^{2}-y^{2}, v=2 x y $$

Sketch the quadric surface. \(x=4 z^{2}-y^{2}\)

We say that two surfaces are normal at a given point if their tangent planes at that point are perpendicular to one another.Show that the pair of surfaces are normal at the given point. \(x^{2}+y^{2}+z^{2}=16\) and \(z^{2}=x^{2}+y^{2} ;(2,2,2 \sqrt{2})\)

Show that the functions \(u\) and \(v\) satisfy the Cauchy-Riemann equations \(u_{x}=v_{y}\) and \(u_{y}=-v_{x}\). $$ u=x^{3}-3 x y^{2}, v=3 x^{2} y-y^{3} $$

Sketch the quadric surface. \(y^{2}-x^{2}=4\)

We say that two surfaces are normal at a given point if their tangent planes at that point are perpendicular to one another.Show that the pair of surfaces are normal at the given point. \(z=x^{2}+4 y^{2}-12\) and \(8 z=4 x+y^{2}+19 ;(-3,-1,1)\)

Show that the functions \(u\) and \(v\) satisfy the Cauchy-Riemann equations \(u_{x}=v_{y}\) and \(u_{y}=-v_{x}\). $$ u=e^{x} \cos y, v=e^{x} \sin y $$

Sketch the quadric surface. \(z^{2}-y^{2}=9\)

Show that the line determined by the intersection of the plane \(z=0\) and the plane tangent to the surface \(z^{2}\left(x^{2}+y^{2}\right)=4\) at a point of the form \((2 \cos \theta, 2 \sin \theta, 1)\) is tangent to the circle \(x^{2}+y^{2}=16\) at the point \((4 \cos \theta, 4 \sin \theta)\).

A function \(z\) satisfies Laplace's equation if $$ \frac{\partial^{2} z}{\partial x^{2}}+\frac{\partial^{2} z}{\partial y^{2}}=0 $$ In Exercises \(52-53\) show that the function satisfies Laplace's equation. $$ z=x^{4}-6 x^{2} y^{2}+y^{4} $$

Sketch the quadric surface. \(z^{2}+4 y^{2}-2 x^{2}=1\)

Show that an equation of the plane tangent to the ellipsoid $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 $$ at the point \(\left(x_{0}, y_{0}, z_{0}\right)\) is $$ \frac{x x_{0}}{a^{2}}+\frac{y y_{0}}{b^{2}}+\frac{z z_{0}}{c^{2}}=1 $$

Sketch the quadric surface. \(4 x^{2}+y^{2}-z^{2}=16\)

Let \(c \neq 0\). Show that an equation of the plane tangent to the paraboloid $$ c z=\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} $$ at the point \(\left(x_{0}, y_{0}, z_{0}\right)\) is $$ c\left(z+z_{0}\right)=\frac{2 x x_{0}}{a^{2}}+\frac{2 y y_{0}}{b^{2}} $$

Sketch the quadric surface. \(z^{2}-4 y^{2}-x^{2}=1\)

Let \(g\) be a differentiable function of one variable and let \(f(x, y)=x g(y / x)\). Show that every plane tangent to the graph of \(f\) passes through the origin.

Sketch the quadric surface. \(x^{2}-9 y^{2}-4 z^{2}=36\)

Show that every line normal to the sphere \(x^{2}+y^{2}+z^{2}=\) 1 passes through the origin.

Let \(f\) be a function of two variables with partials of all orders. Then \(f\) has four second partials, \(f_{x x}, f_{x y}, f_{y x}\), and \(f_{y y} .\) If the second partials are continuous, only three of them can be distinct. a. How many third-order partials does \(f\) have? If these are continuous, what is the maximum number that can be distinct? b. Generalize (a) to \(n\) th-order partials. (Hint: Observe that if all \(n\) th- order partials are continuous, then any \(n\) th-order partial is equal to another \(n\) th-order partial in which all the differentiations with respect to \(x\) are performed first and the differentiations with respect to \(y\) are done last.)

In each of the following, determine a function \(f\) of two variables (different from \(F\) ) and a function \(g\) of one variable such that \(F=g \circ f\). a. \(F(x, y)=\sqrt{4-x^{2}-y^{2}}\) b. \(F(x, y)=e^{x \sqrt{y}}\)

Let \(M\) have continuous partials on a rectangle bounded by \(x=a, x=b, y=c\), and \(y=d\). Show that \(\int_{a}^{b} \frac{\partial M}{\partial x}(x, y) d x=M(b, y)-M(a, y) \quad\) for \(c \leq y \leq d\) and \(\int_{c}^{d} \frac{\partial M}{\partial y}(x, y) d y=M(x, d)-M(x, c) \quad\) for \(a \leq x \leq b\)

Show that there is exactly one plane tangent to the paraboloid \(z=x^{2}+y^{2}\) and parallel to any given nonvertical plane.

Use implicit differentiation to find \(\partial z / \partial x\) and \(\partial z / \partial y\) at the given point. Then find an equation of the plane tangent to the level surface at that point. $$ x^{2}-y^{2}-z^{2}=1 ;(\sqrt{2}, 0,1) $$

If \(c\) is a constant, then an equation of the form $$ \frac{\partial u}{\partial t}=c \frac{\partial^{2} u}{\partial x^{2}} $$ is called a diffusion equation. a. Show that if \(u=e^{a x+b t}\), where \(a\) and \(b\) are constant, then \(u\) satisfies the diffusion equation with \(c=b / a^{2}\). "b. Show that if $$ u=u(x, t)=\frac{1}{\sqrt{t}} e^{-x^{2} / a t} $$ where \(a\) is a constant, then \(u\) satisfies a diffusion equation. [The number \(u(x, t)\) might represent the concentration of a drug at a point \(x\) in a muscle at time \(t\). For each value of \(t\) the graph of \(u\) (considered as a function of \(x\) ) is a bell-shaped curve. As \(t\) increases, the curve becomes flatter (Figure 13.32).]

Use implicit differentiation to find \(\partial z / \partial x\) and \(\partial z / \partial y\) at the given point. Then find an equation of the plane tangent to the level surface at that point. $$ x y z=1 ;\left(2,-3,-\frac{1}{6}\right) $$

Use implicit differentiation to find \(\partial z / \partial x\) and \(\partial z / \partial y\) at the given point. Then find an equation of the plane tangent to the level surface at that point. $$ \ln x+\ln y+\ln z=1 ;(1,1, e) $$

When two resistors having resistances \(R_{1}\) and \(R_{2}\) are connected in parallel, the combined resistance \(R\) is given by \(R=R_{1} R_{2} /\left(R_{1}+R_{2}\right) .\) Show that $$ \frac{\partial^{2} R}{\partial R_{1}^{2}} \frac{\partial^{2} R}{\partial R_{2}^{2}}=\frac{4 R^{2}}{\left(R_{1}+R_{2}\right)^{4}} $$

Express the height \(h\) of a right circular cylinder as a function of the volume \(V\) and radius \(r\).

A mountain climber's oxygen mask is leaking. If the surface of the mountain is represented by \(z=5-x^{2}-2 y^{2}\) and the climber is at \(\left(\frac{1}{2},-\frac{1}{2}, \frac{17}{4}\right)\), in what direction should the climber turn to descend most rapidly?

The kinetic energy \(K\) of a body with mass \(m\) and velocity \(v\) is given by \(K=\frac{1}{2} m v^{2}\). Show that $$ \frac{\partial K}{\partial m} \frac{\partial^{2} K}{\partial v^{2}}=K $$

Express the radius \(r\) of the base of a right circular cone as a function of the volume \(V\) and height \(h\).

Suppose \(T(x, y, z)=x^{3} y+3 x^{2} y^{2} z\). Find the directional derivative of \(T\) at \((1,1,-1)\) in the direction of the gradient.

The ideal gas law states that if \(n\) moles of a gas has volume \(V\) and temperature \(T\) and is under pressure \(p\), then \(p V=n k T\), where \(k\) is the universal gas constant. Show that $$ \frac{\partial V}{\partial T} \frac{\partial T}{\partial p} \frac{\partial p}{\partial V}=-1 $$

Express the surface area \(S\) of a rectangular box with no top as a function of the dimensions \(x, y\), and \(z\).

A metal plate with vertices \((1,1),(5,1),(1,3)\), and \((5,3)\) is heated by a flame at the origin, and the temperature at a point on the plate is inversely proportional to the distance from the origin. If an ant is located at the point \((3,2)\), in what direction should the ant crawl to cool the fastest?

Suppose the quadric surface \(x^{2}-y^{2}=z\) is an equipotential surface. Show that the electric force on a positive unit charge at the origin is perpendicular to the \(x y\) plane.

Express the cost \(C\) of painting a rectangular wall as a function of the dimensions \(x\) and \(y\) (in meters) if the cost per square meter is \(\$ 0.30\).

Express the cost \(C\) of painting a rectangular wall as a function of the dimensions \(x\) and \(y\) (in meters) if the cost per square meter is \(\$ 0.30\) and the wall contains a window 1 square meter in area.

If \(f(x, y)\) is the amount of a commodity produced from \(x\) units of capital and \(y\) units of labor, then \(f\) is called a production function. If $$ f(x, y)=x^{\alpha} y^{\beta} \quad \text { for } x>0 \text { and } y>0 $$ where \(\alpha\) and \(\beta\) are positive constants less than 1, then \(f\) is called a Cobb-Douglas production function. a. Show that \(f(t x, t y)=t^{\alpha+\beta} f(x, y)\). b. If \(z=f(x, y)\), show that $$ \frac{1}{z} \frac{\partial z}{\partial x}=\frac{\alpha}{x} \quad \text { and } \quad \frac{1}{z} \frac{\partial z}{\partial y}=\frac{\beta}{y} $$ and that $$ x \frac{\partial z}{\partial x}+y \frac{\partial z}{\partial y}=(\alpha+\beta) z $$

The strength of the electric field at \((x, y, z)\) due to an infinitely long charged wire lying along the \(z\) axis is given by $$ E(x, y, z)=\frac{c}{\sqrt{x^{2}+y^{2}}} $$ where \(c\) is a positive constant. Describe the level surfaces of \(E\)

The magnitude of the gravitational force exerted on a unit mass at \((x, y, z)\) by a point mass located at the origin is given by $$ F(x, y, z)=\frac{c}{x^{2}+y^{2}+z^{2}} $$ where \(c\) is a positive constant. Describe the level surfaces of \(F\)

It seems reasonable that an increase in taxation on a commodity would decrease the production of that commodity. The following argument supports that claim. Assume that all required derivatives exist. For any \(x \geq 0\), let \(P_{0}(x)\) be the profit before taxes on \(x\) units produced. Let \(P(x, t)\) denote the profit after taxes on \(x\) units produced with \(\operatorname{tax} t\) on each unit. Assume that at any tax rate \(t\) the company will maximize its profits by producing \(f(t)\) units so that $$ \begin{aligned} &\frac{\partial P}{\partial x}(f(t), t)=0 \\ &\frac{\partial^{2} P}{\partial x^{2}}(f(t), t)<0 \end{aligned} $$ (The conditions in (9) and (10) are just those required for the Second Derivative Test.) a. Show that \(P(x, t)=P_{0}(x)-t x\). b. Using (a), show that $$ \frac{\partial P}{\partial x}(x, t)=P_{0}^{\prime}(x)-t \quad \text { and } \quad \frac{\partial^{2} P}{\partial x^{2}}(x, t)=P_{0}^{\prime \prime}(x) $$ c. From \((9)\) and \((\mathrm{b})\), show that \(P_{0}^{\prime}(f(t))-t=0\). d. By differentiating both sides of the equation in (c) and by using (b) and (10), show that $$ f^{\prime}(t)=\frac{1}{P_{0}^{\prime \prime}(f(t))}=\frac{1}{\frac{\partial^{2} P}{\partial x^{2}}(f(t), t)}<0 $$ (Thus the production tends to decrease as the tax rate increases.)

Suppose a thin metal plate occupies the first quadrant of the \(x y\) plane and the temperature at \((x, y)\) is given by $$ T(x, y)=x y $$ Describe the isothermal curves, that is, the level curves of \(T\).

Let \(f(x, y)=(x+1)(y+2)\) for \(x \geq 0\) and \(y \geq 0 .\) Sketch the level curves \(f(x, y)=3\) and \(f(x, y)=4\). (If \(f\) represents a utility function for two competing goods such as beer and wine, then the level curves are called indifference curves.)