Chapter 13

Let \(z=f(y+a x)+g(y-a x)\), with \(a \neq 0\). Show that \(z\) satisfies the wave equation $$ \frac{\partial^{2} z}{\partial x^{2}}=a^{2} \frac{\partial^{2} z}{\partial y^{2}} $$

Find an equation of the plane tangent to the graph of the given function at the indicated point(s). $$ f(x, y)=(2+x-y)^{2} ;(3,-1,36) $$

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

Let \(x, y\), and \(z\) be positive numbers. Show that the geometric mean \((x y z)^{1 / 3}\) of \(x, y\), and \(z\) is less than or equal to their arithmetic mean \((x+y+z) / 3\). (Hint: Maximize \((x y z)^{1 / 3}\) subject to the constraint \(x+y+z=c\), where \(c\) is a fixed number.)

Let \(z=f(x, y), x=r \cos \theta\), and \(y=r \sin \theta\). a. Show that \(\frac{\partial z}{\partial x}=\frac{\partial z}{\partial r} \cos \theta-\frac{\partial z}{\partial \theta} \frac{\sin \theta}{r}\) and \(\frac{\partial z}{\partial y}=\frac{\partial z}{\partial r} \sin \theta+\frac{\partial z}{\partial \theta} \frac{\cos \theta}{r}\).

Find an equation of the plane tangent to the graph of the given function at the indicated point(s). $$ g(x, y)=4 x^{2}+y^{2}-1 ;(2,1,16) $$

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

Sketch the level surface \(f(x, y, z)=c\). \(f(x, y, z)=x^{2}+y^{2}-z^{2} ; c=0\)

Let \(w=f(x, y), x=e^{s} \cos t\), and \(y=e^{s} \sin t .\) Ass that the second partials of \(f\) exist, show that $$ \frac{\partial^{2} w}{\partial x^{2}}+\frac{\partial^{2} w}{\partial y^{2}}=e^{-2 s}\left(\frac{\partial^{2} w}{\partial s^{2}}+\frac{\partial^{2} w}{\partial t^{2}}\right) $$

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

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

Determine whether \(f\) is continuous on the given region \(R\). \(f(x, y)=\left\\{\begin{array}{ll}e^{-\left(1+x^{2}\right) / y} & \text { for } y \neq 0 \\ 0 & \text { for } y=0\end{array}\right.\) \(R\) is the upper half plane \(y \geq 0\).

Sketch the level surface \(f(x, y, z)=c\). \(f(x, y, z)=z-1-x^{2}-y^{2} ; c=2\)

A function \(f\) of two variables is homogeneous of degree \(\boldsymbol{n}\) if for any real number \(t\) we have $$ f(t x, t y)=t^{n} f(x, y) $$ Show that in this case $$ x f_{x}(x, y)+y f_{y}(x, y)=n f(x, y) $$ (Hint: Differentiate both sides of \((8)\) with respect to \(t\), and then set \(t=1 .\) )

Let \(f(x, y)=A x^{2}+2 B x y+C y^{2}\), as in (1). Assume that \(A \neq 0\) and \(A C-B^{2}<0\). Verify that there are points \((x, y)\) as close to \((0,0)\) as one wishes such that \(f(x, y)>0\), and other points \((x, y)\) as close to \((0,0)\) as one wishes such that \(f(x, y)<0 .\) Conclude that \(f\) has a saddle point at \((0,0) .\) (Hint: Consider points of the form \((x, 0)\) and \((-B y / A, y) .)\)

Find an equation of the plane tangent to the graph of the given function at the indicated point(s). $$ f(x, y)=\left\\{\begin{array}{ll} \frac{x^{3}-y^{3}}{x^{2}+y^{2}} & \text { for }(x, y) \neq(0,0) \\ 0 & \text { for }(x, y)=(0,0) \end{array} ; \quad(1,0,1)\right. $$

Find \(f_{x x}, f_{y y}\), and \(f_{z z}\) (where applicable). $$ f(x, y, z)=e^{x^{2}} \sin y z+\ln \left(x^{2}+y^{2}+z^{2}\right) $$

a. Give a definition of the boundary of a set \(R\) in space. b. Give a definition of the limit of a function at a boundary point \(P\) of a given set \(R\) in space. c. Give a definition of continuity of a function on a set \(R\) in space.

Sketch the quadric surface. \(\frac{x^{2}}{4}+y^{2}+\frac{z^{2}}{9}=1\)

Find an equation of the plane tangent to the given surface at the given point. $$ x^{2}+y^{2}+z^{2}=1 ;\left(\frac{1}{2},-\frac{1}{2},-1 / \sqrt{2}\right) $$

Find symmetric equations for the line that lies in the plane \(y=1\) and is tangent to the intersection of the plane and the paraboloid \(z=x^{2}+16 y^{2}\) at \((-3,1,25)\).

When \(x\) moles of sulfuric acid are mixed with \(y\) moles of water, the heat \(Q(x, y)\) produced is given by $$ Q(x, y)=\frac{17,860 x y}{(1.798) x+y} \quad \text { for } x>0 \text { and } y>0 $$ Determine whether \(Q\) has a limit at \((0,0)\), and if so, compute its value.

Sketch the quadric surface. \(x^{2}+2 y^{2}+3 z^{2}=6\)

Show that if \(f\) is differentiable at \(\left(x_{0}, y_{0}\right)\), then \(f\) is continuous at \(\left(x_{0}, y_{0}\right) .\) (Hint: Using (7), show that \(\left.\lim _{(x, y) \rightarrow\left(x_{0}, y_{0}\right)} f(x, y)=f\left(x_{0}, y_{0}\right) .\right)\)

A textbook warehouse in the shape of a rectangular parallelepiped with volume 960,000 cubic feet is to be erected. Assume that because of decorations, the front wall will cost twice as much per square foot as the side and back walls and the floor, and the roof will cost \(\frac{3}{2}\) as much as the side walls. Find the dimensions of the warehouse that will minimize the cost.

Find an equation of the plane tangent to the given surface at the given point. $$ \frac{x^{2}}{4}+\frac{y^{2}}{9}+\frac{z^{2}}{16}=1 ;(0,0,-4) $$

Find symmetric equations for the line that lies in the plane \(x=2\) and is tangent to the intersection of the plane and the cone \(z=\sqrt{x^{2}+y^{2}}\) at \((2,2 \sqrt{3}, 4)\).

The function \(f\) defined by $$ f(x, y)=\frac{100 a x}{a x+b y} \quad \text { for } x>0 \text { and } y>0 $$ where \(a\) and \(b\) are positive constants, appears in the study of the relationship of blood flow through the right lung to the total blood flow in the system. Determine whether \(f\) has a limit at \((0,0)\).

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

Find an equation of the plane tangent to the given surface at the given point. $$ x y z=1 ;\left(\frac{1}{2},-2,-1\right) $$

Find \(\sqrt{f_{x}^{2}+f_{y}^{2}+1}\) $$ f(x, y)=1-x $$

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

A tree trunk may be considered a circular cylinder. Suppose the diameter of the trunk increases 1 inch per year and the height of the trunk increases 6 inches per year. How fast is the volume of wood in the trunk increasing when it is 100 inches high and 5 inches in diameter?

Find an equation of the plane tangent to the given surface at the given point. $$ y e^{x y}+z^{2}=0 ;(0,-1,1) $$

Find \(\sqrt{f_{x}^{2}+f_{y}^{2}+1}\) $$ f(x, y)=4-y^{2} $$

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

The time rate \(Q\) of flow of fluid through a cylindrical tube (such as a windpipe) with radius \(r\) and height \(l\) is given by $$ Q=\frac{\pi p r^{4}}{8 l \eta} $$ where \(\eta\) is the viscosity of the fluid and \(p\) is the difference in pressure at the two ends of the tube. Suppose the length of the tube remains constant, while the radius increases at the rate of \(\frac{1}{10}\) and the pressure decreases at the rate of \(\frac{1}{5}\). Find the rate of change of \(Q\) with respect to time.

Find an equation of the plane tangent to the given surface at the given point. $$ \sin (x y)=2-z^{2} ;\left(\pi, \frac{1}{2},-1\right) $$

Find \(\sqrt{f_{x}^{2}+f_{y}^{2}+1}\) $$ f(x, y)=\sqrt{x^{2}+y^{2}} $$

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

A road is perpendicular to a train track. Suppose a car approaches the intersection of the road and the track at 20 miles per hour, while a train approaches at 100 miles per hour. At what rate is the distance between the car and the train changing when the car is \(0.5\) miles from the intersection and the train is \(1.2\) miles from the intersection?

Find \(\sqrt{f_{x}^{2}+f_{y}^{2}+1}\) $$ f(x, y)=\sqrt{1-x^{2}-y^{2}} $$

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

The mass of a rocket lifting off from earth is decreasing (due to fuel consumption) at the rate of 40 kilograms per second. How fast is the magnitude \(F\) of the force of gravity decreasing when the rocket is 6400 kilometers from the center of the earth and is rising with a velocity of 100 kilometers per second? (Hint: By Newton's Law of Gravitation, \(F=G M m / r^{2}\), where \(G\) is the universal gravitational constant, \(M\) is the mass of the earth, \(m\) is the mass of the rocket, and \(r\) is the distance between the rocket and the center of the earth.)

Find an equation of the plane tangent to the given surface at the given point. $$ z=\ln \sqrt{x^{2}+1} ;(0,2,0) $$

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

Find the point on the hyperbolic paraboloid \(z=x^{2}-3 y^{2}\) at which the tangent plane is parallel to the plane \(8 x+\) \(3 y-z=4\).

Rectangular and polar coordinates in the plane are related by the equations \(x=r \cos \theta, y=r \sin \theta, r=\sqrt{x^{2}+y^{2}}\) \(\theta=\tan ^{-1} y / x\). Find the following partial derivatives. a. \(\frac{\partial x}{\partial r}\) b. \(\frac{\partial x}{\partial \theta}\) c. \(\frac{\partial y}{\partial r}\) d. \(\frac{\partial y}{\partial \theta}\) e. \(\frac{\partial r}{\partial x}\) f. \(\frac{\partial r}{\partial y}\) g. \(\frac{\partial \theta}{\partial x}\) h. \(\frac{\partial \theta}{\partial y}\)

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

Find the point on the paraboloid \(z=9-4 x^{2}-y^{2}\) at which the tangent plane is parallel to the plane \(z=4 y\).