Chapter 14

Use implicit differentiation to find \(\partial z / \partial x\) and \(\partial z / \partial y\) $$x-z=\arctan (y z)$$

If \(f(x, y)=x y,\) find the gradient vector \(\nabla f(3,2)\) and use it to find the tangent line to the level curve \(f(x, y)=6\) at the point \((3,2) .\) Sketch the level curve, the tangent line, and the gradient vector.

Sketch both a contour map and a graph of the function and compare them. $$f(x, y)=x^{2}+9 y^{2}$$

Find the dimensions of the rectangular box with largest volume if the total surface area is given as 64 \(\mathrm{cm}^{2}\) .

\(45-48\) Assume that all the given functions are differentiable. If \(z=f(x, y),\) where \(x=s+t\) and \(y=s-t,\) show that $$\left(\frac{\partial z}{\partial x}\right)^{2}-\left(\frac{\partial z}{\partial y}\right)^{2}=\frac{\partial z}{\partial s} \frac{\partial z}{\partial t}$$

Use implicit differentiation to find \(\partial z / \partial x\) and \(\partial z / \partial y\) $$\sin (x y z)=x+2 y+3 z$$

Sketch both a contour map and a graph of the function and compare them. $$f(x, y)=\sqrt{36-9 x^{2}-4 y^{2}}$$

If \(g(x, y)=x^{2}+y^{2}-4 x,\) find the gradient vector \(\nabla g(1,2)\) and use it to find the tangent line to the level curve \(g(x, y)=1\) at the point \((1,2) .\) Sketch the level curve, the tangent line, and the gradient vector.

Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its 12 edges is a constant \(c .\)

\(49-54\) Assume that all the given functions have continuous second-order partial derivatives. Show that any function of the form $$z=f(x+a t)+g(x-a t)$$ is a solution of the wave equation $$\frac{\partial^{2} z}{\partial t^{2}}=a^{2} \frac{\partial^{2} z}{\partial x^{2}}$$ \([\)Hint: Let \(u=x+a t, v=x-a t]\)

Find $$\partial z / \partial x \text { and } \partial z / \partial y$$ (a) \(z$$=f(x)+g(y)\) (b) \(z=f(x+y)\)

A thin metal plate, located in the \(x y\) -plane, has temperature \(T(x, y)\) at the point \((x, y) .\) The level curves of \(T\) are called isothermals because at all points on an isothermal the temperature is the same. Sketch some isothermals if the temperature function is given by $$T(x, y)=100 /\left(1+x^{2}+2 y^{2}\right)$$

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

\(49-54\) Assume that all the given functions have continuous second-order partial derivatives. If \(u=f(x, y),\) where \(x=e^{s} \cos t\) and \(y=e^{s} \sin t,\) show that $$\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=e^{-2 s}\left[\frac{\partial^{2} u}{\partial s^{2}}+\frac{\partial^{2} u}{\partial t^{2}}\right]$$

Find $$\partial z / \partial x \text { and } \partial z / \partial y$$ (a) \(z=f(x) g(y) \quad\) (b) \(z=f(x y)\) (c) \(z=f(x / y)\)

If \(V(x, y)\) is the electric potential at a point \((x, y)\) in the xy-plane, then the level curves of \(V\) are called equipotential curves because at all points on such a curve the electric potential is the same. Sketch some equipotential curves if \(V(x, y)=c / \sqrt{r^{2}-x^{2}-y^{2}},\) where \(c\) is a positive constant.

The base of an aquarium with given volume \(V\) is made of slate and the sides are made of glass. If slate costs five times as much (per unit area) as glass, find the dimensions of the aquar- ium that minimize the cost of the materials.

Find all the second partial derivatives. $$f(x, y)=x^{3} y^{5}+2 x^{4} y$$

Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph. $$f(x, y)=e^{-x^{2}}+e^{-2 y^{2}}$$

Show that the equation of the tangent plane to the elliptic paraboloid \(z / c=x^{2} / a^{2}+y^{2} / b^{2}\) at the point \(\left(x_{0}, y_{0}, z_{0}\right)\) can be written as $$\frac{2 x X_{0}}{a^{2}}+\frac{2 y y_{0}}{b^{2}}=\frac{z+z_{0}}{c}$$

A cardboard box without a lid is to have a volume of \(32,000 \mathrm{cm}^{3} .\) Find the dimensions that minimize the amount of cardboard used.

A rectangular building is being designed to minimize heat loss. The east and west walls lose heat at a rate of 10 units/m \(^{2}\) per day, the north and south walls at a rate of 8 units/m \(^{2}\) per day, the floor at a rate of 1 unit/m \(^{2}\) per day, and the roof at a rate of 5 units/m \(^{2}\) per day. Each wall must be at least 30 m long, the height must be at least \(4 \mathrm{m},\) and the volume must be exactly 4000 \(\mathrm{m}^{3} .\) (a) Find and sketch the domain of the heat loss as a function of the lengths of the sides. (b) Find the dimensions that minimize heat loss. (Check both the critical points and the points on the boundary of the domain. (c) Could you design a building with even less heat loss if the restrictions on the lengths of the walls were removed?

\(49-54\) Assume that all the given functions have continuous second-order partial derivatives. If \(z=f(x, y),\) where \(x=r \cos \theta\) and \(y=r \sin \theta,\) find (a) \(\partial z / \partial r,\) (b) \(\partial z / \partial \theta,\) and \((\mathrm{c}) \partial^{2} z / \partial r \partial \theta\)

Find all the second partial derivatives. $$f(x, y)=\sin ^{2}(m x+n y)$$

Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph. $$f(x, y)=\left(1-3 x^{2}+y^{2}\right) e^{1-x^{2}-y^{2}}$$

At what point on the paraboloid \(y=x^{2}+z^{2}\) is the tangent plane parallel to the plane \(x+2 y+3 z=1 ?\)

\(49-54\) Assume that all the given functions have continuous second-order partial derivatives. If \(z=f(x, y),\) where \(x=r \cos \theta\) and \(y=r \sin \theta,\) show that $$\frac{\partial^{2} z}{\partial x^{2}}+\frac{\partial^{2} z}{\partial y^{2}}=\frac{\partial^{2} z}{\partial r^{2}}+\frac{1}{r^{2}} \frac{\partial^{2} z}{\partial \theta^{2}}+\frac{1}{r} \frac{\partial z}{\partial r}$$

Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph. \(f(x, y)=x y^{2}-x^{3} \quad \)(monkey saddle)

Are there any points on the hyperboloid \(x^{2}-y^{2}-z^{2}=1\) where the tangent plane is parallel to the plane \(z=x+y ?\)

If the length of the diagonal of a rectangular box must be \(L,\) what is the largest possible volume?

Find all the second partial derivatives. $$w=\sqrt{u^{2}+v^{2}}$$

Three alleles (alternative versions of a gene) \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{O}\) determine the four blood types \(\mathrm{A}(\mathrm{AA}\) or \(\mathrm{AO}), \mathrm{B}(\mathrm{BB}\) or \(\mathrm{BO})\) \(\mathrm{O}(\mathrm{OO}),\) and \(\mathrm{AB} .\) The Hardy-Weinberg Law states that the pro- portion of individuals in a population who carry two different alleles is $$P=2 p q+2 p r+2 r q$$ where \(p, q,\) and \(r\) are the proportions of \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{O}\) in the population. Use the fact that \(p+q+r=1\) to show that \(P\) is at most \(\frac{2}{3} .\)

\(49-54\) Assume that all the given functions have continuous second-order partial derivatives. Suppose \(z=f(x, y),\) where \(x=g(s, t)\) and \(y=h(s, t) .\) (a) Show that $$\begin{aligned} \frac{\partial^{2} z}{\partial t^{2}}=\frac{\partial^{2} z}{\partial x^{2}} &\left(\frac{\partial x}{\partial t}\right)^{2}+2 \frac{\partial^{2} z}{\partial X \partial y} \frac{\partial x}{\partial t} \frac{\partial y}{\partial t}+\frac{\partial^{2} z}{\partial y^{2}}\left(\frac{\partial y}{\partial t}\right)^{2} \\ &+\frac{\partial z}{\partial x} \frac{\partial^{2} x}{\partial t^{2}}+\frac{\partial z}{\partial y} \frac{\partial^{2} y}{\partial t^{2}} \end{aligned}$$ (b) Find a similar formula for \(\partial^{2} z / \partial s \partial t\)

Find all the second partial derivatives. $$v=\frac{x y}{x-y}$$

Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph. \(f(x, y)=x y^{3}-y x^{3} \quad(\) dog saddle \()\)

Show that the ellipsoid \(3 x^{2}+2 y^{2}+z^{2}=9\) and the sphere \(x^{2}+y^{2}+z^{2}-8 x-6 y-8 z+24=0\) are tangent to each other at the point \((1,1,2)\) . (This means that they have a common tangent plane at the point.)

Suppose that a scientist has reason to believe that two quan- tities \(x\) and \(y\) are related linearly, that is, \(y=m x+b,\) at least approximately, for some values of \(m\) and \(b\) . The scientist performs an experiment and collects data in the form of points \(\left(X_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) and then plots these points. The points don't lie exactly on a straight line, so the scientist wants $$ m \text { and } b \text { so that the line } y=m x+b$$ points as well as possible. (See the figure.) Let \(d_{i}=y_{i}-\left(m x_{i}+b\right)\) be the vertical deviation of the point \(\left(x_{i}, y_{i}\right)\) from the line. The method of least squares determines \(m\) and \(b\) so as to minimize \(\Sigma_{1-1}^{n} d_{i}^{2}\) , the sum of the squares of these deviations. Show that, according to this method, the line of best fit is obtained when $$\begin{array}{c}{m \sum_{i=1}^{n} x_{i}+b n=\sum_{i=1}^{n} y_{i}} \\ {m \sum_{i=1}^{n} x_{i}^{2}+b \sum_{i=1}^{n} x_{i}=\sum_{i=1}^{n} x_{i} y_{i}}\end{array}$$ Thus the line is found by solving these two equations in the two unknowns \(m\) and \(b\) . See Section 1.2 for a further discus- sion and applications of the method of least squares.)

A function \(f\) is called homogeneous of degree \(n\) if it satisfies the equation \(f(t X, t y)=t^{n} f(x, y)\) for all \(t\) , where \(n\) is a positive integer and \(f\) has continuous second-order partial derivatives. (a) Verify that \(f(x, y)=x^{2} y+2 x y^{2}+5 y^{3}\) is homogeneous of degree \(3 .\) (b) Show that if \(f\) is homogeneous of degree \(n,\) then $$x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)$$ I Hint: Use the Chain Rule to differentiate \(f(t x, t y)\) with respect to \(t ]\)

Show that every plane that is tangent to the cone \(x^{2}+y^{2}=z^{2}\) passes through the origin.

Find an equation of the plane that passes through the point \((1,2,3)\) and cuts off the smallest volume in the first octant.

If \(f\) is homogeneous of degree \(n,\) show that $$x^{2} \frac{\partial^{2} f}{\partial x^{2}}+2 x y \frac{\partial^{2} f}{\partial x \partial y}+y^{2} \frac{\partial^{2} f}{\partial y^{2}}=n(n-1) f(x, y)$$

Find all the second partial derivatives. $$v=e^{x e^{y}}$$

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

Verify that the conclusion of Clairaut's Theorem holds, that is, \(u_{x y}=u_{y x}\) $$u=x \sin (x+2 y)$$

Show that the sum of the \(x^{-}, y_{-}\) and \(z\) -intercepts of any tangent plane to the surface \(\sqrt{x}+\sqrt{y}+\sqrt{z}=\sqrt{c}\) is a constant.

Suppose that the equation \(F(x, y, z)=0\) implicitly defines each of the three variables \(x, y,\) and \(z\) as functions of the other two: \(z=f(x, y), y=g(x, z), x=h(y, z) .\) If \(F\) is differentiable and \(F_{x}, F_{y},\) and \(F_{z}\) are all nonzero, show that $$\frac{\partial z}{\partial x} \frac{\partial x}{\partial y} \frac{\partial y}{\partial z}=-1$$

Verify that the conclusion of Clairaut's Theorem holds, that is, \(u_{x y}=u_{y x}\) $$u=x^{4} y^{2}-2 x y^{5}$$

Show that the pyramids cut off from the first octant by any tangent planes to the surface \(x y z=1\) at points in the first octant must all have the same volume.

Verify that the conclusion of Clairaut's Theorem holds, that is, \(u_{x y}=u_{y x}\) $$u=\ln \sqrt{x^{2}+y^{2}}$$

Find parametric equations for the tangent line to the curve of intersection of the paraboloid \(z=x^{2}+y^{2}\) and the ellipsoid \(4 x^{2}+y^{2}+z^{2}=9\) at the point \((-1,1,2)\) .