Chapter 14

The voltage \(V\) in a simple electrical circuit is slowly decreasing as the battery wears out. The resistance \(R\) is slowly increasing as the resistor heats up. Use Ohm's Law, \(V=I R,\) to find how the current \(I\) is changing at the moment when \(R=400 \Omega\) \(I=0.08 \mathrm{A}, d V / d t=-0.01 \mathrm{V} / \mathrm{s},\) and \(d R / d t=0.03 \Omega / \mathrm{s}\)

\(39-41\) Use polar coordinates to find the limit. [If \((r, \theta)\) are polar coordinates of the point \((x, y)\) with \(r \geqslant 0,\) note that \(r \rightarrow 0^{+}\) as \((x, y) \rightarrow(0,0) . ]\) $$\lim _{(x, y) \rightarrow(0,0)}\left(x^{2}+y^{2}\right) \ln \left(x^{2}+y^{2}\right)$$

Find the indicated partial derivatives. $$f(x, y)=\arctan (y / x) ; \quad f_{x}(2,3)$$

\(39-44\) Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. $$y=x^{2}-z^{2}, \quad(4,7,3)$$

Draw a contour map of the function showing several level curves. $$f(x, y)=x^{3}-y$$

Four positive numbers, each less than 50 , are rounded to the first decimal place and then multiplied together. Use differen- tials to estimate the maximum possible error in the computed product that might result from the rounding.

Find the points on the cone \(z^{2}=x^{2}+y^{2}\) that are closest to the point \((4,2,0) .\)

The plane \(x+y+2 z=2\) intersects the paraboloid \(z=x^{2}+y^{2}\) in an ellipse. Find the points on this ellipse that are nearest to and farthest from the origin.

\(39-41\) Use polar coordinates to find the limit. If \((r, \theta)\) are polar coordinates of the point \((x, y)\) with \(r \geqslant 0\) , note that \(r \rightarrow 0^{+}\) as \((x, y) \rightarrow(0,0) . ]\) $$\lim _{(x, y \rightarrow \rightarrow 0,0)} \frac{e^{-x^{2}-y^{2}}-1}{x^{2}+y^{2}}$$

Find the indicated partial derivatives. $$f(x, y, z)=\frac{y}{x+y+z} ; \quad f_{y}(2,1,-1)$$

A model for the surface area of a human body is given by \(S=0.1091 w^{0.425} h^{0.725}\) , where \(w\) is the weight (in pounds), \(h\) is the height (in inches), and \(S\) is measured in square feet. If the errors in measurement of \(w\) and \(h\) are at most \(2 \%,\) use differ- entials to estimate the maximum percentage error in the calculated surface area

\(39-44\) Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. $$x^{2}-2 y^{2}+z^{2}+y z=2, \quad(2,1,-1)$$

Draw a contour map of the function showing several level curves. $$f(x, y)=x^{3}-y$$

The plane \(4 x-3 y+8 z=5\) intersects the cone \(z^{2}=x^{2}+y^{2}\) in an ellipse. (a) Graph the cone, the plane, and the ellipse. (b) Use Lagrange multipliers to find the highest and lowest points on the ellipse.

Car A is traveling north on Highway 16 and car \(B\) is traveling west on Highway \(83 .\) Each car is approaching the intersection of these highways. At a certain moment, car \(A\) is 0.3 \(\mathrm{km}\) from the intersection and traveling at 90 \(\mathrm{km} / \mathrm{h}\) while car \(\mathrm{B}\) is 0.4 \(\mathrm{km}\) from the intersection and traveling at 80 \(\mathrm{km} / \mathrm{h} .\) How fast is the distance between the cars changing at that moment?

At the beginning of this section we considered the function $$f(x, y)=\frac{\sin \left(x^{2}+y^{2}\right)}{x^{2}+y^{2}}$$ and guessed that \(f(x, y) \rightarrow 1\) as \((x, y) \rightarrow(0,0)\) on the basis of numerical evidence. Use polar coordinates to confirm the value of the limit. Then graph the function.

Find the indicated partial derivatives. $$f(x, y, z)=\sqrt{\sin ^{2} x+\sin ^{2} y+\sin ^{2} z} ; \quad f_{z}(0,0, \pi / 4)$$

Suppose you need to know an equation of the tangent plane to a surface \(S\) at the point \(P(2,1,3) .\) You don't have an equation for \(S\) but you know that the curves $$\begin{aligned} \mathbf{r}_{1}(t) &=\left\langle 2+3 t, 1-t^{2}, 3-4 t+t^{2}\right\rangle \\ \mathbf{r}_{2}(u) &=\left\langle 1+u^{2}, 2 u^{3}-1,2 u+1\right\rangle \end{aligned}$$ both lie on \(S .\) Find an equation of the tangent plane at \(P .\)

\(39-44\) Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. $$x-z=4 \arctan (y z), \quad(1+\pi, 1,1)$$

Draw a contour map of the function showing several level curves. $$f(x, y)=e^{y / x}$$

Find the points on the surface \(y^{2}=9+x z\) that are closest to the origin.

Find three positive numbers whose sum is 100 and whose product is a maximum.

Find the maximum and minimum values of \(f\) subject to the given constraints. Use a computer algebra system to solve the system of equations that arises in using Lagrange multipliers. (If your CAS finds only one solution, you may need to use additional commands.) \(f(x, y, z)=y e^{x-z} ; \quad 9 x^{2}+4 y^{2}+36 z^{2}=36, x y+y z=1\)

One side of a triangle is increasing at a rate of 3 \(\mathrm{cm} / \mathrm{s}\) and a second side is decreasing at a rate of 2 \(\mathrm{cm} / \mathrm{s} .\) If the area of the triangle remains constant, at what rate does the angle between the sides change when the first side is 20 \(\mathrm{cm}\) long, the second side is \(30 \mathrm{cm},\) and the angle is \(\pi / 6 ?\)

Graph and discuss the continuity of the function $$f(x, y)=\left\\{\begin{array}{ll}{\frac{\sin x y}{x y}} & {\text { if } x y \neq 0} \\ {1} & {\text { if } x y=0}\end{array}\right.$$

Use the definition of partial derivatives as limits \((4)\) to find \(f_{x}(x, y)\) and \(f_{y}(x, y) .\) $$f(x, y)=x y^{2}-x^{3} y$$

\(39-44\) Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. $$z+1=x e^{y} \cos z, \quad(1,0,0)$$

Draw a contour map of the function showing several level curves. $$f(x, y)=y e^{x}$$

Find three positive numbers whose sum is 12 and the sum of whose squares is as small as possible.

Find the maximum and minimum values of \(f\) subject to the given constraints. Use a computer algebra system to solve the system of equations that arises in using Lagrange multipliers. (If your CAS finds only one solution, you may need to use additional commands.) \(f(x, y, z)=x+y+z ; \quad x^{2}-y^{2}=z, x^{2}+z^{2}=4\)

Let $$f(x, y)=\left\\{\begin{array}{ll}{0} & {\text { if } y \leqslant 0 \quad \text { or } \quad y \geqslant x^{4}} \\ {1} & {\text { if } 0 < y < x^{4}}\end{array}\right.$$ (a) Show that \(f(x, y) \rightarrow 0\) as \((x, y) \rightarrow(0,0)\) along any path through \((0,0)\) of the form \(y=m X^{a}\) with \(a<4 .\) (b) Despite part (a), show that \(f\) is discontinuous at \((0,0)\) (c) Show that \(f\) is discontinuous on two entire curves.

Use the definition of partial derivatives as limits \((4)\) to find \(f_{x}(x, y)\) and \(f_{y}(x, y) .\) $$f(x, y)=\frac{x}{x+y^{2}}$$

\(39-44\) Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. $$y z=\ln (x+z), \quad(0,0,1)$$

Draw a contour map of the function showing several level curves. $$f(x, y)=y \sec x$$

If a sound with frequency \(f_{s}\) is produced by a source traveling along a line with speed \(v_{s}\) and an observer is traveling with speed \(v_{o}\) along the same line from the opposite direction toward the source, then the frequency of the sound heard by the observer is $$f_{o}=\left(\frac{c+v_{o}}{c-v_{s}}\right) f_{s}$$ where \(c\) is the speed of sound, about 332 \(\mathrm{m} / \mathrm{s}\) . (This is the Doppler effect.) Suppose that, at a particular moment, you are in a train traveling at 34 \(\mathrm{m} / \mathrm{s}\) and accelerating at 1.2 \(\mathrm{m} / \mathrm{s}^{2}\) A train is approaching you from the opposite direction on the other track at 40 \(\mathrm{m} / \mathrm{s}\) , accelerating at 1.4 \(\mathrm{m} / \mathrm{s}^{2}\) , and sounds its whistle, which has a frequency of 460 \(\mathrm{Hz}\) . At that instant, what is the perceived frequency that you hear and how fast is it changing?

Find the maximum volume of a rectangular box that is inscribed in a sphere of radius \(r .\)

Show that the function \(f\) given by \(f(\mathbf{x})=|\mathbf{x}|\) is continuous on \(\mathbb{R}^{n}\) [Hint: Consider \(|\mathbf{x}-\mathbf{a}|^{2}=(\mathbf{x}-\mathbf{a}) \cdot(\mathbf{x}-\mathbf{a}) . ]\)

\(45-48\) Assume that all the given functions are differentiable. If \(z=f(x, y),\) where \(x=r \cos \theta\) and \(y=r \sin \theta,(a)\) find \(\partial z / \partial r\) and \(\partial z / \partial \theta\) and \((b)\) show that $$\left(\frac{\partial z}{\partial x}\right)^{2}+\left(\frac{\partial z}{\partial y}\right)^{2}=\left(\frac{\partial z}{\partial r}\right)^{2}+\frac{1}{r^{2}}\left(\frac{\partial z}{\partial \theta}\right)^{2}$$

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

Prove that if \(f\) is a function of two variables that is differen- tiable at \((a, b),\) then \(f\) is continuous at \((a, b) .\) Hint: Show that $$\lim _{(\Delta x \Delta y \rightarrow(0,0)} f(a+\Delta x, b+\Delta y)=f(a, b)$$

\(45-46\) Use a computer to graph the surface, the tangent plane, and the normal line on the same screen. Choose the domain carefully so that you avoid extraneous vertical planes. Choose the viewpoint so that you get a good view of all three objects. $$x y+y z+z x=3, \quad(1,1,1)$$

Draw a contour map of the function showing several level curves. $$f(x, y)=\sqrt{y^{2}-x^{2}}$$

If \(\mathbf{c} \in V_{n}\) , show that the function \(f\) given by \(f(\mathbf{x})=\mathbf{c} \cdot \mathbf{x}\) is continuous on \(\mathbb{R}^{n} .\)

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

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

Draw a contour map of the function showing several level curves. $$f(x, y)=y /\left(x^{2}+y^{2}\right)$$

Find the dimensions of the box with volume 1000 \(\mathrm{cm}^{3}\) that has minimal surface area.

(a) Maximize \(\sum_{t=1}^{n} X_{i} y_{i}\) subject to the constraints \(\sum_{t=1}^{n} x_{t}^{2}=1\) and \(\Sigma_{i=1}^{n} y_{i}^{2}=1.\) (b) Put $$x_{i}=\frac{a_{i}}{\sqrt{\Sigma a_{j}^{2}}} \quad \text { and } \quad y_{i}=\frac{b_{i}}{\sqrt{\Sigma b_{j}^{2}}}$$ to show that $$\sum a_{i} b_{i} \leqslant \sqrt{\Sigma a_{j}^{2}} \sqrt{\Sigma b_{j}^{2}}$$ for any numbers \(a_{1}, \ldots, a_{n} b_{1}, \ldots, b_{n}\) . This inequality is known as the Cauchy-Schwarz Inequality.

Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane \(x+2 y+3 z=6\)

\(45-48\) Assume that all the given functions are differentiable. If \(z=f(x-y),\) show that \(\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}=0\)