Chapter 13

Consider the following functions \(f.\) a. Is \(f\) continuous at (0,0)\(?\) b. Is \(f\) differentiable at (0,0)\(?\) c. If possible, evaluate \(f_{x}(0,0)\) and \(f_{y}(0,0)\) d. Determine whether \(f_{x}\) and \(f_{y}\) are continuous at \((0,0)\) e. Explain why Theorems 5 and 6 are consistent with the results in parts \((a)-(d).\) $$f(x, y)=\left\\{\begin{array}{cc}-\frac{x y}{x^{2}+y^{2}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{array}\right.$$

Production functions Economists model the output of manufacturing systems using production functions that have many of the same properties as utility functions. The family of Cobb-Douglas production functions has the form \(P=f(K, L)=C K^{a} L^{1-a},\) where K represents capital, L represents labor, and C and a are positive real numbers with \(0

If possible, find the absolute maximum and minimum values of the following functions on the set \(R\). $$f(x, y)=x^{2}-y^{2} ; R=\\{(x, y) ;|x|<1,|y|<1\\}$$

Evaluate the following limits. $$\lim _{(x, y, z) \rightarrow(0,1,0)} \ln e^{x z}(1+y)$$

Consider the following equations of quadric surfaces. a. Find the intercepts with the three coordinate axes, when they exist. b. Find the equations of the \(x y-, x z^{-}\), and \(y z\) -traces, when they exist. c. Sketch a graph of the surface. $$2 y-\frac{x^{2}}{8}-\frac{z^{2}}{18}=0$$

a. Determine the domain and range of the following functions. b. Graph each function using a graphing utility. Be sure to experiment with the window and orientation to give the best perspective of the surface. $$g(x, y)=e^{-x y}.$$

Heron's formula The area of a triangle with sides of length \(a, b\), and \(c\) is given by a formula from antiquity called Heron's formula: $$A=\sqrt{s(s-a)(s-b)(s-c)}$$ where \(s=(a+b+c) / 2\) is the semiperimeter of the triangle. a. Find the partial derivatives \(A_{a}, A_{b},\) and \(A_{c}\). b. A triangle has sides of length \(a=2, b=4\), and \(c=5\). Estimate the change in the area when \(a\) increases by \(0.03, b\) decreases by \(0.08\), and \(c\) increases by \(0.6\). c. For an equilateral triangle with \(a=b=c\), estimate the percent change in the area when all sides increase in length by \(p \%\).

Consider the following surfaces specified in the form \(z=f(x, y)\) and the curve \(C\) in the \(x y\) -plane given parametrically in the form \(x=g(t), y=h(t)\). a. In each case, find \(z^{\prime}(t)\). b. Imagine that you are walking on the surface directly above the curve \(C\) in the direction of increasing t. Find the values of \(t\) for which you are walking uphill (that is, \(z\) is increasing). $$z=4 x^{2}-y^{2}+1, C: x=\cos t, y=\sin t ; 0 \leq t \leq 2 \pi$$

Consider the following functions \(f.\) a. Is \(f\) continuous at (0,0)\(?\) b. Is \(f\) differentiable at (0,0)\(?\) c. If possible, evaluate \(f_{x}(0,0)\) and \(f_{y}(0,0)\) d. Determine whether \(f_{x}\) and \(f_{y}\) are continuous at \((0,0)\) e. Explain why Theorems 5 and 6 are consistent with the results in parts \((a)-(d).\) $$f(x, y)=\left\\{\begin{array}{cc}\frac{2 x y^{2}}{x^{2}+y^{4}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{array}\right.$$

Production functions Economists model the output of manufacturing systems using production functions that have many of the same properties as utility functions. The family of Cobb-Douglas production functions has the form \(P=f(K, L)=C K^{a} L^{1-a},\) where K represents capital, L represents labor, and C and a are positive real numbers with \(0

Evaluate the following limits. $$\lim _{(x, y, z) \rightarrow(1,1,1)} \frac{y z-x y-x z-x^{2}}{y z+x y+x z-y^{2}}$$

Consider the following equations of quadric surfaces. a. Find the intercepts with the three coordinate axes, when they exist. b. Find the equations of the \(x y-, x z^{-}\), and \(y z\) -traces, when they exist. c. Sketch a graph of the surface. $$\frac{x^{2}}{25}+\frac{y^{2}}{9}-z^{2}=1$$

a. Determine the domain and range of the following functions. b. Graph each function using a graphing utility. Be sure to experiment with the window and orientation to give the best perspective of the surface. $$f(x, y)=|x y|.$$

Surface area of a cone A cone with height \(h\) and radius \(r\) has a lateral surface area (the curved surface only, excluding the base) of \(S=\pi r \sqrt{r^{2}+h^{2}}\) a. Estimate the change in the surface area when \(r\) increases from \(r=2.50\) to \(r=2.55\) and \(h\) decreases from \(h=0.60\) to \(h=0.58\) b. When \(r=100\) and \(h=200,\) is the surface area more sensitive to a small change in \(r\) or a small change in \(h ?\) Explain.

Determine whether the following statements are true and give an explanation or counterexample. a. \(\frac{\partial}{\partial x}\left(y^{10}\right)=10 y^{9}\) b. \(\frac{\partial^{2}}{\partial x \partial y}(\sqrt{x y})=\frac{1}{\sqrt{x y}}\) c. If \(f\) has continuous partial derivatives of all orders, then \(f_{x x y}=f_{y x x}\)

Temperature of an elliptical plate The temperature of points on an elliptical plate \(x^{2}+y^{2}+x y \leq 1\) is given by \(T(x,y)=25\left(x^{2}+y^{2}\right) .\) Find the hottest and coldest temperatures on the edge of the elliptical plate.

Evaluate the following limits. $$\lim _{(x, y, z) \rightarrow(1,1,1)} \frac{x-\sqrt{x z}-\sqrt{x y}+\sqrt{y z}}{x-\sqrt{x z}+\sqrt{x y}-\sqrt{y z}}$$

Gradients in three dimensions Consider the following functions \(f,\) points \(P,\) and unit vectors \(\mathbf{u}\) a. Compute the gradient of \(f\) and evaluate it at \(P\). b. Find the unit vector in the direction of maximum increase of \(f\) at \(P\). c. Find the rate of change of the function in the direction of maximum increase at \(P\) d. Find the directional derivative at \(P\) in the direction of the given vector. $$f(x, y, z)=4-x^{2}+3 y^{2}+\frac{z^{2}}{2} ; P(0,2,-1) ;\left\langle 0, \frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right\rangle$$

Line tangent to an intersection curve Consider the paraboloid \(z=x^{2}+3 y^{2}\) and the plane \(z=x+y+4,\) which intersects the paraboloid in a curve \(C\) at (2,1,7) (see figure). Find the equation of the line tangent to \(C\) at the point \((2,1,7) .\) Proceed as follows. a. Find a vector normal to the plane at (2,1,7) b. Find a vector normal to the plane tangent to the paraboloid at (2,1,7) c. Argue that the line tangent to \(C\) at (2,1,7) is orthogonal to both normal vectors found in parts (a) and (b). Use this fact to find a direction vector for the tangent line. d. Knowing a point on the tangent line and the direction of the tangent line, write an equation of the tangent line in parametric form.

Consider the following surfaces specified in the form \(z=f(x, y)\) and the curve \(C\) in the \(x y\) -plane given parametrically in the form \(x=g(t), y=h(t)\). a. In each case, find \(z^{\prime}(t)\). b. Imagine that you are walking on the surface directly above the curve \(C\) in the direction of increasing t. Find the values of \(t\) for which you are walking uphill (that is, \(z\) is increasing). $$z=2 x^{2}+y^{2}+1, C: x=1+\cos t, y=\sin t ; 0 \leq t \leq 2 \pi$$

Maximizing a sum Find the maximum value of \(x_{1}+x_{2}+x_{3}+x_{4}\) subject to the condition that \(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=16\).

Find the point on the surface curve \(y=x^{2}\) nearest the line \(y=x-1 .\) Identify the point on the line.

Evaluate the following limits. $$\lim _{(x, y, z) \rightarrow(1,1,1)} \frac{x^{2}+x y-x z-y z}{x-z}$$

a. Determine the domain and range of the following functions. b. Graph each function using a graphing utility. Be sure to experiment with the window and orientation to give the best perspective of the surface. $$h(x, y)=(x+y) /(x-y).$$

Gradients in three dimensions Consider the following functions \(f,\) points \(P,\) and unit vectors \(\mathbf{u}\) a. Compute the gradient of \(f\) and evaluate it at \(P\). b. Find the unit vector in the direction of maximum increase of \(f\) at \(P\). c. Find the rate of change of the function in the direction of maximum increase at \(P\) d. Find the directional derivative at \(P\) in the direction of the given vector. $$f(x, y, z)=1+4 x y z ; P(1,-1,-1) ;\left\langle\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}\right\rangle$$

Batting averages Batting averages in baseball are defined by \(A=x / y,\) where \(x \geq 0\) is the total number of hits and \(y>0\) is the total number of at- bats. Treat \(x\) and \(y\) as positive real numbers and note that \(0 \leq A \leq 1\) a. Use differentials to estimate the change in the batting average if the number of hits increases from 60 to 62 and the number of at-bats increases from 175 to 180 . b. If a batter currently has a batting average of \(A=0.350,\) does the average decrease if the batter fails to get a hit more than it increases if the batter gets a hit? c. Does the answer to part (b) depend on the current batting average? Explain.

A projectile with mass \(m\) is launched into the air on a parabolic trajectory. For \(t \geq 0,\) its horizontal and vertical coordinates are \(x(t)=u_{0} t\) and \(y(t)=-\frac{1}{2} g t^{2}+v_{0} t\), respectively, where \(u_{0}\) is the initial horizontal velocity, \(v_{0}\) is the initial vertical velocity, and \(g\) is the acceleration due to gravity. Recalling that \(u(t)=x^{\prime}(t)\) and \(v(t)=y^{\prime}(t)\) are the components of the velocity, the energy of the projectile (kinetic plus potential) is \(E(t)=\frac{1}{2} m\left(u^{2}+v^{2}\right)+m g y\). Use the Chain Rule to compute \(E^{\prime}(t)\) and show that \(E^{\prime}(t)=0\), for all \(t \geq 0 .\) Interpret the result.

Rectangular boxes with a volume of \(10 \mathrm{m}^{3}\) are made of two materials. The material for the top and bottom of the box costs \(\$ 10 / \mathrm{m}^{2}\) and the material for the sides of the box costs \(\$ 1 / \mathrm{m}^{2}\). What are the dimensions of the box that minimize the cost of the box?

Evaluate the following limits. $$\lim _{(x, y, z) \rightarrow(1,-1,1)} \frac{x z+5 x+y z+5 y}{x+y}$$

a. Determine the domain and range of the following functions. b. Graph each function using a graphing utility. Be sure to experiment with the window and orientation to give the best perspective of the surface. $$G(x, y)=\ln (2+\sin (x+y)).$$

Gradients in three dimensions Consider the following functions \(f,\) points \(P,\) and unit vectors \(\mathbf{u}\) a. Compute the gradient of \(f\) and evaluate it at \(P\). b. Find the unit vector in the direction of maximum increase of \(f\) at \(P\). c. Find the rate of change of the function in the direction of maximum increase at \(P\) d. Find the directional derivative at \(P\) in the direction of the given vector. $$f(x, y, z)=x y+y z+x z+4 ; P(2,-2,1) ;\left\langle 0,-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right\rangle$$

Economists use utility functions to describe consumers' relative preference for two or more commodities (for example, vanilla vs. chocolate ice cream or leisure time vs. material goods). The Cobb-Douglas family of utility functions has the form \(U(x, y)=x^{a} y^{1-a},\) where \(x\) and \(y\) are the amounts of two commodities and \(0

Water-level changes A conical tank with radius \(0.50 \mathrm{m}\) and height \(2.00 \mathrm{m}\) is filled with water (see figure). Water is released from the tank, and the water level drops by \(0.05 \mathrm{m}\) (from \(2.00 \mathrm{m}\) to \(1.95 \mathrm{m}\) ). Approximate the change in the volume of water in the tank. (Hint: When the water level drops, both the radius and height of the cone of water change.)

Determine whether the following statements are true and give an explanation or counterexample. a. If the limits $$\lim _{(x, 0) \rightarrow(0,0)} f(x, 0) \text { and } \lim _{(0, y) \rightarrow(0,0)} f(0, y)$$ exist and equal \(L,\) then $$\lim _{(x, y) \rightarrow(0,0)} f(x, y)=L$$ b. If $$\lim _{(x, y) \rightarrow(a, b)} f(x, y)=L$$, then \(f\) is continuous at \((a, b)\) c. If \(f\) is continuous at \((a, b),\) then $$\lim _{(x, y) \rightarrow(a, b)} f(x, y) \text { exists. }$$ d. If \(P\) is a boundary point of the domain of \(f,\) then \(P\) is in the domain of \(f\)

Consider the following equations of quadric surfaces. a. Find the intercepts with the three coordinate axes, when they exist. b. Find the equations of the \(x y-, x z^{-}\), and \(y z\) -traces, when they exist. c. Sketch a graph of the surface. $$z=\frac{x^{2}}{9}-y^{2}$$

Gradients in three dimensions Consider the following functions \(f,\) points \(P,\) and unit vectors \(\mathbf{u}\) a. Compute the gradient of \(f\) and evaluate it at \(P\). b. Find the unit vector in the direction of maximum increase of \(f\) at \(P\). c. Find the rate of change of the function in the direction of maximum increase at \(P\) d. Find the directional derivative at \(P\) in the direction of the given vector. $$f(x, y, z)=1+\sin (x+2 y-z) ; P\left(\frac{\pi}{6}, \frac{\pi}{6},-\frac{\pi}{6}\right) ;\left\langle\frac{1}{3}, \frac{2}{3}, \frac{2}{3}\right\rangle$$

The volume of a solid torus (a bagel or doughnut) is given by \(V=\left(\pi^{2} / 4\right)(R+r)(R-r)^{2},\) where \(r\) and \(R\) are the inner and outer radii and \(R>r\) (see figure). a. If \(R\) and \(r\) increase at the same rate, does the volume of the torus increase, decrease, or remain constant? b. If \(R\) and \(r\) decrease at the same rate, does the volume of the torus increase, decrease, or remain constant?

Flow in a cylinder Poiseuille's Law is a fundamental law of fluid dynamics that describes the flow velocity of a viscous incompressible fluid in a cylinder (it is used to model blood flow through veins and arteries). It says that in a cylinder of radius \(R\) and length \(L\), the velocity of the fluid \(r \leq R\) units from the centerline of the cylinder is \(V=\frac{P}{4 L \nu}\left(R^{2}-r^{2}\right),\) where \(P\) is the difference in the pressure between the ends of the cylinder and \(\nu\) is the viscosity of the fluid (see figure). Assuming that \(P\) and \(\nu\) are constant, the velocity \(V\) along the centerline of the cylinder \((r=0)\) is \(V=k R^{2} / L,\) where \(k\) is a constant that we will take to be \(k=1\) a. Estimate the change in the centerline velocity \((r=0)\) if the radius of the flow cylinder increases from \(R=3 \mathrm{cm}\) to \(R=3.05 \mathrm{cm}\) and the length increases from \(L=50 \mathrm{cm}\) to \(L=50.5 \mathrm{cm}\) b. Estimate the percent change in the centerline velocity if the radius of the flow cylinder \(R\) decreases by \(1 \%\) and the length \(L\) increases by \(2 \%\) c. Complete the following sentence: If the radius of the cylinder increases by \(p \%,\) then the length of the cylinder must increase by approximately \- \% in order for the velocity to remain constant.

Maximizing a sum Geometric and arithmetic means Prove that the geometric mean of a set of positive numbers \(\left(x_{1} x_{2} \cdots x_{n}\right)^{1 / n}\) is no greater than the arithmetic mean \(\left(x_{1}+\cdots+x_{n}\right) / n\) in the following cases. a. Find the maximum value of \(x y z,\) subject to \(x+y+z=k\) where \(k\) is a real number and \(x>0, y>0,\) and \(z>0 .\) Use the result to prove that $$(x y z)^{1 / 3} \leq \frac{x+y+z}{3}$$ b. Generalize part (a) and show that $$\left(x_{1} x_{2} \cdots x_{n}\right)^{1 / n} \leq \frac{x_{1}+\cdots+x_{n}}{n}$$

Use the method of your choice to ate the following limits. $$\lim _{(x, y) \rightarrow(0,0)} \frac{y^{2}}{x^{8}+y^{2}}$$

Gradients in three dimensions Consider the following functions \(f,\) points \(P,\) and unit vectors \(\mathbf{u}\) a. Compute the gradient of \(f\) and evaluate it at \(P\). b. Find the unit vector in the direction of maximum increase of \(f\) at \(P\). c. Find the rate of change of the function in the direction of maximum increase at \(P\) d. Find the directional derivative at \(P\) in the direction of the given vector. $$f(x, y, z)=e^{x y-1} ; P(0,1,-1) ;\left\langle-\frac{2}{3}, \frac{2}{3},-\frac{1}{3}\right\rangle$$

One of several empirical formulas that relates the surface area \(S\) of a human body to the height \(h\) and weight \(w\) of the body is the Mosteller formula \(S(h, w)=\frac{1}{60} \sqrt{h w},\) where \(h\) is measured in centimeters, \(w\) is measured in kilograms, and \(S\) is measured in square meters. Suppose that \(h\) and \(w\) are functions of \(t\). a. Find \(S^{\prime}(t)\). b. Show that the condition that the surface area remains constant as \(h\) and \(w\) change is \(w h^{\prime}(t)+h w^{\prime}(t)=0\). c. Show that part (b) implies that for constant surface area, \(h\) and \(w\) must be inversely related; that is, \(h=C / w,\) where \(C\) is a constant.

Compute the first partial derivatives of the following functions. $$f(x, y)=\ln \left(1+e^{-x y}\right)$$

Problems with two constraints Given a differentiable function \(w=f(x, y, z),\) the goal is to find its maximum and minimum values subject to the constraints \(g(x, y, z)=0\) and \(h(x, y, z)=0\) where \(g\) and \(h\) are also differentiable. a. Imagine a level surface of the function \(f\) and the constraint surfaces \(g(x, y, z)=0\) and \(h(x, y, z)=0 .\) Note that \(g\) and \(h\) intersect (in general) in a curve \(C\) on which maximum and minimum values of \(f\) must be found. Explain why \(\nabla g\) and \(\nabla h\) are orthogonal to their respective surfaces. b. Explain why \(\nabla f\) lies in the plane formed by \(\nabla g\) and \(\nabla h\) at a point of \(C\) where \(f\) has a maximum or minimum value. c. Explain why part (b) implies that \(\nabla f=\lambda \nabla g+\mu \nabla h\) at a point of \(C\) where \(f\) has a maximum or minimum value, where \(\lambda\) and \(\mu\) (the Lagrange multipliers) are real numbers. d. Conclude from part (c) that the equations that must be solved for maximum or minimum values of \(f\) subject to two constraints are \(\nabla f=\lambda \nabla g+\mu \nabla h, g(x, y, z)=0,\) and \(h(x, y, z)=0\)

Use the method of your choice to ate the following limits. $$\lim _{(x, y) \rightarrow(0,1)} \frac{y \sin x}{x(y+1)}$$

Gradients in three dimensions Consider the following functions \(f,\) points \(P,\) and unit vectors \(\mathbf{u}\) a. Compute the gradient of \(f\) and evaluate it at \(P\). b. Find the unit vector in the direction of maximum increase of \(f\) at \(P\). c. Find the rate of change of the function in the direction of maximum increase at \(P\) d. Find the directional derivative at \(P\) in the direction of the given vector. $$f(x, y, z)=\ln \left(1+x^{2}+y^{2}+z^{2}\right) ; P(1,1,-1) ;\left\langle\frac{2}{3}, \frac{2}{3},-\frac{1}{3}\right\rangle$$

The pressure, temperature, and volume of an ideal gas are related by \(P V=k T,\) where \(k>0\) is a constant. Any two of the variables may be considered independent, which determines the third variable. a. Use implicit differentiation to compute the partial derivatives \(\frac{\partial P}{\partial V}, \frac{\partial T}{\partial P},\) and \(\frac{\partial V}{\partial T}\). b. Show that \(\frac{\partial P}{\partial V} \frac{\partial T}{\partial P} \frac{\partial V}{\partial T}=-1 .\) (See Exercise 67 for a generalization.)

Compute the first partial derivatives of the following functions. $$f(x, y)=1-\tan ^{-1}\left(x^{2}+y^{2}\right)$$

Probability of at least one encounter Suppose that in a large group of people a fraction \(0 \leq r \leq 1\) of the people have flu. The probability that in \(n\) random encounters, you will meet at least one person with flu is \(P=f(n, r)=1-(1-r)^{n} .\) Although \(n\) is a positive integer, regard it as a positive real number. a. Compute \(f_{r}\) and \(f_{n}\) b. How sensitive is the probability \(P\) to the flu rate \(r ?\) Suppose you meet \(n=20\) people. Approximately how much does the probability \(P\) increase if the flu rate increases from \(r=0.1\) to \(r=0.11\) (with \(n\) fixed)? c. Approximately how much does the probability \(P\) increase if the flu rate increases from \(r=0.9\) to \(r=0.91\) with \(n=20 ?\) d. Interpret the results of parts (b) and (c).

Find the dimensions of the rectangular box with maximum volume in the first octant with one vertex at the origin and the opposite vertex on the ellipsoid \(36 x^{2}+4 y^{2}+9 z^{2}=36\).