Chapter 10

Calculate all four second-order partial derivatives of \(f(x, y)=(2 x+4 y) e^{y}\) \(f_{x x}(x, y)=\) ___________. \(f_{x y}(x, y)=\) ___________. \(f_{y x}(x, y)=\) ___________. \(f_{y y}(x, y)=\) ___________.

Find the partial derivatives of the function $$f(x, y)=\int_{y}^{x} \cos \left(7 t^{2}+8 t-2\right) d t$$ \(f_{x}(x, y)=\) ___________. \(f_{y}(x, y)=\) ___________.

Find the limit, if it exists, or type \(\mathrm{N}\) if it does not exist. $$\lim _{(x, y, z) \rightarrow(0,0,0)} \frac{4 x y+2 y z+2 x z}{16 x^{2}+4 y^{2}+4 z^{2}}=$$ ___________.

Find the maximum and minimum values of the function \(f(x, y, z)=\) \(x^{2} y^{2} z^{2}\) subject to the constraint \(x^{2}+y^{2}+z^{2}=64 .\) Maximum value is ____________, occuring at points (positive integer or "infinitely many"). Minimum value is ____________, occuring at points (positive integer or "infinitely many").

Find \(A\) and \(B\) so that \(f(x, y)=x^{2}+A x+y^{2}+B\) has a local minimum at the point \((7,0),\) with \(z\) -coordinate 5 . \(A=\) ___________. \(B=\) ___________.

The radius of a right circular cone is increasing at a rate of 3 inches per second and its height is decreasing at a rate of 5 inches per second. At what rate is the volume of the cone changing when the radius is 40 inches and the height is 40 inches? _________ cubic inches per second.

One mole of ammonia gas is contained in a vessel which is capable of changing its volume (a compartment sealed by a piston, for example). The total energy \(U\) (in Joules) of the ammonia is a function of the volume \(V\) (in cubic meters) of the container, and the temperature \(T\) (in degrees Kelvin) of the gas. The differential \(d U\) is given by \(d U=840 d V+27.32 d T\). (a) How does the energy change if the volume is held constant and the temperature is decreased slightly? \(\odot\) it increases slightly \(\odot\) it does not change \(\odot\) it decreases slightly (b) How does the energy change if the temperature is held constant and the volume is increased slightly? \(\odot\) it does not change \(\odot\) it increases slightly \(\odot\) it decreases slightly (c) Find the approximate change in energy if the gas is compressed by 150 cubic centimeters and heated by 3 degrees Kelvin. Change in energy = ___________. Please include units in your answer.

Let \(f(x, y)=(-(2 x+y))^{6}\). Then \(\frac{\partial^{2} f}{\partial x \partial y}\) __________. \(\frac{\partial^{3} f}{\partial x \partial y \partial x}=\) __________. \(\frac{\partial^{3} f}{\partial x^{2} \partial y}=\) __________.

Let \(f(x, y)=e^{-2 x} \sin (4 y)\) (a) Using difference quotients with \(\Delta x=0.1\) and \(\Delta y=0.1,\) we estimate \(f_{x}(2,-2) \approx\) _________. \(f_{y}(2,-2) \approx\) _________. (b) Using difference quotients with \(\Delta x=0.01\) and \(\Delta y=0.01\), we find better estimates: \(f_{x}(2,-2) \approx\) __________. \(f_{y}(2,-2) \approx\) __________.

Find the limit, if it exists, or type \(\mathrm{N}\) if it does not exist. $$\lim _{(x, y, z) \rightarrow(5,1,4)} \frac{2 z e^{x^{2}+y^{2}}}{5 x^{2}+y^{2}+4 z^{2}}=$$ _____________.

Find the maximum and minimum values of the function \(f(x, y, z, t)=\) \(x+y+z+t\) subject to the constraint \(x^{2}+y^{2}+z^{2}+t^{2}=100 .\) Maximum value is ________, occuring at points (positive integer or "infinitely many"). Minimum value is ________,, occuring at points (positive integer or "infinitely many").

Suppose that you are climbing a hill whose shape is given by \(z=902-\) \(0.07 x^{2}-0.1 y^{2},\) and that you are at the point (40,70,300) In which direction (unit vector) should you proceed initially in order to reach the top of the hill fastest? (_________________) If you climb in that direction, at what angle above the horizontal will you be climbing initially (radian measure)?

In a simple electric circuit, Ohm's law states that \(V=I R,\) where \(\mathrm{V}\) is the voltage in volts, I is the current in amperes, and \(\mathrm{R}\) is the resistance in ohms. Assume that, as the battery wears out, the voltage decreases at 0.03 volts per second and, as the resistor heats up, the resistance is increasing at 0.02 ohms per second. When the resistance is 100 ohms and the current is 0.02 amperes, at what rate is the current changing? ___________ amperes per second .

An unevenly heated metal plate has temperature \(T(x, y)\) in degrees Celsius at a point \((x, y) .\) If \(T(2,1)=119, T_{x}(2,1)=19,\) and \(T_{y}(2,1)=-14,\) estimate the temperature at the point (2.04,0.96) . \(T(2.04,0.96) \approx\) ________.

If \(z_{x y}=5 y\) and all of the second order partial derivatives of \(z\) are continuous, then (a) \(z_{y x}=\) __________. (b) \(z_{x y x}=\) __________. (c) \(z_{x y y}=\) __________.

Find the limit (enter 'DNE' if the limit does not exist) Hint: rationalize the denominator. $$\lim _{(x, y) \rightarrow(0,0)} \frac{\left(9 x^{2}+2 y^{2}\right)}{\sqrt{\left(9 x^{2}+2 y^{2}+1\right)}-1}$$ __________.

Find the maximum and minimum volumes of a rectangular box whose surface area equals 7000 square \(\mathrm{cm}\) and whose edge length (sum of lengths of all edges) is \(440 \mathrm{~cm}\). Maximum value is ________________. occuring at ( ____________, ______________) Manimum value is ________________. occuring at ( ____________, ______________)

Consider the three points \((5,1),(7,3),\) and (9,4) . (a) Supposed that at (5,1) , we know that \(f_{x}=f_{y}=0\) and \(f_{x x}=0\), \(f_{y y}=0\), and \(f_{x y}>0\). What can we conclude about the behavior of this function near the point (5,1)\(? \quad(\square(5,1)\) is a local maximum \(\square(5,1)\) is a local minimum \(\square(5,1)\) is a saddle point \(\square(5,1)\) is a none of these) (b) Supposed that at \((7,3),\) we know that \(f_{x}=f_{y}=0\) and \(f_{x x}>0\), \(f_{y y}<0,\) and \(f_{x y}=0 .\) What can we conclude about the behavior of this function near the point (7,3)\(? \quad(\square(7,3)\) is a local maximum \(\square(7,3)\) is a local minimum \(\square(7,3)\) is a saddle point \(\square(7,3)\) is a none of these) (c) Supposed that at (9,4) , we know that \(f_{x}=f_{y}=0\) and \(f_{x x}<0\), \(f_{y y}=0,\) and \(f_{x y}>0 .\) What can we conclude about the behavior of this function near the point (9,4)\(? \quad(\square(9,4)\) is a local maximum \(\square(9,4)\) is a local minimum \(\square(9,4)\) is a saddle point \(\square(9,4)\) is a none of these) Using this information, on a separate sheet of paper sketch a possible contour diagram for \(f\).

Are the following statements true or false? (a) The gradient vector \(\nabla f(a, b)\) is tangent to the contour of \(f\) at \((a, b)\). (b) \(f_{\vec{u}}(a, b)=\|\nabla f(a, b)\| .\) (c) \(f_{\vec{u}}(a, b)\) is parallel to \(\vec{u}\). (d) If \(\vec{u}\) is perpendicular to \(\nabla f(a, b),\) then \(f_{\vec{u}}(a, b)=\langle 0,0\rangle\). (e) If \(\vec{u}\) is a unit vector, then \(f_{\vec{u}}(a, b)\) is a vector. (f) Suppose \(f_{x}(a, b)\) and \(f_{y}(a, b)\) both exist. Then there is always a direction in which the rate of change of \(f\) at \((a, b)\) is zero. (g) If \(f(x, y)\) has \(f_{x}(a, b)=0\) and \(f_{y}(a, b)=0\) at the point \((a, b)\), then \(f\) is constant everywhere. (h) \(\nabla f(a, b)\) is a vector in 3 -dimensional space.

Suppose \(z=x^{2} \sin y, x=2 s^{2}+1 t^{2}, y=-6 s t\). A. Use the chain rule to find \(\frac{\partial z}{\partial s}\) and \(\frac{\partial z}{\partial t}\) as functions of \(\mathrm{x}, \mathrm{y}, \mathrm{s}\) and \(\mathrm{t}\). \(\frac{\partial z}{\partial s}=\) ________. \(\frac{\partial z}{\partial t}=\) ________. B. Find the numerical values of \(\frac{\partial z}{\partial s}\) and \(\frac{\partial z}{\partial t}\) when \((s, t)=(-3,3)\). \(\frac{\partial z}{\partial s}(-3,3)=\) ________. \(\frac{\partial z}{\partial t}(-3,3)=\) ________.

Let \(f\) be the function defined by \(f(x, y)=2 x^{2}+3 y^{3}\). a. Find the equation of the tangent plane to \(f\) at the point (1,2) . b. Use the linearization to approximate the values of \(f\) at the points (1.1,2.05) and (1.3,2.2) c. Compare the approximations form part (b) to the exact values of \(f(1.1,2.05)\) and \(f(1.3,2.2) .\) Which approximation is more accurate. Explain why this should be expected.

If \(z=f(x)+y g(x),\) what can we say about \(z_{y y} ?\) \(\odot z_{y y}=0\) \(\odot z_{y y}=y\) \(\odot z_{y y}=z_{x x}\) \(\odot z_{y y}=g(x)\) \(\odot\) We cannot say anything

Your monthly car payment in dollars is \(P=f\left(P_{0}, t, r\right),\) where \(\$ P_{0}\) is the amount you borrowed, t is the number of months it takes to pay off the loan, and r percent is the interest rate. (a) Is \(\partial P / \partial t positive or negative? \)\quad(\square\( positive \)\square\( negative) Suppose that your bank tells you that the magnitude of \)\partial P / \partial t is 15 . What are the units of this value? _________ (b) Is \(\partial P / \partial r positive or negative? \)\quad(\square\( positive \)\square$ negative) . What are the units of this value? __________.

The largest set on which the function \(f(x, y)=1 /\left(3-x^{2}-y^{2}\right)\) is continuous is __________. A. All of the xy-plane B. The interior of the circle \(x^{2}+y^{2}=3\) C. The exterior of the circle \(x^{2}+y^{2}=3\) D. The interior of the circle \(x^{2}+y^{2}=3,\) plus the circle E. All of the xy-plane except the circle \(x^{2}+y^{2}=3\)

Find three positive real numbers whose sum is 94 and whose product is a maximum. Enter the three numbers separated by commas: ____________.

Let \(E(x, y)=\frac{100}{1+(x-5)^{2}+4(y-2.5)^{2}}\) represent the elevation on a land mass at location \((x, y)\). Suppose that \(E, x,\) and \(y\) are all measured in meters. a. Find \(E_{x}(x, y)\) and \(E_{y}(x, y)\). b. Let \(\mathbf{u}\) be a unit vector in the direction of \(\langle-4,3\rangle .\) Determine \(D_{\mathbf{u}} E(3,4)\). What is the practical meaning of \(D_{\mathbf{u}} E(3,4)\) and what are its units? c. Find the direction of greatest increase in \(E\) at the point (3,4) . d. Find the instantaneous rate of change of \(E\) in the direction of greatest decrease at the point \((3,4) .\) Include units on your answer. e. At the point \((3,4),\) find a direction \(\mathbf{w}\) in which the instantaneous rate of change of \(E\) is \(0 .\)

Find the indicated derivative. In each case, state the version of the Chain Rule that you are using. a. \(\frac{d f}{d t},\) if \(f(x, y)=2 x^{2} y, x=\cos (t),\) and \(y=\ln (t) .\) b. \(\frac{\partial f}{\partial w},\) if \(f(x, y)=2 x^{2} y, x=w+z^{2},\) and \(y=\frac{2 z+1}{w}\) c. \(\frac{\partial f}{\partial v},\) if \(f(x, y, z)=2 x^{2} y+z^{3}, x=u-v+2 w, y=w 2^{v}-u^{3},\) and \(z=u^{2}-v\)

Shown in Figure 10.3 .9 is a contour plot of a function \(f\) with the values of \(f\) labeled on the contours. The point (2,1) is highlighted in red. a. Estimate the partial derivatives \(f_{x}(2,1)\) and \(f_{y}(2,1)\). b. Determine whether the second-order partial derivative \(f_{x x}(2,1)\) is positive or negative, and explain your thinking. c. Determine whether the second-order partial derivative \(f_{y y}(2,1)\) is positive or negative, and explain your thinking. d. Determine whether the second-order partial derivative \(f_{x y}(2,1)\) is positive or negative, and explain your thinking. e. Determine whether the second-order partial derivative \(f_{y x}(2,1)\) is positive or negative, and explain your thinking. f. Consider a function \(g\) of the variables \(x\) and \(y\) for which \(g_{x}(2,2)>0\) and \(g_{x x}(2,2)<0 .\) Sketch possible behavior of some contours around (2,2) on the left axes in Figure 10.3 .10 . g. Consider a function \(h\) of the variables \(x\) and \(y\) for which \(h_{x}(2,2)>0\) and \(h_{x y}(2,2)<0\). Sketch possible behavior of some contour lines around (2,2) on the right axes in Figure \(10.3 .10 .\)

An experiment to measure the toxicity of formaldehyde yielded the data in the table below. The values show the percent, \(P=f(t, c),\) of rats surviving an exposure to formaldehyde at a concentration of \(c\) (in parts \(f\). $$\begin{array}{|l|l|l|l|l|l|l|}\hline & t=14 & t=16 & t=18 & t=20 & t=22 & t=24 \\\\\hline c=0 & 100 & 100 & 100 & 99 & 97 & 95 \\\\\hline c=2 & 100 & 99 & 98 & 97 & 95 & 92 \\\\\hline c=6 & 96 & 95 & 93 & 90 & 86 & 80 \\\\\hline c=15 & 96 & 93 & 82 & 70 & 58 & 36 \\\\\hline\end{array}$$\ (a) Estimate \(f_{t}(18,6):\) \(f_{t}(18,6) \approx\) __________. (b) Estimate \(f_{c}(18,6)\) \(f_{c}(18,6) \approx\) __________.

Consider the function \(f\) defined by \(f(x, y)=\frac{x y}{x^{2}+y^{2}+1}\) a. What is the domain of \(f ?\) b. Evaluate limit of \(f\) at (0,0) along the following paths: \(x=0, y=0\), \(y=x,\) and \(y=x^{2}\) c. What do you conjecture is the value of \(\lim _{(x, y) \rightarrow(0,0)} f(x, y) ?\) d. Is \(f\) continuous at (0,0)\(?\) Why or why not? e. Use appropriate technology to sketch both surface and contour plots of \(f\) near (0,0) . Write several sentences to say how your plots affirm your findings in (a) - (d).

The Cobb-Douglas production function is used in economics to model production levels based on labor and equipment. Suppose we have a specific Cobb-Douglas function of the form $$f(x, y)=50 x^{0.4} y^{0.6}$$ where \(x\) is the dollar amount spent on labor and \(y\) the dollar amount spent on equipment. Use the method of Lagrange multipliers to determine how much should be spent on labor and how much on equipment to maximize productivity if we have a total of 1.5 million dollars to invest in labor and equipment.

Find all directions in which the directional derivative of \(f(x, y)=y e^{-x y}\) is 1 at the point (0,2) .

Let \(z=u^{2}-v^{2}\) and suppose that $$\begin{array}{l}u=e^{x} \cos (y) \\\v=e^{x} \sin (y)\end{array}$$ a. Find the values of \(u\) and \(v\) that correspond to \(x=0\) and \(y=2 \pi / 3\). b. Use the Chain Rule to find the general partial derivatives $$\frac{\partial z}{\partial x} \text { and } \frac{\partial z}{\partial y}$$ and then determine both \(\left.\frac{\partial z}{\partial x}\right|_{\left(0, \frac{2 \pi}{3}\right)}\) and \(\left.\frac{\partial z}{\partial y}\right|_{\left(0, \frac{2 \pi}{3}\right)}\).

Let \(g\) be a function that is differentiable at (-2,5) and suppose that its tangent plane at this point is given by \(z=-7+4(x+2)-3(y-5)\). a. Determine the values of \(g(-2,5), g_{x}(-2,5),\) and \(g_{y}(-2,5) .\) Write one sentence to explain your thinking. b. Estimate the value of \(g(-1.8,4.7)\). Clearly show your work and thinking. c. Given changes of \(d x=-0.34\) and \(d y=0.21\), estimate the corresponding change in \(g\) that is given by its differential, \(d g\). d. Suppose that another function \(h\) is also differentiable at \((-2,5),\) but that its tangent plane at (-2,5) is given by \(3 x+2 y-4 z=9\) Determine the values of \(h(-2,5), h_{x}(-2,5),\) and \(h_{y}(-2,5),\) and then estimate the value of \(h(-1.8,4.7) .\) Clearly show your work and thinking.

The Heat Index, \(I,\) (measured in apparent degrees \(F\) ) is a function of the actual temperature \(T\) outside (in degrees \(\mathrm{F}\) ) and the relative humidity \(H\) (measured as a percentage). A portion of the table which gives values for this function, \(I(T, H),\) is reproduced in Table 10.3 .11 . $$\begin{array}{ccccc}\hline T \downarrow \backslash H \rightarrow & 70 & 75 & 80 & 85 \\ \hline 90 & 106 & 109 & 112 & 115 \\\\\hline 92 & 112 & 115 & 119 & 123 \\ \hline 94 & 118 & 122 & 127 & 132 \\\\\hline 96 & 125 & 130 & 135 & 141 \\\\\hline\end{array}$$ a. State the limit definition of the value \(I_{T T}(94,75)\). Then, estimate \(I_{T T}(94,75),\) and write one complete sentence that carefully explains the meaning of this value, including units. b. State the limit definition of the value \(I_{H H}(94,75) .\) Then, estimate \(I_{H H}(94,75),\) and write one complete sentence that carefully explains the meaning of this value, including units. c. Finally, do likewise to estimate \(I_{H T}(94,75),\) and write a sentence to explain the meaning of the value you found.

Consider the function \(g\) defined by \(g(x, y)=\frac{x y}{x^{2}+y^{2}}\). a. What is the domain of \(g ?\) b. Evaluate limit of \(g\) at (0,0) along the following paths: \(x=0, y=x\), and \(y=2 x\). c. What can you now say about the value of \(\lim _{(x, y) \rightarrow(0,0)} g(x, y) ?\) d. Is \(g\) continuous at (0,0)\(?\) Why or why not? e. Use appropriate technology to sketch both surface and contour plots of \(g\) near (0,0) . Write several sentences to say how your plots affirm your findings in (a) - (d).

Use the method of Lagrange multipliers to find the point on the line \(x-2 y=5\) that is closest to the point \((1,3) .\) To do so, respond to the following prompts. a. Write the function \(f=f(x, y)\) that measures the square of the distance from \((x, y)\) to (1,3) . (The extrema of this function are the same as the extrema of the distance function, but \(f(x, y)\) is simpler to work with.) b. What is the constraint \(g(x, y)=c ?\) c. Write the equations resulting from \(\nabla f=\lambda \nabla g\) and the constraint. Find all the points \((x, y)\) satisfying these equations. d. Test all the points you found to determine the extrema.

Find, if possible, a function \(f\) such that $$\nabla f=\left\langle\sin (y z), x z \cos (y z)+2 y, x y \cos (y z)+\frac{5}{z}\right\rangle .$$ If not possible, explain why.

Suppose that \(T=x^{2}+y^{2}-2 z\) where $$\begin{array}{l}x=\rho \sin (\phi) \cos (\theta) \\\y=\rho \sin (\phi) \sin (\theta) \\\z=\rho \cos (\phi)\end{array}$$ a. Construct a tree diagram representing the dependencies among the variables. b. Apply the chain rule to find the partial derivatives $$\frac{\partial T}{\partial \rho}, \frac{\partial T}{\partial \phi}, \text { and } \frac{\partial T}{\partial \theta} .$$

In the following questions, we determine and apply the linearization for several different functions. a. Find the linearization \(L(x, y)\) for the function \(f\) defined by \(f(x, y)=\) \(\cos (x)\left(2 e^{2 y}+e^{-2 y}\right)\) at the point \(\left(x_{0}, y_{0}\right)=(0,0) .\) Use the linearization to estimate the value of \(f(0.1,0.2)\). Compare your estimate to the actual value of \(f(0.1,0.2)\) b. The Heat Index, \(I,\) (measured in apparent degrees \(\mathrm{F}\) ) is a function of the actual temperature \(T\) outside (in degrees \(\mathrm{F}\) ) and the relative humidity \(H\) (measured as a percentage). A portion of the table which gives values for this function, \(I=I(T, H),\) is provided in Table 10.4 .13 $$\begin{array}{ccccc}\hline T \downarrow \backslash H \rightarrow & 70 & 75 & 80 & 85 \\ \hline 90 & 106 & 109 & 112 & 115 \\\\\hline 92 & 112 & 115 & 119 & 123 \\ \hline 94 & 118 & 122 & 127 & 132 \\\\\hline 96 & 125 & 130 & 135 & 141 \\\\\hline\end{array}$$ Suppose you are given that \(I_{T}(94,75)=3.75\) and \(I_{H}(94,75)=0.9\). Use this given information and one other value from the table to estimate the value of \(I(93.1,77)\) using the linearization at (94,75) . Using proper terminology and notation, explain your work and thinking. c. Just as we can find a local linearization for a differentiable function of two variables, we can do so for functions of three or more variables. By extending the concept of the local linearization from two to three variables, find the linearization of the function \(h(x, y, z)=e^{2 x}(y+\) \(z^{2}\) ) at the point \(\left(x_{0}, y_{0}, z_{0}\right)=(0,1,-2) .\) Then, use the linearization to estimate the value of \(h(-0.1,0.9,-1.8)\).

The temperature on a heated metal plate positioned in the first quadrant of the \(x y\) -plane is given by $$C(x, y)=25 e^{-(x-1)^{2}-(y-1)^{3}}$$ Assume that temperature is measured in degrees Celsius and that \(x\) and \(y\) are each measured in inches. a. Determine \(C_{x x}(x, y)\) and \(C_{y y}(x, y) .\) Do not do any additional work to algebraically simplify your results. b. Calculate \(C_{x x}(1.1,1.2) .\) Suppose that an ant is walking past the point (1.1,1.2) along the line \(y=1.2 .\) Write a sentence to explain the meaning of the value of \(C_{x x}(1.1,1.2)\), including units. c. Calculate \(C_{y y}(1.1,1.2) .\) Suppose instead that an ant is walking past the point (1.1,1.2) along the line \(x=1.1 .\) Write a sentence to explain the meaning of the value of \(C_{y y}(1.1,1.2),\) including units. d. Determine \(C_{x y}(x, y)\) and hence compute \(C_{x y}(1.1,1.2) .\) What is the meaning of this value? Explain, in terms of an ant walking on the heated metal plate.

The Heat Index, \(I,\) (measured in apparent degrees \(F)\) is a function of the actual temperature \(T\) outside (in degrees \(\mathrm{F}\) ) and the relative humidity \(H\) (measured as a percentage). A portion of the table which gives values for this function, \(I=I(T, H),\) is reproduced in Table \(10.2 .10 .\) $$\begin{array}{ccccc}\hline T \downarrow \backslash H \rightarrow & 70 & 75 & 80 & 85 \\ \hline 90 & 106 & 109 & 112 & 115 \\\\\hline 92 & 112 & 115 & 119 & 123 \\\\\hline 94 & 118 & 122 & 127 & 132 \\\\\hline 96 & 125 & 130 & 135 & 141 \\\\\hline\end{array}$$ a. State the limit definition of the value \(I_{T}(94,75)\). Then, estimate \(I_{T}(94,75),\) and write one complete sentence that carefully explains the meaning of this value, including its units. b. State the limit definition of the value \(I_{H}(94,75)\). Then, estimate \(I_{H}(94,75),\) and write one complete sentence that carefully explains the meaning of this value, including its units. c. Suppose you are given that \(I_{T}(92,80)=3.75\) and \(I_{H}(92,80)=0.8\). Estimate the values of \(I(91,80)\) and \(I(92,78)\). Explain how the partial derivatives are relevant to your thinking. d. On a certain day, at 1 p.m. the temperature is 92 degrees and the relative humidity is \(85 \%\). At 3 p.m., the temperature is 96 degrees and the relative humidity \(75 \% .\) What is the average rate of change of the heat index over this time period, and what are the units on your answer? Write a sentence to explain your thinking.

Consider the function \(h\) defined by \(h(x, y)=\frac{2 x^{2} y}{x^{4}+y^{2}}\). a. What is the domain of \(h ?\) b. Evaluate the limit of \(h\) at (0,0) along all linear paths the contain the origin. What does this tell us about \(\lim _{(x, y) \rightarrow(0,0)} h(x, y) ?\) (Hint: A non-vertical line throught the origin has the form \(y=m x\) for some constant \(m .)\) c. Does \(\lim _{(x, y) \rightarrow(0,0)} h(x, y)\) exist? Verify your answer. Check by using appropriate technology to sketch both surface and contour plots of \(h\) near \((0,0) .\) Write several sentences to say how your plots affirm your findings about \(\lim _{(x, y) \rightarrow(0,0)} h(x, y)\).

Apply the Method of Lagrange Multipliers solve each of the following constrained optimization problems. a. Determine the absolute maximum and absolute minimum values of \(f(x, y)=(x-1)^{2}+(y-2)^{2}\) subject to the constraint that \(x^{2}+y^{2}=\) 16 b. Determine the points on the sphere \(x^{2}+y^{2}+z^{2}=4\) that are closest to and farthest from the point (3,1,-1) . (As in the preceding exercise, you may find it simpler to work with the square of the distance formula, rather than the distance formula itself.) c. Find the absolute maximum and minimum of \(f(x, y, z)=x^{2}+y^{2}+z^{2}\) subject to the constraint that \((x-3)^{2}+(y+2)^{2}+(z-5)^{2} \leq 16\). (Hint: here the constraint is a closed, bounded region. Use the boundary of that region for applying Lagrange Multipliers, but don't forget to also test any critical values of the function that lie in the interior of the region.)

What is the shortest distance from the surface \(x y+6 x+z^{2}=41\) to the origin? distance \(=\) ___________.

Suppose that the temperature on a metal plate is given by the function \(T\) with $$T(x, y)=100-\left(x^{2}+4 y^{2}\right)$$ where the temperature is measured in degrees Fahrenheit and \(x\) and \(y\) are each measured in feet. Now suppose that an ant is walking on the metal plate in such a way that it walks in a straight line from the point (1,4) to the point (5,6) . a. Find parametric equations \((x(t), y(t))\) for the ant's coordinates as it walks the line from (1,4) to (5,6) b. What can you say about \(\frac{d x}{d t}\) and \(\frac{d y}{d t}\) for every value of \(t ?\) c. Determine the instantaneous rate of change in temperature with respect to \(t\) that the ant is experiencing at the moment it is halfway from (1,4) to \((5,6),\) using your parametric equations for \(x\) and \(y\). Include units on your answer.

In the following questions, we investigate two different applied settings using the differential. a. Let \(f\) represent the vertical displacement in centimeters from the rest position of a string (like a guitar string) as a function of the distance \(x\) in centimeters from the fixed left end of the string and \(y\) the time in seconds after the string has been plucked. A simple model for \(f\) could be $$f(x, y)=\cos (x) \sin (2 y)$$ Use the differential to approximate how much more this vibrating string is vertically displaced from its position at \((a, b)=\left(\frac{\pi}{4}, \frac{\pi}{3}\right)\) if we decrease \(a\) by \(0.01 \mathrm{~cm}\) and increase the time by 0.1 seconds. Compare to the value of \(f\) at the point \(\left(\frac{\pi}{4}-0.01, \frac{\pi}{3}+0.1\right)\). b. Resistors used in electrical circuits have colored bands painted on them to indicate the amount of resistance and the possible error in the resistance. When three resistors, whose resistances are \(R_{1}, R_{2},\) and \(R_{3},\) are connected in parallel, the total resistance \(R\) is given by $$\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}$$ Suppose that the resistances are \(R_{1}=25 \Omega, R_{2}=40 \Omega,\) and \(R_{3}=\) \(50 \Omega\). Find the total resistance \(R\). If you know each of \(R_{1}, R_{2}\), and \(R_{3}\) with a possible error of \(0.5 \%\), estimate the maximum error in your calculation of \(R\).

Let \(f(x, y)=8-x^{2}-y^{2}\) and \(g(x, y)=8-x^{2}+4 x y-y^{2}\). a. Determine \(f_{x}, f_{y}, f_{x x}, f_{y y}, f_{x y},\) and \(f_{y x}\) b. Evaluate each of the partial derivatives in (a) at the point (0,0) . c. What do the values in (b) suggest about the behavior of \(f\) near (0,0)\(?\) Plot a graph of \(f\) and compare what you see visually to what the values suggest. d. Determine \(g_{x}, g_{y}, g_{x x}, g_{y y}, g_{x y},\) and \(g_{y x}\) e. Evaluate each of the partial derivatives in (d) at the point (0,0) . f. What do the values in (e) suggest about the behavior of \(g\) near (0,0)\(?\) Plot a graph of \(g\) and compare what you see visually to what the values suggest. g. What do the functions \(f\) and \(g\) have in common at (0,0)\(?\) What is different? What do your observations tell you regarding the importance of a certain second-order partial derivative?

For each of the following prompts, provide an example of a function of two variables with the desired properties (with justification), or explain why such a function does not exist. a. A function \(p\) that is defined at \((0,0),\) but \(\lim _{(x, y) \rightarrow(0,0)} p(x, y)\) does not exist. b. A function \(q\) that does not have a limit at \((0,0),\) but that has the same limiting value along any line \(y=m x\) as \(x \rightarrow 0\). c. A function \(r\) that is continuous at \((0,0),\) but \(\lim _{(x, y) \rightarrow(0,0)} r(x, y)\) does not exist. d. A function \(s\) such that \(\lim _{(x, x) \rightarrow(0,0)} s(x, x)=3\) and \(\lim _{(x, 2 x) \rightarrow(0,0)} s(x, 2 x)=6\) for which \(\lim _{(x, y) \rightarrow(0,0)} s(x, y)\) exists. e. A function \(t\) that is not defined at (1,1) but \(\lim _{(x, y) \rightarrow(1,1)} t(x, y)\) does exist.

In this exercise we consider how to apply the Method of Lagrange Multipliers to optimize functions of three variable subject to two constraints. Suppose we want to optimize \(f=f(x, y, z)\) subject to the constraints \(g(x, y, z)=c\) and \(h(x, y, z)=k\). Also suppose that the two level surfaces \(g(x, y, z)=c\) and \(h(x, y, z)=k\) intersect at a curve \(C\). The optimum point \(P=\left(x_{0}, y_{0}, z_{0}\right)\) will then lie on \(C\). a. Assume that \(C\) can be represented parametrically by a vector-valued function \(\mathbf{r}=\mathbf{r}(t) .\) Let \(\overrightarrow{O P}=\mathbf{r}\left(t_{0}\right) .\) Use the Chain Rule applied to \(f(\mathbf{r}(t)), g(\mathbf{r}(t)),\) and \(h(\mathbf{r}(t)),\) to explain why $$\begin{array}{l} \nabla f\left(x_{0}, y_{0}, z_{0}\right) \cdot \mathbf{r}^{\prime}\left(t_{0}\right)=0 \\ \nabla g\left(x_{0}, y_{0}, z_{0}\right) \cdot \mathbf{r}^{\prime}\left(t_{0}\right)=0, \text { and } \\ \nabla h\left(x_{0}, y_{0}, z_{0}\right) \cdot \mathbf{r}^{\prime}\left(t_{0}\right)=0\end{array}$$ Explain how this shows that \(\nabla f\left(x_{0}, y_{0}, z_{0}\right), \nabla g\left(x_{0}, y_{0}, z_{0}\right),\) and \(\nabla h\left(x_{0}, y_{0}, z_{0}\right)\) are all orthogonal to \(C\) at \(P\). This shows that \(\nabla f\left(x_{0}, y_{0}, z_{0}\right), \nabla g\left(x_{0}, y_{0}, z_{0}\right),\) and \(\nabla h\left(x_{0}, y_{0}, z_{0}\right)\) all lie in the same plane. b. Assuming that \(\nabla g\left(x_{0}, y_{0}, z_{0}\right)\) and \(\nabla h\left(x_{0}, y_{0}, z_{0}\right)\) are nonzero and not parallel, explain why every point in the plane determined by \(\nabla g\left(x_{0}, y_{0}, z_{0}\right)\) and \(\nabla h\left(x_{0}, y_{0}, z_{0}\right)\) has the form \(s \nabla g\left(x_{0}, y_{0}, z_{0}\right)+t \nabla h\left(x_{0}, y_{0}, z_{0}\right)\) for some scalars \(s\) and \(t\). c. Parts (a.) and (b.) show that there must exist scalars \(\lambda\) and \(\mu\) such that $$\nabla f\left(x_{0}, y_{0}, z_{0}\right)=\lambda \nabla g\left(x_{0}, y_{0}, z_{0}\right)+\mu \nabla h\left(x_{0}, y_{0}, z_{0}\right)$$ So to optimize \(f=f(x, y, z)\) subject to the constraints \(g(x, y, z)=c\) and \(h(x, y, z)=k\) we must solve the system of equations $$\nabla f(x, y, z)=\lambda \nabla g(x, y, z)+\mu \nabla h(x, y, z)$$ $$\begin{array}{l}g(x, y, z)=c, \text { and } \\\h(x, y, z)=k .\end{array}$$ for \(x, y, z, \lambda,\) and \(\mu .\) Use this idea to find the maximum and minimum values of \(f(x, y, z)=\) \(x+2 y\) subject to the constraints \(y^{2}+z^{2}=8\) and \(x+y+z=10\).