Chapter 13

Consider the following functions \(f\) and points \(P .\) Sketch the \(x y\) -plane showing \(P\) and the level curve through \(P\). Indicate (as in Figure 70 ) the directions of maximum increase, maximum decrease, and no change for \(f\). $$f(x, y)=8+4 x^{2}+2 y^{2} ; P(2,-4)$$

Write the differential \(d w\) in terms of the differentials of the independent variables. $$w=f(x, y, z)=x y^{2}+z x^{2}+y z^{2}$$

A volume function The volume of a right circular cone of radius \(r\) and height \(h\) is \(V(r, h)=\pi r^{2} h / 3.\) a. Graph the function in the window \([0,5] \times[0,5] \times[0,150].\) b. What is the domain of the volume function? c. What is the relationship between the values of \(r\) and \(h\) when \(V=100 ?\)

Determine whether the following statements are true and give an explanation or counterexample. Assume all partial derivatives exist. a. If \(z=(x+y) \sin x y,\) where \(x\) and \(y\) are functions of \(s,\) then $$\frac{\partial z}{\partial s}=\frac{d z}{d x} \frac{d x}{d s}.$$ b. Given that \(w=f(x(s, t), y(s, t), z(s, t)),\) the rate of change of \(w\) with respect to \(t\) is \(d w / d t\).

Verify that \(f_{x y}=f_{y x}\) for the following functions. $$f(x, y)=e^{x+y}$$

Show that the Second Derivative Test is inconclusive when applied to the following functions at \((0,0) .\) Describe the behavior of the function at the critical point. $$f(x, y)=x^{2} y-3$$

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$f(x, y)=\left\\{\begin{array}{ll}\frac{y^{4}-2 x^{2}}{y^{4}+x^{2}} & \text { if }(x, y) \neq(0,0) \\\0 & \text { if }(x, y)=(0,0)\end{array}\right.$$

Consider the following cylinders in \(\mathbb{R}^{3}\). a. Identify the coordinate axis to which the cylinder is parallel. b. Sketch the cylinder. $$x^{2}+4 y^{2}=4$$

Consider the following functions \(f\) and points \(P .\) Sketch the \(x y\) -plane showing \(P\) and the level curve through \(P\). Indicate (as in Figure 70 ) the directions of maximum increase, maximum decrease, and no change for \(f\). $$f(x, y)=-4+6 x^{2}+3 y^{2} ; P(-1,-2)$$

Write the differential \(d w\) in terms of the differentials of the independent variables. $$w=f(x, y, z)=\sin (x+y-z)$$

Find the indicated derivative in two ways: a. Replace \(x\) and \(y\) to write \(z\) as a function of \(t\) and differentiate. b. Use the Chain Rule. \(z^{\prime}(t),\) where \(z=\ln (x+y), x=t e^{t},\) and \(y=e^{t}\)

Verify that \(f_{x y}=f_{y x}\) for the following functions. $$f(x, y)=\sqrt{x y}$$

Show that the Second Derivative Test is inconclusive when applied to the following functions at \((0,0) .\) Describe the behavior of the function at the critical point. $$f(x, y)=x^{4} y^{2}$$

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$f(x, y)=\sqrt{x^{2}+y^{2}}$$

Consider the following cylinders in \(\mathbb{R}^{3}\). a. Identify the coordinate axis to which the cylinder is parallel. b. Sketch the cylinder. $$x^{2}+z^{2}=4$$

Consider the following functions \(f\) and points \(P .\) Sketch the \(x y\) -plane showing \(P\) and the level curve through \(P\). Indicate (as in Figure 70 ) the directions of maximum increase, maximum decrease, and no change for \(f\). $$f(x, y)=x^{2}+x y+y^{2}+7 ; P(-3,3)$$

The electric potential function for two positive charges, one at (0,1) with twice the strength as the charge at \((0,-1),\) is given by $$\varphi(x, y)=\frac{2}{\sqrt{x^{2}+(y-1)^{2}}}+\frac{1}{\sqrt{x^{2}+(y+1)^{2}}}.$$ a. Graph the electric potential using the window \([-5,5] \times[-5,5] \times[0,10].\) b. For what values of \(x\) and \(y\) is the potential \(\varphi\) defined? c. Is the electric potential greater at (3,2) or (2,3)\(?\) d. Describe how the electric potential varies along the line \(y=x.\)

Find the indicated derivative in two ways: a. Replace \(x\) and \(y\) to write \(z\) as a function of \(t\) and differentiate. b. Use the Chain Rule. \(z^{\prime}(t),\) where \(z=\frac{1}{x}+\frac{1}{y}, x=t^{2}+2 t,\) and \(y=t^{3}-2\)

Find the first partial derivatives of the following functions. $$f(x, y, z)=x y+x z+y z$$

Show that the Second Derivative Test is inconclusive when applied to the following functions at \((0,0) .\) Describe the behavior of the function at the critical point. $$f(x, y)=\sin \left(x^{2} y^{2}\right)$$

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$f(x, y)=e^{x^{2}+y^{2}}$$

Cobb-Douglas production function The output \(Q\) of an economic system subject to two inputs, such as labor \(L\) and capital \(K\), is often modeled by the Cobb-Douglas production function \(Q(L, K)=c L^{a} K^{b},\) where \(a, b,\) and \(c\) are positive real numbers. When \(a+b=1,\) the case is called constant returns to scale. Suppose \(a=\frac{1}{3}, b=\frac{2}{3},\) and \(c=40.\) a. Graph the output function using the window \([0,20] \times[0,20] \times[0,500].\) b. If \(L\) is held constant at \(L=10,\) write the function that gives the dependence of \(Q\) on \(K.\) c. If \(K\) is held constant at \(K=15,\) write the function that gives the dependence of \(Q\) on \(L.\)

Consider the following functions \(f\) and points \(P .\) Sketch the \(x y\) -plane showing \(P\) and the level curve through \(P\). Indicate (as in Figure 70 ) the directions of maximum increase, maximum decrease, and no change for \(f\). $$f(x, y)=\tan (2 x+2 y) ; P(\pi / 16, \pi / 16)$$

Find the indicated derivative for the following functions. \(\partial z / \partial p,\) where \(z=x / y, x=p+q,\) and \(y=p-q\)

Find the first partial derivatives of the following functions. $$g(x, y, z)=2 x^{2} y-3 x z^{4}+10 y^{2} z^{2}$$

Find the absolute maximum and minimum values of the following functions on the given set \(R\). $$f(x, y)=x^{2}+y^{2}-2 y+1 ; R=\left\\{(x, y): x^{2}+y^{2} \leq 4\right\\}$$

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$f(x, y)=\sin x y$$

Resistors in parallel Two resistors wired in parallel in an electrical circuit give an effective resistance of \(R(x, y)=\frac{x y}{x+y},\) where \(x\) and \(y\) are the positive resistances of the individual resistors (typically measured in ohms).a. Graph the resistance function using the window \([0,10] \times[0,10] \times[0,5].\) b. Estimate the maximum value of \(R\), for \(0

Law of cosines The side lengths of any triangle are related by the Law of Cosines, $$c^{2}=a^{2}+b^{2}-2 a b \cos \theta$$ a. Estimate the change in the side length \(c\) when \(a\) changes from \(a=2\) to \(a=2.03, b\) changes from \(b=4.00\) to \(b=3.96\) and \(\theta\) changes from \(\theta=\pi / 3\) to \(\theta=\pi / 3+\pi / 90\) b. If \(a\) changes from \(a=2\) to \(a=2.03\) and \(b\) changes from \(b=4.00\) to \(b=3.96,\) is the resulting change in \(c\) greater in magnitude when \(\theta=\pi / 20\) (small angle) or when \(\theta=9 \pi / 20\) (close to a right angle)?

Find the indicated derivative for the following functions. \(d w / d t,\) where \(w=x y z, x=2 t^{4}, y=3 t^{-1},\) and \(z=4 t^{-3}\)

Find the first partial derivatives of the following functions. $$h(x, y, z)=\cos (x+y+z)$$

Find the absolute maximum and minimum values of the following functions on the given set \(R\). $$f(x, y)=2 x^{2}+y^{2} ; R=\left\\{(x, y): x^{2}+y^{2} \leq 16\right\\}$$

A snapshot of a water wave moving toward shore is described by the function \(z=10 \sin (2 x-3 y),\) where \(z\) is the height of the water surface above (or below) the \(x y\) -plane, which is the level of undisturbed water. a. Graph the height function using the window \([-5,5] \times[-5,5] \times[-15,15].\) b. For what values of \(x\) and \(y\) is \(z\) defined? c. What are the maximum and minimum values of the water height? d. Give a vector in the \(x y\) -plane that is orthogonal to the level curves of the crests and troughs of the wave (which is parallel to the direction of wave propagation).

Consider the paraboloid \(f(x, y)=\) \(16-x^{2} / 4-y^{2} / 16\) and the point \(P\) on the given level curve of \(f\) Compute the slope of the line tangent to the level curve at \(P\) and verify that the tangent line is orthogonal to the gradient at that point. $$f(x, y)=0 ; P(8,0)$$

Travel cost The cost of a trip that is \(L\) miles long, driving a car that gets \(m\) miles per gallon, with gas costs of \(\$ p /\) gal is \(C=L p / m\) dollars. Suppose you plan a trip of \(L=1500 \mathrm{mi}\) in a car that gets \(m=32 \mathrm{mi} / \mathrm{gal},\) with gas costs of \(p=\$ 3.80 / \mathrm{gal}\) a. Explain how the cost function is derived. b. Compute the partial derivatives \(C_{L}, C_{m},\) and \(C_{p} .\) Explain the meaning of the signs of the derivatives in the context of this problem. c. Estimate the change in the total cost of the trip if \(L\) changes from \(L=1500\) to \(L=1520, m\) changes from \(m=32\) to 31 and \(p\) changes from \(\$ 3.80\) to \(\$ 3.85\) d. Is the total cost of the trip (with \(L=1500 \mathrm{mi}\) \(m=32 \mathrm{mi} / \mathrm{gal},\) and \(p=\$ 3.80\) ) more sensitive to a \(1 \%\) change in \(L, m,\) or \(p\) (assuming the other two variables are fixed)? Explain.

Find the indicated derivative for the following functions. \(\partial w / \partial x,\) where \(w=\cos z-\cos x \cos y+\sin x \sin y,\) and \(z=x+y\)

Find the first partial derivatives of the following functions. $$Q(x, y, z)=\tan x y z$$

Find the absolute maximum and minimum values of the following functions on the given set \(R\). $$\begin{aligned} &f(x, y)=4+2 x^{2}+y^{2}\\\ &R=\\{(x, y):-1 \leq x \leq 1,-1 \leq y \leq 1\\} \end{aligned}$$

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$h(x, y)=\cos (x+y)$$

Approximate mountains Suppose the elevation of Earth's surface over a \(16-\mathrm{mi}\) by \(16-\mathrm{mi}\) region is approximated by the function $$z=10 e^{-\left(x^{2}+y^{2}\right)}+5 e^{-\left((x+5)^{2}+(y-3)^{2}\right) / 10}+4 e^{-2\left((x-4)^{2}+(y+1)^{2}\right)}.$$ a. Graph the height function using the window \([-8,8] \times[-8,8] \times[0,15].\) b. Approximate the points \((x, y)\) where the peaks in the landscape appear. c. What are the approximate elevations of the peaks?

Consider the paraboloid \(f(x, y)=\) \(16-x^{2} / 4-y^{2} / 16\) and the point \(P\) on the given level curve of \(f\) Compute the slope of the line tangent to the level curve at \(P\) and verify that the tangent line is orthogonal to the gradient at that point. $$f(x, y)=12 ; P(4,0)$$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The planes tangent to the cylinder \(x^{2}+z^{2}=1\) in \(\mathbb{R}^{3}\) all have the form \(a x+b z+c=0\) b. Suppose \(w=x y / z,\) for \(x>0, y>0,\) and \(z>0 .\) A decrease in \(z\) with \(x\) and \(y\) fixed results in an increase in \(w\) c. The gradient \(\nabla F(a, b, c)\) lies in the plane tangent to the surface \(F(x, y, z)=0\) at \((a, b, c)\)

Find the indicated derivative for the following functions. \(\frac{\partial z}{\partial x},\) where \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1\)

Find the first partial derivatives of the following functions. $$F(u, v, w)=\frac{u}{v+w}$$

Absolute maximum and minimum values Find the absolute maximum and minimum values of the following functions over the given regions \(R\). Use Lagrange multipliers to check for extreme points on the boundary. $$f(x, y)=x^{2}+4 y^{2}+1 ; R=\left\\{(x, y): x^{2}+4 y^{2} \leq 1\right\\}$$

Find the absolute maximum and minimum values of the following functions on the given set \(R\). $$\begin{aligned} &f(x, y)=6-x^{2}-4 y^{2}\\\ &R=\\{(x, y):-2 \leq x \leq 2,-1 \leq y \leq 1\\} \end{aligned}$$

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$p(x, y)=e^{x-y}$$

Find the domain of the following functions. If possible, give a description of the domains (for example, all points outside a sphere of radius 1 centered at the origin ).$$f(x, y, z)=2 x y z-3 x z+4 y z.$$

Consider the paraboloid \(f(x, y)=\) \(16-x^{2} / 4-y^{2} / 16\) and the point \(P\) on the given level curve of \(f\) Compute the slope of the line tangent to the level curve at \(P\) and verify that the tangent line is orthogonal to the gradient at that point. $$f(x, y)=12 ; P(2 \sqrt{3}, 4)$$

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