Chapter 13

Use differentials to approximate the change in \(z\) for the given changes in the independent variables. \(z=-x^{2}+3 y^{2}+2\) when \((x, y)\) changes from (-1,2) to (-1.05,1.9)

Graph several level curves of the following functions using the given window. Label at least two level curves with their \(z\) -values. $$z=2 x-y ;[-2,2] \times[-2,2].$$

Given the following equations, evaluate \(d y / d x .\) Assume that each equation implicitly defines \(y\) as a differentiable function of \(x\). $$x^{3}+3 x y^{2}-y^{5}=0$$

Find the four second partial derivatives of the following functions. $$Q(r, s)=r / s$$

Applications of Lagrange multipliers Use Lagrange multipliers in the following problems. When the domain of the objective function is unbounded or open, explain why you have found an absolute maximum or minimum value. Extreme distances to a sphere Find the minimum and maximum distances between the sphere \(x^{2}+y^{2}+z^{2}=9\) and the point (2,3,4).

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$f(x, y)=x^{2}+2 x y-y^{3}$$

Find an equation of the plane parallel to the plane \(Q\) passing through the point \(P_{0}\). $$Q: 4 x+3 y-2 z=12 ; P_{0}(1,-1,3)$$

Interpreting directional derivatives A function \(f\) and a point \(P\) are given. Let \(\theta\) correspond to the direction of the directional derivative. a. Find the gradient and evaluate it at \(P\). b. Find the angles \(\theta\) (with respect to the positive \(x\) -axis) associated with the directions of maximum increase, maximum decrease, and zero change. c. Write the directional derivative at \(P\) as a function of \(\theta ;\) call this function \(g(\theta)\) d. Find the value of \(\theta\) that maximizes \(g(\theta)\) and find the maximum value. e. Verify that the value of \(\theta\) that maximizes \(g\) corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient. $$f(x, y)=10-2 x^{2}-3 y^{2} ; P(3,2)$$

Use differentials to approximate the change in \(z\) for the given changes in the independent variables. \(z=e^{x+y}\) when \((x, y)\) changes from (0,0) to (0.1,-0.05)

Graph several level curves of the following functions using the given window. Label at least two level curves with their \(z\) -values. $$z=\sqrt{x^{2}+4 y^{2}} ;[-8,8] \times[-8,8].$$

Given the following equations, evaluate \(d y / d x .\) Assume that each equation implicitly defines \(y\) as a differentiable function of \(x\). $$2 \sin x y=1$$

Find the four second partial derivatives of the following functions. $$F(r, s)=r e^{s}$$

Applications of Lagrange multipliers Use Lagrange multipliers in the following problems. When the domain of the objective function is unbounded or open, explain why you have found an absolute maximum or minimum value. Maximum volume cylinder in a sphere Find the dimensions of a right circular cylinder of maximum volume that can be inscribed in a sphere of radius 16.

Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. Confirm your results using a graphing utility. $$f(x, y)=\sin (2 \pi x) \cos (\pi y), \text { for }|x| \leq \frac{1}{2} \text { and }|y| \leq \frac{1}{2}$$

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$f(x, y)=\frac{x y}{x^{2} y^{2}+1}$$

Find an equation of the plane parallel to the plane \(Q\) passing through the point \(P_{0}\). $$Q: x-5 y-2 z=1 ; P_{0}(1,2,0)$$

Interpreting directional derivatives A function \(f\) and a point \(P\) are given. Let \(\theta\) correspond to the direction of the directional derivative. a. Find the gradient and evaluate it at \(P\). b. Find the angles \(\theta\) (with respect to the positive \(x\) -axis) associated with the directions of maximum increase, maximum decrease, and zero change. c. Write the directional derivative at \(P\) as a function of \(\theta ;\) call this function \(g(\theta)\) d. Find the value of \(\theta\) that maximizes \(g(\theta)\) and find the maximum value. e. Verify that the value of \(\theta\) that maximizes \(g\) corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient. $$f(x, y)=8+x^{2}+3 y^{2} ; P(-3,-1)$$

Use differentials to approximate the change in \(z\) for the given changes in the independent variables. \(z=\ln (1+x+y)\) when \((x, y)\) changes from (0,0) to (-0.1,0.03)

Graph several level curves of the following functions using the given window. Label at least two level curves with their \(z\) -values. $$z=e^{-x^{2}-2 y^{2}} ;[-2,2] \times[-2,2].$$

Given the following equations, evaluate \(d y / d x .\) Assume that each equation implicitly defines \(y\) as a differentiable function of \(x\). $$y e^{x y}-2=0$$

Find the four second partial derivatives of the following functions. $$H(x, y)=\sqrt{4+x^{2}+y^{2}}$$

A shipping company handles rectangular boxes provided the sum of the height and the girth of the box does not exceed 96 in. (The girth is the perimeter of the smallest base of the box.) Find the dimensions of the box that meets this condition and has the largest volume.

Find an equation of the line of intersection of the planes \(Q\) and \(R\). $$Q:-x+2 y+z=1 ; R: x+y+z=0$$

Changes in torus surface area The surface area of a torus (an ideal bagel or doughnut) with an inner radius \(r\) and an outer radius \(R>r\) is \(S=4 \pi^{2}\left(R^{2}-r^{2}\right)\) a. If \(r\) increases and \(R\) decreases, does \(S\) increase or decrease, or is it impossible to say? b. If \(r\) increases and \(R\) increases, does \(S\) increase or decrease, or is it impossible to say? c. Estimate the change in the surface area of the torus when \(r\) changes from \(r=3.00\) to \(r=3.05\) and \(R\) changes from \(R=5.50\) to \(R=5.65\) d. Estimate the change in the surface area of the torus when \(r\) changes from \(r=3.00\) to \(r=2.95\) and \(R\) changes from \(R=7.00\) to \(R=7.04\) e. Find the relationship between the changes in \(r\) and \(R\) that leaves the surface area (approximately) unchanged.

Graph several level curves of the following functions using the given window. Label at least two level curves with their \(z\) -values. $$z=\sqrt{25-x^{2}-y^{2}} ;[-6,6] \times[-6,6].$$

Given the following equations, evaluate \(d y / d x .\) Assume that each equation implicitly defines \(y\) as a differentiable function of \(x\). $$\sqrt{x^{2}+2 x y+y^{4}}=3$$

Verify that \(f_{x y}=f_{y x}\) for the following functions. $$f(x, y)=2 x^{3}+3 y^{2}+1$$

Maximizing utility functions Find the values of \(\ell\) and \(g\) with \(\ell \geq 0\) and \(g \geq 0\) that maximize the following utility functions subject to the given constraints. Give the value of the utility function at the optimal point. $$U=f(\ell, g)=32 \ell^{2 / 3} g^{1 / 3} \text { subject to } 4 \ell+2 g=12$$

A lidless box is to be made using \(2 \mathrm{m}^{2}\) of cardboard. Find the dimensions of the box with the largest possible volume.

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$S(x, y)=\frac{4 x^{2} y^{2}}{x^{2}+y^{2}}$$

Find an equation of the line of intersection of the planes \(Q\) and \(R\). $$Q: x+2 y-z=1 ; R: x+y+z=1$$

Changes in cone volume The volume of a right circular cone with radius \(r\) and height \(h\) is \(V=\pi r^{2} h / 3\) a. Approximate the change in the volume of the cone when the radius changes from \(r=6.5\) to \(r=6.6\) and the height changes from \(h=4.20\) to \(h=4.15\) b. Approximate the change in the volume of the cone when the radius changes from \(r=5.40\) to \(r=5.37\) and the height changes from \(h=12.0\) to \(h=11.96\)

Graph several level curves of the following functions using the given window. Label at least two level curves with their \(z\) -values. $$z=\sqrt{y-x^{2}-1} ;[-5,5] \times[-5,5].$$

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

Maximizing utility functions Find the values of \(\ell\) and \(g\) with \(\ell \geq 0\) and \(g \geq 0\) that maximize the following utility functions subject to the given constraints. Give the value of the utility function at the optimal point. $$U=f(\ell, g)=8 \ell^{4 / 5} g^{1 / 5} \text { subject to } 10 \ell+8 g=40$$

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$f(x, y)=\frac{2}{x\left(y^{2}+1\right)}$$

Find an equation of the line of intersection of the planes \(Q\) and \(R\). $$Q: 2 x-y+3 z-1=0 ; R:-x+3 y+z-4=0$$

Interpreting directional derivatives A function \(f\) and a point \(P\) are given. Let \(\theta\) correspond to the direction of the directional derivative. a. Find the gradient and evaluate it at \(P\). b. Find the angles \(\theta\) (with respect to the positive \(x\) -axis) associated with the directions of maximum increase, maximum decrease, and zero change. c. Write the directional derivative at \(P\) as a function of \(\theta ;\) call this function \(g(\theta)\) d. Find the value of \(\theta\) that maximizes \(g(\theta)\) and find the maximum value. e. Verify that the value of \(\theta\) that maximizes \(g\) corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient. $$f(x, y)=e^{-x^{2}-2 y^{2}} ; P(-1,0)$$

Area of an ellipse The area of an ellipse with axes of length \(2 a\) and \(2 b\) is \(A=\pi a b\). Approximate the percent change in the area when \(a\) increases by \(2 \%\) and \(b\) increases by \(1.5 \%\)

Graph several level curves of the following functions using the given window. Label at least two level curves with their \(z\) -values. $$z=3 \cos (2 x+y) ;[-2,2] \times[-2,2].$$

The \(x\) - and \(y\) -components of a fluid moving in two dimensions are given by the following functions u and \(v\). The speed of the fluid at \((x, y)\) is \(s(x, y)=\sqrt{u(x, y)^{2}+v(x, y)^{2}}\). Use the Chain Rule to find \(\partial s / \partial x\) and \(\partial s / \partial y\). \(u(x, y)=2 y\) and \(v(x, y)=-2 x ; x \geq 0\) and \(y \geq 0\)

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

Maximizing utility functions Find the values of \(\ell\) and \(g\) with \(\ell \geq 0\) and \(g \geq 0\) that maximize the following utility functions subject to the given constraints. Give the value of the utility function at the optimal point. $$U=f(\ell, g)=\ell^{1 / 6} g^{5 / 6} \text { subject to } 4 \ell+5 g=20$$

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$f(x, y)=\frac{x^{2}+y^{2}}{x\left(y^{2}-1\right)}$$

Find an equation of the line of intersection of the planes \(Q\) and \(R\). $$Q: x-y-2 z=1 ; R: x+y+z=-1$$

Volume of a paraboloid The volume of a segment of a circular paraboloid (see figure) with radius \(r\) and height \(h\) is \(V=\pi r^{2} h / 2\) Approximate the percent change in the volume when the radius decreases by \(1.5 \%\) and the height increases by \(2.2 \%\)

Verify that \(f_{x y}=f_{y x}\) for the following functions. $$f(x, y)=3 x^{2} y^{-1}-2 x^{-1} y^{2}$$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. Suppose you are standing at the center of a sphere looking at a point \(P\) on the surface of the sphere. Your line of sight to \(P\) is orthogonal to the plane tangent to the sphere at \(P\). b. At a point that maximizes \(f\) on the curve \(g(x, y)=0,\) the dot product \(\nabla f \cdot \nabla g\) is zero.

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. $$f(x, y)=4+x^{4}+3 y^{4}$$

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