Chapter 13

Find the first partial derivatives of the following functions. $$G(r, s, t)=\sqrt{r s+r t+s t}$$

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. $$f(x, y)=x^{2}-4 y^{2}+x y ; R=\left\\{(x, y): 4 x^{2}+9 y^{2} \leq 36\right\\}$$

Find the absolute maximum and minimum values of the following functions on the given set \(R\). $$\begin{aligned} &f(x, y)=2 x^{2}-4 x+3 y^{2}+2\\\ &R=\left\\{(x, y):(x-1)^{2}+y^{2} \leq 1\right\\} \end{aligned}$$

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

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. $$x^{2}+\frac{y^{2}}{4}+\frac{z^{2}}{9}=1$$

Consider the upper half of the ellipsoid \(f(x, y)=\sqrt{1-\frac{x^{2}}{4}-\frac{y^{2}}{16}}\) 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. $$f(x, y)=\sqrt{3} / 2 ; P(1 / 2, \sqrt{3})$$

Find an equation of the plane tangent to the following surfaces at the given point. $$z=\tan ^{-1}(x y) ;(1,1, \pi / 4)$$

Suppose \(w=f(x, y, z)\) and \(\ell\) is the line \(\mathbf{r}(t)=\langle a t, b t, c t\rangle,\) for \(-\infty

Find the first partial derivatives of the following functions. $$f(w, x, y, z)=w^{2} x y^{2}+x y^{3} z^{2}$$

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. $$f(x, y)=2 x^{2}+y^{2}+2 x-3 y ; R=\left\\{(x, y): x^{2}+y^{2} \leq 1\right\\}$$

Find the absolute maximum and minimum values of the following functions on the given set \(R\). \(f(x, y)=x^{2}+y^{2}-2 x-2 y ; R\) is the closed set bounded by the triangle with vertices \((0,0),(2,0),\) and (0,2)

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

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. $$4 x^{2}+y^{2}+\frac{z^{2}}{2}=1$$

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 ).$$p(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}-9}.$$

Consider the upper half of the ellipsoid \(f(x, y)=\sqrt{1-\frac{x^{2}}{4}-\frac{y^{2}}{16}}\) 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. $$f(x, y)=1 / \sqrt{2} ; P(0, \sqrt{8})$$

Assume that \(F(x, y, z(x, y))=0\) implicitly defines \(z\) as a differentiable function of \(x\) and \(y .\) Extend Theorem 9 to show that \(\frac{\partial z}{\partial x}=-\frac{F_{x}}{F_{z}}\) and \(\frac{\partial z}{\partial y}=-\frac{F_{y}}{F_{z}}\).

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

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. $$f(x, y)=(x-1)^{2}+(y+1)^{2} ; R=\left\\{(x, y): x^{2}+y^{2} \leq 4\right\\}$$

Find the absolute maximum and minimum values of the following functions on the given set \(R\). $$\begin{aligned} &f(x, y)=-2 x^{2}+4 x-3 y^{2}-6 y-1\\\ &R=\left\\{(x, y):(x-1)^{2}+(y+1)^{2} \leq 1\right\\} \end{aligned}$$

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$g(x, y)=\sqrt[3]{x^{2}+y^{2}-9}$$

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. $$\frac{x^{2}}{3}+3 y^{2}+\frac{z^{2}}{12}=3$$

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 ).$$f(x, y, z)=\sqrt{y-z}.$$

Consider the upper half of the ellipsoid \(f(x, y)=\sqrt{1-\frac{x^{2}}{4}-\frac{y^{2}}{16}}\) 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. $$f(x, y)=1 / \sqrt{2} ; P(\sqrt{2}, 0)$$

Find an equation of the plane tangent to the following surfaces at the given point. $$\sin x y z=\frac{1}{2} ;\left(\pi, 1, \frac{1}{6}\right)$$

Find the first partial derivatives of the following functions. $$h(w, x, y, z)=\frac{w z}{x y}$$

Find the absolute maximum and minimum values of the following functions on the given set \(R\). \(f(x, y)=\sqrt{x^{2}+y^{2}-2 x+2} ; R\) is the closed half disk \(\left\\{(x, y): x^{2}+y^{2} \leq 4 \text { with } y \geq 0\right\\}\)

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

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 ).$$Q(x, y, z)=\frac{10}{1+x^{2}+y^{2}+4 z^{2}}.$$

Consider the upper half of the ellipsoid \(f(x, y)=\sqrt{1-\frac{x^{2}}{4}-\frac{y^{2}}{16}}\) 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. $$f(x, y)=1 / \sqrt{2} ; P(1,2)$$

Find the points at which the following surfaces have horizontal tangent planes. $$z=\sin (x-y) \text { in the region }-2 \pi \leq x \leq 2 \pi,-2 \pi \leq y \leq 2 \pi$$

Find the first partial derivatives of the following functions. $$F(w, x, y, z)=w \sqrt{x+2 y+3 z}$$

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

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$f(x, y)=\left\\{\begin{array}{ll}\frac{\sin \left(x^{2}+y^{2}\right)}{x^{2}+y^{2}} & \text { if }(x, y) \neq(0,0) \\\1 & \text { if }(x, y)=(0,0)\end{array}\right.$$

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. $$x=y^{2}+z^{2}$$

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 ).$$F(x, y, z)=\sqrt{y-x^{2}}.$$

Find the points at which the following surfaces have horizontal tangent planes. $$x^{2}+y^{2}-z^{2}-2 x+2 y+3=0$$

Use the result of Exercise 48 to evaluate \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) for the following relations. $$x y z+x+y-z=0$$

Consider the Ideal Gas Law \(P V=k T\), where \(k>0\) is a constant. Solve this equation for \(V\) in terms of \(P\) and \(T\) a. Determine the rate of change of the volume with respect to the pressure at constant temperature. Interpret the result. b. Determine the rate of change of the volume with respect to the temperature at constant pressure. Interpret the result. c. Assuming \(k=1,\) draw several level curves of the volume function and interpret the results as in Example 5

If possible, find the absolute maximum and minimum values of the following functions on the set \(R\). $$f(x, y)=x+3 y ; R=\\{(x, y):|x|<1,|y|<2\\}$$

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$f(x, y)=\left\\{\begin{array}{ll}\frac{1-\cos \left(x^{2}+y^{2}\right)}{x^{2}+y^{2}} & \text { if }(x, y)\neq(0,0)\\\0 & \text { if }(x, y)=(0,0)\end{array}\right.$$

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. $$z=\frac{x^{2}}{4}+\frac{y^{2}}{9}$$

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 ).$$f(w, x, y, z)=\sqrt{1-w^{2}-x^{2}-y^{2}-z^{2}}.$$

Find the points at which the following surfaces have horizontal tangent planes. $$x^{2}+2 y^{2}+z^{2}-2 x-2 z-2=0$$

A box with a square base of length \(x\) and height \(h\) has a volume \(V=x^{2} h\) a. Compute the partial derivatives \(V_{x}\) and \(V_{h}\) b. For a box with \(h=1.5 \mathrm{m},\) use linear approximation to estimate the change in volume if \(x\) increases from \(x=0.5 \mathrm{m}\) to \(x=0.51 \mathrm{m}\) c. For a box with \(x=0.5 \mathrm{m},\) use linear approximation to estimate the change in volume if \(h\) decreases from \(h=1.5 \mathrm{m}\) to \(h=1.49 \mathrm{m}\) d. For a fixed height, does a \(10 \%\) change in \(x\) always produce (approximately) a \(10 \%\) change in \(V\) ? Explain. e. For a fixed base length, does a \(10 \%\) change in \(h\) always produce (approximately) a \(10 \%\) change in \(V\) ? Explain.

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 \(R\). $$f(x, y)=2 e^{-x-y} ; R=\\{(x, y): x \geq 0, y \geq 0\\}$$

Evaluate the following limits. $$\lim _{(x, y, z) \rightarrow(1, \ln 2,3)} z e^{x y}$$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The domain of the function \(f(x, y)=1-|x-y|\) is \(\\{(x, y): x \geq y\\}.\) b. The domain of the function \(Q=g(w, x, y, z)\) is a region in \(\mathbb{R}^{3}.\) c. All level curves of the plane \(z=2 x-3 y\) are lines.

Find the points at which the following surfaces have horizontal tangent planes. $$z=\cos 2 x \sin y \text { in the region }-\pi \leq x \leq \pi,-\pi \leq y \leq \pi$$

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). $$z=x^{2}+4 y^{2}+1, C: x=\cos t, y=\sin t ; 0 \leq t \leq 2 \pi$$