Chapter 14

\(1-6\) Use the Chain Rule to find \(d z / d t\) or \(d w / d t\) $$z=x^{2}+y^{2}+x y, \quad x=\sin t, \quad y=e^{t}$$

The temperature \(T\) at a location in the Northern Hemisphere depends on the longitude \(x,\) latitude \(y,\) and time \(t,\) so we can write \(T=f(x, y, t) .\) Let's measure time in hours from the beginning of January. (a) What are the meanings of the partial derivatives \(\partial T / \partial x\) \(\partial T / \partial y,\) and \(\partial T / \partial t ?\) (b) Honolulu has longitude \(158^{\circ} \mathrm{W}\) and latitude \(21^{\circ} \mathrm{N}\) . Suppose that at \(9 : 00\) AM on January 1 the wind is blowing hot air to the northeast, so the air to the west and south is warm and the air to the north and east is cooler. Would you expect \(f_{x}(158,21,9), f_{1}(158,21,9),\) and \(f_{l}(158,21,9)\) to be positive or negative? Explain.

Suppose that \(\lim _{(x, y \rightarrow(3,1)} f(x, y)=6 .\) What can you say about the value of \(f(3,1) ?\) What if \(f\) is continuous?

I-6 Find an equation of the tangent plane to the given surface at the specified point. $$ z=4 x^{2}-y^{2}+2 y, \quad(-1,2,4) $$

Suppose \((1,1)\) is a critical point of a function \(f\) with contin- uous second derivatives. In each case, what can you say about \(f ?\) $$\begin{array}{ll}{\text { (a) } f_{x x}(1,1)=4,} & {f_{x y}(1,1)=1, \quad f_{y}(1,1)=2} \\ {\text { (b) } f_{x x}(1,1)=4,} & {f_{x y}(1,1)=3, \quad f_{y}(1,1)=2}\end{array}$$

Level curves for barometric pressure (in millibars) are shown for \(6 : 00\) AM on November \(10,1998 .\) A deep low with pressure 972 \(\mathrm{mb}\) is moving over northeast Iowa. The distance along the red line from K (Kearney, Nebraska) to S (Sioux City, Iowa) is 300 \(\mathrm{km} .\) Estimate the value of the directional derivative of the pressure function at Kearney in the direction of Sioux City. What are the units of the directional derivative?

(a) Use a graphing calculator or computer to graph the circle \(x^{2}+y^{2}=1 .\) On the same screen, graph several curves of the form \(x^{2}+y=c\) until you find two that just touch the circle. What is the significance of the values of \(c\) for these two curves? (b) Use Lagrange multipliers to find the extreme values of \(f(x, y)=x^{2}+y\) subject to the constraint \(x^{2}+y^{2}=1\) Compare your answers with those in part (a).

\(1-6\) Use the Chain Rule to find \(d z / d t\) or \(d w / d t\) $$z=\cos (x+4 y), \quad x=5 t^{4}, \quad y=1 / t$$

Explain why each function is continuous or discontinuous. (a) The outdoor temperature as a function of longitude, latitude, and time (b) Elevation (height above sea level) as a function of longitude, latitude, and time (c) The cost of a taxi ride as a function of distance traveled and time

I-6 Find an equation of the tangent plane to the given surface at the specified point. $$ z=3(x-1)^{2}+2(y+3)^{2}+7, \quad(2,-2,12) $$

Suppose \((0,2)\) is a critical point of a function \(g\) with continuous second derivatives. In each case, what can you say about \(g\) ? (a) $$g_{x x}(0,2)=-1, \quad g_{x},(0,2)=6, \quad g_{y}(0,2)=1$$ (b) $$g_{x x}(0,2)=-1, \quad g_{x},(0,2)=2, \quad g_{y}(0,2)=-8$$ (c) $$g_{x x}(0,2)=4, \quad g_{x y}(0,2)=6, \quad g_{y y}(0,2)=9$$

Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s). \(f(x, y)=x^{2}+y^{2} ; \quad x y=1\)

\(1-6\) Use the Chain Rule to find \(d z / d t\) or \(d w / d t\) $$z=\sqrt{1+x^{2}+y^{2}}, \quad x=\ln t, \quad y=\cos t$$

The wind-chill index \(W\) is the perceived temperature when the actual temperature is \(T\) and the wind speed is \(v,\) so we can write \(W=f(T, v) .\) The following table of values is an excerpt from Table 1 in Section \(14.1 .\) (a) Estimate the values of \(f_{T}(-15,30)\) and \(f_{i}(-15,30) .\) What are the practical interpretations of these values? (b) In general, what can you say about the signs of \(\partial W / \partial T\) \(\quad\) and \(\partial W / \partial v ?\) (c) What appears to be the value of the following limit? $$\lim _{v \rightarrow \infty} \frac{\partial W}{\partial v}$$

I-6 Find an equation of the tangent plane to the given surface at the specified point. $$ z=\sqrt{x y}, \quad(1,1,1) $$

Verify for the Cobb-Douglas production function $$P(L, K)=1.01 L^{0.75} K^{0.25}$$ discussed in Example 3 that the production will be doubled if both the amount of labor and the amount of capital are doubled. Determine whether this is also true for the general production function $$P(L, K)=b L^{\alpha} K^{1-\alpha}$$

Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s). \(f(x, y)=4 x+6 y, \quad x^{2}+y^{2}=13\)

\(1-6\) Use the Chain Rule to find \(d z / d t\) or \(d w / d t\) $$z=\tan ^{-1}(y / x), \quad x=e^{t}, \quad y=1-e^{-t}$$

\(4-6\) Find the directional derivative of \(f\) at the given point in the direction indicated by the angle \(\theta .\) $$f(x, y)=x^{2} y^{3}-y^{4}, \quad(2,1), \quad \theta=\pi / 4$$

I-6 Find an equation of the tangent plane to the given surface at the specified point. $$ z=y \ln x, \quad(1,4,0) $$

\(3-4\) Use a table of numerical values of \(f(x, y)\) for \((x, y)\) near the origin to make a conjecture about the value of the limit of \(f(x, y)\) as \((x, y) \rightarrow(0,0) .\) Then explain why your guess is correct. $$f(x, y)=\frac{2 x y}{x^{2}+2 y^{2}}$$

Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s). \(f(x, y)=x^{2} y ; \quad x^{2}+2 y^{2}=6\)

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. $$f(x, y)=9-2 x+4 y-x^{2}-4 y^{2}$$

\(1-6\) Use the Chain Rule to find \(d z / d t\) or \(d w / d t\) $$w=x e^{y_{2}}, \quad x=t^{2}, \quad y=1-t, \quad z=1+2 t$$

\(4-6\) Find the directional derivative of \(f\) at the given point in the direction indicated by the angle \(\theta\) . $$f(x, y)=y e^{-x}, \quad(0,4), \quad \theta=2 \pi / 3$$

\(5-22\) Find the limit, if it exists, or show that the limit does not exist. $$\lim _{(x, y) \rightarrow(1,2)}\left(5 x^{3}-x^{2} y^{2}\right)$$

I-6 Find an equation of the tangent plane to the given surface at the specified point. $$ z=y \cos (x-y), \quad(2,2,2) $$

Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s). \(f(x, y)=e^{x y} ; \quad x^{3}+y^{3}=16\)

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. $$f(x, y)=x^{3} y+12 x^{2}-8 y$$

\(4-6\) Find the directional derivative of \(f\) at the given point in the direction indicated by the angle \(\theta\) . $$f(x, y)=x \sin (x y), \quad(2,0), \quad \theta=\pi / 3$$

\(1-6\) Use the Chain Rule to find \(d z / d t\) or \(d w / d t\) $$w=\ln \sqrt{x^{2}+y^{2}+z^{2}}, \quad x=\sin t, \quad y=\cos t, \quad z=\tan t$$

\(5-22\) Find the limit, if it exists, or show that the limit does not exist. $$\lim _{(x, y) \rightarrow(1,-1)} e^{-x y} \cos (x+y)$$

I-6 Find an equation of the tangent plane to the given surface at the specified point. $$ z=e^{x^{2}-y^{2}}, \quad(1,-1,1) $$

Let $$f(x, y)=\ln (x+y-1)$$ (a) Evaluate \(f(1,1), \quad\) (b) Evaluate \(f(e, 1)\) (c) Find and sketch the domain of \(f\) . (d) Find the range of \(f .\)

Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s). \(f(x, y, z)=2 x+6 y+10 z ; \quad x^{2}+y^{2}+z^{2}=35\)

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. $$f(x, y)=x^{4}+y^{4}-4 x y+2$$

\(7-10\) (a) Find the gradient of \(f\) (b) Evaluate the gradient at the point \(P .\) (c) Find the rate of change of \(f\) at \(P\) in the direction of the vector \mathbf{u} . $$f(x, y)=\sin (2 x+3 y), \quad P(-6,4), \quad \mathbf{u}=\frac{1}{2}(\sqrt{3} \mathbf{i}-\mathbf{j})$$

\(7-12\) Use the Chain Rule to find \(\partial z / \partial s / \partial s\) and \(\partial z / \partial t\) $$z=x^{2} y^{3}, \quad x=s \cos t, \quad y=s \sin t$$

\(5-22\) Find the limit, if it exists, or show that the limit does not exist. $$\lim _{(x, y \rightarrow(2,1)} \frac{4-x y}{x^{2}+3 y^{2}}$$

\(7-8\) Graph the surface and the tangent plane at the given point. (Choose the domain and viewpoint so that you get a good view of both the surface and the tangent plane.) Then zoom in until the surface and the tangent plane become indistinguishable. $$z=x^{2}+x y+3 y^{2}, \quad(1,1,5)$$

Let \(f(x, y)=x^{2} e^{3 x y}\) (a) Evaluate \(f(2,0)\) (b) Find the domain of \(f\). (c) Find the range of \(f\)

Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s). \(f(x, y, z)=8 x-4 z ; \quad x^{2}+10 y^{2}+z^{2}=5\)

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. $$f(x, y)=e^{4 y-x^{2}-y^{2}}$$

\(7-10\) (a) Find the gradient of \(f\) (b) Evaluate the gradient at the point \(P\) . (c) Find the rate of change of \(f\) at \(P\) in the direction of the \(\quad\) vector \(\mathbf{u} .\) $$f(x, y)=y^{2} / x, \quad P(1,2), \quad \mathbf{u}=\frac{1}{3}(2 \mathbf{i}+\sqrt{5} \mathbf{j})$$

\(7-12\) Use the Chain Rule to find \(\partial z / \partial s / \partial s\) and \(\partial z / \partial t\) $$z=\arcsin (x-y), \quad x=s^{2}+t^{2}, \quad y=1-2 s t$$

\(5-22\) Find the limit, if it exists, or show that the limit does not exist. $$\lim _{(x, y \rightarrow(1,0)} \ln \left(\frac{1+y^{2}}{x^{2}+x y}\right)$$

Find and sketch the domain of the function \(f(x, y)=\sqrt{1+x-y^{2}} .\) What is the range of \(f ?\)

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. $$f(x, y)=(1+x y)(x+y)$$

Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s). \(f(x, y, z)=x y z ; \quad x^{2}+2 y^{2}+3 z^{2}=6\)

\(7-10\) (a) Find the gradient of \(f\) (b) Evaluate the gradient at the point \(P .\) (c) Find the rate of change of \(f\) at \(P\) in the direction of the vector \mathbf{u} . $$f(x, y, z)=x e^{2 y z}, \quad P(3,0,2), \quad \mathbf{u}=\left\langle\frac{2}{3},-\frac{2}{3}, \frac{1}{3}\right\rangle$$