Chapter 14

\(21-26\) Find the maximum rate of change of \(f\) at the given point and the direction in which it occurs. $$f(p, q)=q e^{-p}+p e^{-q}, \quad(0,0)$$

Find the first partial derivatives of the function. $$f(x, y)=x^{y}$$

Sketch the graph of the function. $$f(x, y)=y$$

Use a graph and/or level curves to estimate the local maximum and minimum values and saddle point(s) of the function. Then use calculus to find these values precisely. $$\begin{array}{l}{f(x, y)=\sin x+\sin y+\sin (x+y)} \\ {0 \leqslant x \leqslant 2 \pi, 0 \leqslant y \leqslant 2 \pi}\end{array}$$

The total production \(P\) of a certain product depends on the amount \(L\) of labor used and the amount \(K\) of capital investment. In Sections 14.1 and 14.3 we discussed how the CobbDouglas model \(P=b L^{\alpha} K^{1-\alpha}\) follows from certain economic assumptions, where \(b\) and \(\alpha\) are positive constants and \(\alpha<1\) If the cost of a unit of labor is \(m\) and the cost of a unit of capital is \(n,\) and the company can spend only \(p\) dollars as its total budget, then maximizing the production \(P\) is subject to the constraint \(m L+n K=p .\) Show that the maximum production occurs when $$L=\frac{\alpha p}{m} \quad \text { and } \quad K=\frac{(1-\alpha) p}{n}$$

\(21-26\) Use the Chain Rule to find the indicated partial derivatives. $$\begin{array}{l}{R=\ln \left(u^{2}+v^{2}+w^{2}\right)} \\ {u=x+2 y, \quad v=2 x-y, \quad w=2 x y} \\ {\frac{\partial R}{\partial x}, \frac{\partial R}{\partial y} \text { when } x=y=1}\end{array}$$

\(23-24\) Use a computer graph of the function to explain why the limit does not exist. $$\lim _{(x, y) \rightarrow(0,0)} \frac{2 x^{2}+3 x y+4 y^{2}}{3 x^{2}+5 y^{2}}$$

\(21-26\) Find the maximum rate of change of \(f\) at the given point and the direction in which it occurs. $$f(x, y)=\sin (x y), \quad(1,0)$$

Sketch the graph of the function. $$f(x, y)=10-4 x-5 y$$

Find the first partial derivatives of the function. $$w=\sin \alpha \cos \beta$$

\(21-26\) Use the Chain Rule to find the indicated partial derivatives. $$\begin{array}{l}{M=x e^{y-z^{2}}, \quad x=2 u v, \quad y=u-v, \quad z=u+v} \\\ {\frac{\partial M}{\partial u}, \frac{\partial M}{\partial v} \text { when } u=3, v=-1}\end{array}$$

\(23-24\) Use a computer graph of the function to explain why the limit does not exist. $$\lim _{(x, y \rightarrow(0,0)} \frac{x y^{3}}{x^{2}+y^{6}}$$

Find the first partial derivatives of the function. $$w=e^{v} /\left(u+v^{2}\right)$$

\(21-26\) Find the maximum rate of change of \(f\) at the given point and the direction in which it occurs. $$f(x, y, z)=(x+y) / z, \quad(1,1,-1)$$

Use a graph and/or level curves to estimate the local maximum and minimum values and saddle point(s) of the function. Then use calculus to find these values precisely. $$\begin{array}{l}{f(x, y)=\sin x+\sin y+\cos (x+y)} \\ {0 \leqslant x \leqslant \pi / 4,0 \leqslant y \leqslant \pi / 4}\end{array}$$

Use Lagrange multipliers to prove that the rectangle with maximum area that has a given perimeter \(p\) is a square.

\(21-26\) Use the Chain Rule to find the indicated partial derivatives. $$\begin{array}{ll}{u=x^{2}+y z,} & {x=p r \cos \theta, \quad y=p r \sin \theta, \quad z=p+r} \\ {\frac{\partial u}{\partial p}, \frac{\partial u}{\partial r}, \frac{\partial u}{\partial \theta}} & {\text { when } p=2, r=3, \theta=0}\end{array}$$

\(25-30\) Find the differential of the function. $$ z=x^{3} \ln \left(y^{2}\right) $$

\(25-26\) Find \(h(x, y)=g(f(x, y))\) and the set on which \(h\) is continuous. $$g(t)=t^{2}+\sqrt{t}, \quad f(x, y)=2 x+3 y-6$$

Find the first partial derivatives of the function. $$f(r, s)=r \ln \left(r^{2}+s^{2}\right)$$

\(21-26\) Find the maximum rate of change of \(f\) at the given point and the direction in which it occurs. \(f(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}}, \quad(3,6,-2)\)

Sketch the graph of the function. $$f(x, y)=y^{2}+1$$

\(21-26\) Use the Chain Rule to find the indicated partial derivatives. $$\begin{array}{l}{Y=w \tan ^{-1}(u v), \quad u=r+s, \quad v=s+t, \quad w=t+r} \\\ {\frac{\partial Y}{\partial r}, \frac{\partial Y}{\partial s}, \frac{\partial Y}{\partial t} \quad \text { when } r=1, s=0, t=1}\end{array}$$

\(25-30\) Find the differential of the function. $$ v=y \cos x y $$

\(25-26\) Find \(h(x, y)=g(f(x, y))\) and the set on which \(h\) is continuous. $$g(t)=t+\ln t, \quad f(x, y)=\frac{1-x y}{1+x^{2} y^{2}}$$

Find the first partial derivatives of the function. $$f(x, t)=\arctan (x \sqrt{t})$$

\(21-26\) Find the maximum rate of change of \(f\) at the given point and the direction in which it occurs. $$f(x, y, z)=\tan (x+2 y+3 z), \quad(-5,1,1)$$

Sketch the graph of the function. $$f(x, y)=3-x^{2}-y^{2}$$

Use a graphing device as in Example 4 (or Newton's method or a rootfinder) to find the critical points of \(f\) correct to three decimal places. Then classify the critical points and find the highest or lowest points on the graph. $$f(x, y)=5-10 x y-4 x^{2}+3 y-y^{4}$$

Use a graphing device as in Example 4 (or Newton's method or a rootfinder) to find the critical points of \(f\) correct to three decimal places. Then classify the critical points and find the highest or lowest points on the graph. $$f(x, y)=2 x+4 x^{2}-y^{2}+2 x y^{2}-x^{4}-y^{4}$$

\(25-30\) Find the differential of the function. $$ m=p^{5} q^{3} $$

\(27-28\) Graph the function and observe where it is discontinuous. Then use the formula to explain what you have observed. $$f(x, y)=e^{1 /(x-y)}$$

(a) Show that a differentiable function \(f\) decreases most rapidly at \(x\) in the direction opposite to the gradient vector, that is, in the direction of \(-\nabla f(\mathbf{x})\) . (b) Use the result of part (a) to find the direction in which the function \(f(x, y)=x^{4} y-x^{2} y^{3}\) decreases fastest at the point \((2,-3) .\)

Sketch the graph of the function. $$f(x, y)=4 x^{2}+y^{2}+1$$

Use a graphing device as in Example 4 (or Newton's method or a rootfinder) to find the critical points of \(f\) correct to three decimal places. Then classify the critical points and find the highest or lowest points on the graph. $$f(x, y)=e^{x}+y^{4}-x^{3}+4 \cos y$$

\(27-28\) Graph the function and observe where it is discontinuous. Then use the formula to explain what you have observed. $$f(x, y)=\frac{1}{1-x^{2}-y^{2}}$$

Find the first partial derivatives of the function. $$f(x, y)=\int_{y}^{x} \cos \left(t^{2}\right) d t$$

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

Sketch the graph of the function. $$f(x, y)=\sqrt{16-x^{2}-16 y^{2}}$$

\(25-30\) Find the differential of the function. $$ T=\frac{v}{1+u v w} $$

Find the absolute maximum and minimum values of \(f\) on the set \(D .\) \(f(x, y)=1+4 x-5 y, \quad D\) is the closed triangular region with vertices \((0,0),(2,0),\) and \((0,3)\)

\(29-38\) Determine the set of points at which the function is continuous. $$F(x, y)=\frac{\sin (x y)}{e^{x}-y^{2}}$$

\(25-30\) Find the differential of the function. $$ R=\alpha \beta^{2} \cos \gamma $$

Find the first partial derivatives of the function. $$f(x, y, z)=x z-5 x^{2} y^{3} z^{4}$$

Find all points at which the direction of fastest change of the function \(f(x, y)=x^{2}+y^{2}-2 x-4 y\) is \(\mathbf{i}+\mathbf{j}\)

Sketch the graph of the function. $$f(x, y)=\sqrt{x^{2}+y^{2}}$$

Find the absolute maximum and minimum values of \(f\) on the set \(D .\) $$\begin{array}{l}{f(x, y)=3+x y-x-2 y, \quad D \text { is the closed triangular }} \\ {\text { region with vertices }(1,0),(5,0), \text { and }(1,4)}\end{array}$$

\(29-38\) Determine the set of points at which the function is continuous. $$F(x, y)=\frac{x-y}{1+x^{2}+y^{2}}$$

\(25-30\) Find the differential of the function. $$ w=x y e^{x z} $$

Find the first partial derivatives of the function. $$f(x, y, z)=x \sin (y-z)$$