Chapter 14

Find and sketch the domain of the function. $$f(x, y)=\sqrt{1-x^{2}}-\sqrt{1-y^{2}}$$

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. $$f(x, y)=e^{y}\left(y^{2}-x^{2}\right)$$

Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s). \(f(x, y, z)=3 x-y-3 z\) \(x+y-z=0, \quad x^{2}+2 z^{2}=1\)

\(11-17\) Find the directional derivative of the function at the given point in the direction of the vector \(\mathbf{v}\) . $$f(x, y, z)=\sqrt{x y z}, \quad(3,2,6), \quad \mathbf{v}=\langle- 1,-2,2\rangle$$

\(11-16\) Explain why the function is differentiable at the given point. Then find the linearization \(L(x, y)\) of the function at that point. $$ f(x, y)=\sin (2 x+3 y), \quad(-3,2) $$

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

Find the first partial derivatives of the function. $$f(x, y)=x^{4} y^{3}+8 x^{2} y$$

Find and sketch the domain of the function. $$f(x, y)=\sqrt{y}+\sqrt{25-x^{2}-y^{2}}$$

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. $$f(x, y)=y^{2}-2 y \cos x, \quad 1 \leqslant x \leqslant 7$$

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

\(11-17\) Find the directional derivative of the function at the given point in the direction of the vector \(\mathbf{v}\) . $$g(x, y, z)=(x+2 y+3 z)^{3 / 2}, \quad(1,1,2), \quad \mathbf{v}=2 \mathbf{j}-\mathbf{k}$$

\(17-20\) Use a tree diagram to write out the Chain Rule for the given case. Assume all functions are differentiable. $$u=f(x, y), \quad \text { where } x=x(r, s, t), y=y(r, s, t)$$

\(17-18 \text { Verify the linear approximation at }(0,0)\) $$ \frac{2 x+3}{4 y+1} \approx 3+2 x-12 y $$

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

Find the first partial derivatives of the function. $$f(x, t)=e^{-t} \cos \pi x$$

Find and sketch the domain of the function. $$f(x, y)=\frac{\sqrt{y-x^{2}}}{1-x^{2}}$$

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. $$f(x, y)=\sin x \sin y$$ $$-\pi< x <\pi, \quad-\pi< y <\pi$$

Find the extreme values of \(f\) on the region described by the inequality. \(f(x, y)=2 x^{2}+3 y^{2}-4 x-5, \quad x^{2}+y^{2} \leqslant 16\)

\(17-20\) Use a tree diagram to write out the Chain Rule for the given case. Assume all functions are differentiable. $$\begin{array}{l}{R=f(x, y, z, t), \quad \text { where } x=x(u, v, w), y=y(u, v, w)} \\ {z=z(u, v, w), t=t(u, v, w)}\end{array}$$

\(5-22\) Find the limit, if it exists, or show that the limit does not exist. $$\lim _{(x, y) \rightarrow(0,0)} \frac{x y^{4}}{x^{2}+y^{8}}$$

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

Find and sketch the domain of the function. $$f(x, y)=\arcsin \left(x^{2}+y^{2}-2\right)$$

Show that \(f(x, y)=x^{2}+4 y^{2}-4 x y+2\) has an infinite number of critical points and that \(D=0\) at each one. Then show that \(f\) has a local (and absolute) minimum at each critical point.

Find the extreme values of \(f\) on the region described by the inequality. \(f(x, y)=e^{-x y}, \quad x^{2}+4 y^{2} \leqslant 1\)

\(17-20\) Use a tree diagram to write out the Chain Rule for the given case. Assume all functions are differentiable. $$w=f(r, s, t), \quad \text { where } r=r(x, y), \quad s=s(x, y), \quad t=t(x, y)$$

Find the linear approximation of the function \(f(x, y)=\sqrt{20-x^{2}-7 y^{2}}\) at \((2,1)\) and use it to approximate \(f(1.95,1.08) .\)

Find the directional derivative of \(f(x, y)=\sqrt{x y}\) at \(P(2,8)\) in the direction of \(Q(5,4)\)

5-22 Find the limit, if it exists, or show that the limit does not exist. $$\lim _{(x, y, z) \rightarrow(3,0,1)} e^{-x y} \sin (\pi z / 2)$$

Find the first partial derivatives of the function. $$z=(2 x+3 y)^{10}$$

Find and sketch the domain of the function. $$f(x, y, z)=\sqrt{1-x^{2}-y^{2}-z^{2}}$$

Show that \(f(x, y)=x^{2} y e^{-x^{2}-y^{2}}\) has maximum values at \((\pm 1,1 / \sqrt{2})\) and minimum values at \((\pm 1,-1 / \sqrt{2}) .\) Show also that \(f\) has infinfinitely many other critical points and \(D=0\) at each of them. Which of them give rise to maximum values? Minimum values? Saddle points?

Consider the problem of maximizing the function \(f(x, y)=2 x+3 y\) subject to the constraint \(\sqrt{x}+\sqrt{y}=5\) (a) Try using Lagrange multipliers to solve the problem. (b) Does \(f(25,0)\) give a larger value than the one in part (a)? (c) Solve the problem by graphing the constraint equation and several level curves of \(f\) . (d) Explain why the method of Lagrange multipliers fails to solve the problem. (e) What is the significance of \(f(9,4) ?\)

Find the linear approximation of the function \(f(x, y)=\ln (x-3 y)\) at \((7,2)\) and use it to approximate \(f(6.9,2.06) .\) Illustrate by graphing \(f\) and the tangent plane.

\(17-20\) Use a tree diagram to write out the Chain Rule for the given case. Assume all functions are differentiable. $$\begin{array}{l}{t=f(u, v, w), \quad \text { where } u=u(p, q, r, s), v=v(p, q, r, s),} \\ {w=w(p, q, r, s)}\end{array}$$

\(5-22\) Find the limit, if it exists, or show that the limit does not exist. $$\lim _{(x, y, z) \rightarrow(0,0,0)} \frac{x^{2}+2 y^{2}+3 z^{2}}{x^{2}+y^{2}+z^{2}}$$

Find the directional derivative of \(f(x, y, z)=x y+y z+z x\) at \(P(1,-1,3)\) in the direction of \(Q(2,4,5)\)

Find the first partial derivatives of the function. $$z=\tan x y$$

Find and sketch the domain of the function. $$f(x, y, z)=\ln \left(16-4 x^{2}-4 y^{2}-z^{2}\right)$$

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. $$f(x, y)=x^{2}+y^{2}+x^{-2} y^{-2}$$

Consider the problem of minimizing the function \(f(x, y)=x\) on the curve \(y^{2}+x^{4}-x^{3}=0\) (a piriform). (a) Try using Lagrange multipliers to solve the problem. (b) Show that the minimum value is \(f(0,0)=0\) but the Lagrange condition \(\nabla f(0,0)=\lambda \nabla g(0,0)\) is not satisfied for any value of \(\lambda .\) (c) Explain why Lagrange multipliers fail to find the minimum value in this case.

\(21-26\) Use the Chain Rule to find the indicated partial derivatives. $$\begin{array}{ll}{z=x^{2}+x y^{3},} & {x=u v^{2}+w^{3}, \quad y=u+v e^{m}} \\\ {\frac{\partial z}{\partial u}, \frac{\partial z}{\partial v}, \frac{\partial z}{\partial w}} & {\text { when } u=2, v=1, w=0}\end{array}$$

Find the linear approximation of the function \(f(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}}\) at \((3,2,6)\) and use it to approximate the number \(\sqrt{(3.02)^{2}+(1.97)^{2}+(5.99)^{2}}\)

\(5-22\) Find the limit, if it exists, or show that the limit does not exist. $$\lim _{(x, y, z) \rightarrow(0,0,0)} \frac{x y+y z^{2}+x z^{2}}{x^{2}+y^{2}+z^{4}}$$

\(21-26\) Find the maximum rate of change of \(f\) at the given point and the direction in which it occurs. $$f(x, y)=y^{2} / x, \quad(2,4)$$

Find the first partial derivatives of the function. $$f(x, y)=\frac{x-y}{x+y}$$

Sketch the graph of the function. $$f(x, y)=3$$

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. $$f(x, y)=x y e^{-x^{2}-y^{2}}$$

(a) If your computer algebra system plots implicitly defined curves, use it to estimate the minimum and maximum values of \(f(x, y)=x^{3}+y^{3}+3 x y\) subject to the constraint \((x-3)^{2}+(y-3)^{2}=9\) by graphical methods. (b) Solve the problem in part (a) with the aid of Lagrange multipliers. Use your CAS to solve the equations numerically. Compare your answers with those in part (a).

\(21-26\) Use the Chain Rule to find the indicated partial derivatives. $$\begin{array}{l}{u=\sqrt{r^{2}+s^{2}}, \quad r=y+x \cos t, \quad s=x+y \sin t} \\ {\frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial u}{\partial t} \quad \text { when } x=1, y=2, t=0}\end{array}$$

\(5-22\) Find the limit, if it exists, or show that the limit does not exist. $$\lim _{(x, y, z) \rightarrow(0,0,0)} \frac{y z}{x^{2}+4 y^{2}+9 z^{2}}$$