Chapter 14

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

\(5-22\) Find the limit, if it exists, or show that the limit does not exist. $$\lim _{(x, y) \rightarrow(0,0)} \frac{y^{4}}{x^{4}+3 y^{4}}$$

\(\begin{array}{l}{9-10 \text { Draw the graph of } f \text { and its tangent plane at the given }} \\ {\text { point. (Use your computer algebra system both to compute the }} \\ {\text { partial derivatives and to graph the surface and its tangent plane. }} \\ {\text { Then zoom in until the surface and the tangent plane become }} \\ {\text { indistinguishable. }}\end{array}\) $$ f(x, y)=\frac{x y \sin (x-y)}{1+x^{2}+y^{2}}, \quad(1,1,0) $$

Let $$f(x, y, z)=e^{\sqrt{1-x^{2}-y^{2}}}$$ (a) Evaluate \(f(2,-1,6), \quad\) (b) Find the domain of \(f\) (c) Find 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. $$f(x, y)=2 x^{3}+x y^{2}+5 x^{2}+y^{2}$$

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

\(7-10\) (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} . $$f(x, y, z)=\sqrt{x+y z}, \quad P(1,3,1), \quad \mathbf{u}=\left\langle\frac{2}{7}, \frac{3}{7}, \frac{6}{7}\right\rangle$$

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

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

\(\begin{array}{l}{9-10 \text { Draw the graph of } f \text { and its tangent plane at the given }} \\ {\text { point. (Use your computer algebra system both to compute the }} \\ {\text { partial derivatives and to graph the surface and its tangent plane. }} \\ {\text { Then zoom in until the surface and the tangent plane become }} \\ {\text { indistinguishable. }}\end{array}\) $$ f(x, y)=e^{-x y / 10}(\sqrt{x}+\sqrt{y}+\sqrt{x y}), \quad\left(1,1,3 e^{-01}\right) $$

Let $$g(x, y, z)=\ln \left(25-x^{2}-y^{2}-z^{2}\right)$$ (a) Evaluate \(g(2,-2,4)\). (b) Find the domain of \(g\). (c) Find the range of \(g\).

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. $$f(x, y)=x^{3}-12 x y+8 y^{3}$$

Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s). \(f(x, y, z)=x^{2}+y^{2}+z^{2} ; \quad x^{4}+y^{4}+z^{4}=1\)

\(11-17\) Find the directional derivative of the function at the given point in the direction of the vector \(\mathbf{v}\) . $$f(x, y)=1+2 x \sqrt{y}, \quad(3,4), \quad \mathbf{v}=\langle 4,-3\rangle$$

\(7-12\) Use the Chain Rule to find \(\partial z / \partial s / \partial s\) and \(\partial z / \partial t\) $$z=e^{r} \cos \theta, \quad r=s t, \quad \theta=\sqrt{s^{2}+t^{2}}$$

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

\(11-16\) Explain why the function is differentiable at the given point. Then find the linearization \(L(x, y)\) of the function at that point. $$ f(x, y)=x \sqrt{y}, \quad(1,4) $$

If \(f(x, y)=16-4 x^{2}-y^{2},\) find \(f_{x}(1,2)\) and \(f_{y}(1,2)\) and interpret these numbers as slopes. Illustrate with either hand-drawn sketches or computer plots.

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

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. $$f(x, y)=x y+\frac{1}{x}+\frac{1}{y}$$

Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s). \(f(x, y, z)=x^{4}+y^{4}+z^{4} ; \quad x^{2}+y^{2}+z^{2}=1\)

II-17 Find the directional derivative of the function at the given point in the direction of the vector \(\mathbf{v}\) . $$f(x, y)=\ln \left(x^{2}+y^{2}\right), \quad(2,1), \quad \mathbf{v}=\langle- 1,2\rangle$$

\(7-12\) Use the Chain Rule to find \(\partial z / \partial s / \partial s\) and \(\partial z / \partial t\) $$z=\tan (u / v), \quad u=2 s+3 t, \quad v=3 s-2 t$$

\(5-22\) Find the limit, if it exists, or show that the limit does not exist. $$\lim _{(x, y \rightarrow(0,0)} \frac{6 x^{3} y}{2 x^{4}+y^{4}}$$

If \(f(x, y)=\sqrt{4-x^{2}-4 y^{2}},\) find \(f_{x}(1,0)\) and \(f_{y}(1,0)\) and interpret these numbers as slopes. Illustrate with either hand drawn sketches or computer plots.

\(11-16\) Explain why the function is differentiable at the given point. Then find the linearization \(L(x, y)\) of the function at that point. $$ f(x, y)=x^{3} y^{4}, \quad(1,1) $$

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

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. $$f(x, y)=e^{x} \cos y$$

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

\(11-17\) Find the directional derivative of the function at the given point in the direction of the vector \(\mathbf{v}\) . $$g(p, q)=p^{4}-p^{2} q^{3}, \quad(2,1), \quad \mathbf{v}=\mathbf{i}+3 \mathbf{j}$$

If \(z=f(x, y),\) where \(f\) is differentiable, and $$\begin{array}{rl}{x=g(t)} & {y=h(t)} \\ {g(3)=2} & {h(3)=7} \\\ {g^{\prime}(3)=5} & {h^{\prime}(3)=-4} \\ {f_{x}(2,7)=6} & {f(2,7)=-8}\end{array}$$ find \(d z / d t\) when \(t=3\)

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

Find \(f_{x}\) and \(f_{y}\) and graph \(f, f_{x}\) and \(f_{y}\) with domains and viewpoints that enable you to see the relationships between them. $$f(x, y)=x^{2}+y^{2}+x^{2} y$$

Find and sketch the domain of the function. $$f(x, y)=\ln \left(9-x^{2}-9 y^{2}\right)$$

\(11-16\) Explain why the function is differentiable at the given point. Then find the linearization \(L(x, y)\) of the function at that point. $$ f(x, y)=\frac{x}{x+y}, \quad(2,1) $$

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. $$f(x, y)=y \cos x$$

Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s). \(f\left(x_{1}, x_{2}, \ldots, x_{n}\right)=x_{1}+x_{2}+\cdots+x_{n}\) \(x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=1\)

\(11-17\) Find the directional derivative of the function at the given point in the direction of the vector \(\mathbf{v}\) . $$g(r, s)=\tan ^{-1}(r s), \quad(1,2), \quad \mathbf{v}=5 \mathbf{i}+10 \mathbf{j}$$

Let \(W(s, t)=F(u(s, t), v(s, t)),\) where \(F, u,\) and \(v\) are differentiable, and $$\begin{array}{ll}{u(1,0)=2} & {v(1,0)=3} \\ {u_{s}(1,0)=-2} & {v_{s}(1,0)=5} \\ {u_{t}(1,0)=6} & {v_{t}(1,0)=4} \\ {F_{v}(2,3)=-1} & {F_{r}(2,3)=10}\end{array}$$ Find \(W_{s}(1,0)\) and \(W_{t}(1,0)\)

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

Find \(f_{x}\) and \(f_{y}\) and graph \(f, f_{x}\) and \(f_{y}\) with domains and viewpoints that enable you to see the relationships between them. $$f(x, y)=x e^{-x^{t}-y^{2}}$$

\(11-16\) Explain why the function is differentiable at the given point. Then find the linearization \(L(x, y)\) of the function at that point. $$ f(x, y)=\sqrt{x+e^{4 y}}, \quad(3,0) $$

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

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

Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s). \(f(x, y, z)=x+2 y ; \quad x+y+z=1, \quad y^{2}+z^{2}=4\)

\(11-17\) Find the directional derivative of the function at the given point in the direction of the vector v. $$f(x, y, z)=x e^{y}+y e^{z}+z e^{x}, \quad(0,0,0), \quad \mathbf{v}=\langle 5,1,-2\rangle$$

Suppose \(f\) is a differentiable function of \(x\) and \(y,\) and \(g(u, v)=f\left(e^{u}+\sin v, e^{u}+\cos v\right) .\) Use the table of values to calculate \(g_{d}(0,0)\) and \(g_{c}(0,0)\) $$\begin{array}{|c|c|c|c|c|}\hline & {f} & {g} & {f_{x}} & {f_{y}} \\\ \hline(0,0) & {3} & {6} & {4} & {8} \\ \hline(1,2) & {6} & {3} & {2} & {5} \\\ \hline\end{array}$$

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

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

\(11-16\) Explain why the function is differentiable at the given point. Then find the linearization \(L(x, y)\) of the function at that point. $$ f(x, y)=e^{-x y} \cos y, \quad(\pi, 0) $$