Chapter 13

Find the directional derivative of \(f\) at the point \(P\) in the direction of a. $$ f(x, y, z)=e^{x^{2}+y^{2}+z^{2}} ; P=(0,0,0) ; \mathbf{a}=-\mathbf{i}+\mathbf{j}-\mathbf{k} $$

Find the first partial derivatives of the function. $$ z=x^{y} $$

Evaluate the limit. $$ \lim _{(x, y) \rightarrow(0,0)} \frac{y\left(e^{x}-1\right)}{\sqrt{x^{2}+y^{2}}} $$

Find the domain of the function. \(f(x, y, z)=\frac{x y z}{(x+y)^{3}-(x+z)^{3}}\)

Find the extreme values of \(f\) in the region described by the given inequalities. In each case assume that the extreme values exist. $$ f(x, y)=x y ; 2 x^{2}+y^{2} \leq 4 $$

Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ f(x, y)=x^{2}-e^{y^{2}} $$

Compute \(\partial z / \partial r\) and \(\partial z / \partial s\). $$ \mathrm{z}=u e^{v}+v e^{-u} ; u=\ln r, v=s \ln r $$

Find the gradient of the function at the given point. $$ f(x, y, z)=z e^{-x} \tan y ;(0, \pi,-2) $$

Find the directional derivative of \(f\) at the point \(P\) in the direction of a. $$ f(x, y, z)=y z 2^{x} ; P=(1,-1,1) ; \mathbf{a}=2 \mathbf{j}-\mathbf{k} $$

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

Evaluate the limit. $$ \lim _{(x, y) \rightarrow(0,0)} x y \frac{x^{2}-y^{2}}{x^{2}+y^{2}} $$

Sketch the level curve \(f(x, y)=c\). \(f(x, y)=3 x-y ; c=2,3\)

Find the extreme values of \(f\) in the region described by the given inequalities. In each case assume that the extreme values exist. $$ f(x, y)=16-x^{2}-4 y^{2} ; x^{4}+2 y^{4} \leq 1 $$

Determine \(d f\). $$ f(x, y)=y \ln \frac{1+x}{1-x} $$

Find the gradient of the function at the given point. $$ g(x, y, z)=e^{x}(\sin y+\sin z) ;(1, \pi / 2, \pi / 2) $$

Compute \(\partial z / \partial r\) and \(\partial z / \partial s\). $$ z=2^{u-v} ; u=r \cos s, v=r \sin s $$

Find the directional derivative of \(f\) at the point \(P\) in the direction of a. $$ \begin{aligned} &f(x, y, z)=y \sin ^{-1} x z ; P=(1 / \sqrt{2}, 0,1 / \sqrt{2}) \\ &\mathbf{a}=-2 \mathbf{i}-2 \mathbf{j}-2 \mathbf{k} \end{aligned} $$

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

Evaluate the limit. $$ \lim _{(x, y) \rightarrow(0,0)} \frac{x y}{\sqrt{x^{2}+y^{2}}} $$

Sketch the level curve \(f(x, y)=c\). \(f(x, y)=6 x^{2} ; c=6,24\)

Find the points on the surface \(x^{2}-y z=1\) that are closest to the origin.

Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ k(x, y)=e^{x} \sin y $$

Compute \(d w / d t\) $$ \mathrm{w}=\frac{x}{y}-\frac{z}{x} ; x=\sin t, y=\cos t, z=\tan t $$

Determine \(d f\). $$ f(x, y)=x^{2}+y^{2} $$

Find the direction in which \(f\) increases most rapidly at the given point, and find the maximal directional derivative at that point. $$ f(x, y)=e^{x}(\cos y+\sin y) ;(0,0) $$

Let \(f(x, y)=\sinh (y \ln x)\). Find the directional derivative of \(f\) at \((2,-1)\) in the direction away from the origin.

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

Evaluate the limit. $$ \lim _{(x, y, z) \rightarrow(0,0,0)} \frac{x^{3}+y^{3}+z^{3}}{x^{2}+y^{2}+z^{2}} $$

Sketch the level curve \(f(x, y)=c\). \(f(x, y)=x^{2}+4 y^{2} ; c=1,4\)

Find the points on the sphere \(x^{2}+y^{2}+z^{2}=1\) that are closest to or farthest from the point \((4,2,1)\).

Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ f(x, y)=e^{x}(\sin y-1) $$

Compute \(d w / d t\) $$ w=\frac{z}{x y^{2}}-3 ; x=\frac{1}{t^{2}}, y=-5 t, z=\sqrt{t} $$

Determine \(d f\). $$ f(x, y)=\cos (x+y)+\cos (x-y) $$

Find the direction in which \(f\) increases most rapidly at the given point, and find the maximal directional derivative at that point. $$ f(x, y)=e^{2 x}(\cos y-\sin y) ;\left(\frac{1}{6},-\pi / 2\right) $$

Let \(f(x, y)=2 x+x^{2} y+y \sin y\) and let \(\mathbf{u}=a \mathbf{i}+b \mathbf{j}\) be a unit vector. a. Express \(D_{\mathbf{u}} f(1,0)\) in terms of \(a\) and \(b\). b. Using the result of part (a), find the values of \(a\) and \(b\) for which \(D_{\mathbf{u}} f(1,0)\) is maximum.

Find the first partial derivatives of the function. $$ f(u, v, w)=\frac{1}{\sqrt{u^{2}+v^{2}+w^{2}}} $$

Determine whether \(\lim _{(x, y) \rightarrow(0,0)} \frac{x y}{x^{2}+y^{2}}\) exists.

Sketch the level curve \(f(x, y)=c\). \(f(x, y)=x^{2}-y ; c=-2,2\)

A triangle is to be inscribed in the ellipse \(\frac{1}{4} x^{2}+y^{2}=1\) with one vertex of the triangle at \((-2,0)\) and the opposite side perpendicular to the \(x\) axis. Find the largest possible area of the triangle.

Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ f(u, v)=|u|+|v| $$

Compute \(d w / d t\) $$ w=\sqrt{x^{2}+y^{2}+z^{2}} ; x=e^{t}, y=e^{-t}, z=2 t $$

Determine \(d f\). $$ f(x, y)=x \tan y+y \cot x $$

Find the direction in which \(f\) increases most rapidly at the given point, and find the maximal directional derivative at that point. $$ f(x, y)=3 x^{2}+4 y^{2} ;(-1,1) $$

Find the first partial derivatives of the function. $$ w=e^{x}(\cos y+\sin z) $$

Determine whether \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}}{x^{2}+y^{2}}\) exists.

Sketch the level curve \(f(x, y)=c\). \(f(x, y)=x^{2}-y^{2} ; c=-1,0,1\)

Let \(x\) and \(y\) denote the acute angles of a right triangle. Find the maximum value of \(\sin x \sin y\).

Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ g(u, v)=3-|u-2|+|v+1| $$

Compute \(d w / d t\) $$ w=\ln \left(x^{2}+y^{2}+z^{2}\right) ; x=\sin t, y=\cos t, z=e^{-t^{2}} $$

Find the direction in which \(f\) increases most rapidly at the given point, and find the maximal directional derivative at that point. $$ f(x, y, z)=\ln \left(x^{2}+y^{2}+z^{2}\right) ;(2,0,1) $$