Chapter 19

In Exercises 1 through 8 , do each of the following: (a) Find \(D_{11} f(x, y)\); (b) find \(D_{22} f(x, y) ;\) (c) show that \(D_{12} f(x, y)=D_{21} f(x, y) .\) $$ f(x, y)=\frac{x^{2}}{y}-\frac{y}{x^{2}} $$

In Exercises 1 through 4 , find the indicated partial derivative by two methods: (a) Use the chain rule; (b) make the substitutions for \(x\) and \(y\) before differentiating. $$ u=x^{2}-y^{2} ; x=3 r-s ; y=r+2 s ; \frac{\partial u}{\partial r} ; \frac{\partial u}{\partial s} $$

If \(f(x, y)=3 x^{2}+2 x y-y^{2}, \Delta x=0.03\), and \(\Delta y=-0.02\), find (a) the increment of \(f\) at \((1,4)\) and (b) the total differential of \(f\) at \((1,4)\).

Let the function \(f\) of two variables \(x\) and \(y\) be the set of all ordered pairs of the form \((P, z)\) such that \(z=(x+y) /(x-y)\). Find: (a) \(f(-3,4) ;\) (b) \(f\left(x^{2}, y^{2}\right) ;\) (c) \([f(x, y)]^{2} ;\) (d) \(f(-x, y)-f(x,-y) ;\) (e) the domain of \(f ;(\mathrm{f})\) the range of \(f\). Draw a sketch showing as a shaded region in \(R^{2}\) the set of points not in the domain of \(f .\)

In Exercises 1 through 8 , do each of the following: (a) Find \(D_{11} f(x, y)\); (b) find \(D_{22} f(x, y) ;\) (c) show that \(D_{12} f(x, y)=D_{21} f(x, y) .\) $$ f(x, y)=2 x^{3}-3 x^{2} y+x y^{2} $$

If \(f(x, y)=x y e^{x y}, \Delta x=-0.1\), and \(\Delta y=0.2\), find (a) the increment of \(f\) at \((2,-4)\) and (b) the total differential of \(f\) at \((2,-4)\).

Let the function \(g\) of three variables, \(x, y\), and \(z\), be the set of all ordered pairs of the form \((P, w)\) such that \(w=\) \(\sqrt{4-x^{2}-y^{2}-z^{2}}\). Find: (a) \(g(1,-1,-1) ;\) (b) \(g\left(-a, 2 b, \frac{1}{2} c\right) ;\) (c) \(g(y,-x,-y) ;\) (d) the domain of \(g ;\) (e) the range of \(g\) (f) \([g(x, y, z)]^{2}-[g(x+2, y+2, z)]^{2}\). Draw a sketch showing as a shaded solid in \(R^{3}\) the set of points in the domain of \(g\).

In Exercises 1 through 8 , do each of the following: (a) Find \(D_{11} f(x, y)\); (b) find \(D_{22} f(x, y) ;\) (c) show that \(D_{12} f(x, y)=D_{21} f(x, y) .\) $$ f(x, y)=e^{2 x} \sin y $$

In Exercises 1 through 8 , do each of the following: (a) Find \(D_{11} f(x, y)\); (b) find \(D_{22} f(x, y) ;\) (c) show that \(D_{12} f(x, y)=D_{21} f(x, y) .\) $$ f(x, y)=e^{-x / y}+\ln \frac{y}{x} $$

In Exercises 1 through 4 , find the indicated partial derivative by two methods: (a) Use the chain rule; (b) make the substitutions for \(x\) and \(y\) before differentiating. $$ u=x^{2}+y^{2} ; x=\cosh r \cos t ; y=\sinh r \sin t ; \frac{\partial u}{\partial r} ; \frac{\partial u}{\partial t} $$

If \(f(x, y, z)=x^{2} y+2 x y z-z^{3}, \Delta x=0.01, \Delta y=0.03\), and \(\Delta z=-0.01\), find (a) the increment of \(f\) at \((-3,0,2)\) and (b) the total differential of \(f\) at \((-3,0,2)\).

In Exercises 1 through 6 , discuss the continuity of \(f\). \(f(x, y)= \begin{cases}\frac{x+y}{x^{2}+y^{2}} & \text { if }(x, y) \neq(0,0) \\\ 0 & \text { if }(x, y)=(0,0)\end{cases}\)

In Exercises 3 through 11 , find the domain and range of the function \(f\) and draw a sketch showing as a shaded region in \(R^{2}\) the set of points in the domain of \(f\). $$ f(x, y)=x \sqrt{25-x^{2}-y^{2}} $$

In Exercises 1 through 8 , do each of the following: (a) Find \(D_{11} f(x, y)\); (b) find \(D_{22} f(x, y) ;\) (c) show that \(D_{12} f(x, y)=D_{21} f(x, y) .\) $$ f(x, y)=\left(x^{2}+y^{2}\right) \tan ^{-1} \frac{y}{x} $$

In Exercises 5 through 10, find the indicated partial derivative by using the chain rule. \(u=\sin ^{-1}(3 x+y) ; x=r^{2} e^{s} ; y=\sin r s ; \frac{\partial u}{\partial r} ; \frac{\partial u}{\partial s}\)

In Exercises 5 through 8 , prove that \(f\) is differentiable at all points in its domain by doing each of the following: (a) Find \(\Delta f\left(x_{0}, y_{0}\right)\) for the given function; (b) find an \(\epsilon_{1}\) and an \(\epsilon_{2}\) so that Eq. (3) holds; (c) show that the \(\epsilon_{1}\) and the \(\epsilon_{2}\) found in part (b) both approach zero as \((\Delta x, \Delta y) \rightarrow(0,0)\). $$ f(x, y)=x^{2} y-2 x y $$

In Exercises 1 through 6 , discuss the continuity of \(f\). \(f(x, y)= \begin{cases}\frac{x y}{|x|+|y|} & \text { if }(x, y) \neq(0,0) \\\ 0 & \text { if }(x, y)=(0,0)\end{cases}\)

In Exercises 3 through 11 , find the domain and range of the function \(f\) and draw a sketch showing as a shaded region in \(R^{2}\) the set of points in the domain of \(f\). $$ f(x, y)=\frac{x}{\sqrt{25-x^{2}-y^{2}}} $$

In Exercises 1 through 8 , do each of the following: (a) Find \(D_{11} f(x, y)\); (b) find \(D_{22} f(x, y) ;\) (c) show that \(D_{12} f(x, y)=D_{21} f(x, y) .\) $$ f(x, y)=\sin ^{-1} \frac{3 y}{x^{2}} $$

In Exercises 5 through 10, find the indicated partial derivative by using the chain rule. $$ u=x e^{-y} ; x=\tan ^{-1}(r s t) ; y=\ln (3 r s+5 s t) ; \frac{\partial u}{\partial r} ; \frac{\partial u}{\partial s} ; \frac{\partial u}{\partial t} $$

In Exercises 5 through 8 , prove that \(f\) is differentiable at all points in its domain by doing each of the following: (a) Find \(\Delta f\left(x_{0}, y_{0}\right)\) for the given function; (b) find an \(\epsilon_{1}\) and an \(\epsilon_{2}\) so that Eq. (3) holds; (c) show that the \(\epsilon_{1}\) and the \(\epsilon_{2}\) found in part (b) both approach zero as \((\Delta x, \Delta y) \rightarrow(0,0)\). $$ f(x, y)=2 x^{2}+3 y^{2} $$

In Exercises 1 through 6 , discuss the continuity of \(f\). \(f(x, y)= \begin{cases}\frac{x^{2} y^{2}}{\left|x^{3}\right|+\left|y^{3}\right|} & \text { if }(x, y) \neq(0,0) \\\ 0 & \text { if }(x, y)=(0,0)\end{cases}\)

In Exercises 3 through 11 , find the domain and range of the function \(f\) and draw a sketch showing as a shaded region in \(R^{2}\) the set of points in the domain of \(f\). $$ f(x, y)=\sqrt{\frac{x-y}{x+y}} $$

In Exercises 1 through 8 , do each of the following: (a) Find \(D_{11} f(x, y)\); (b) find \(D_{22} f(x, y) ;\) (c) show that \(D_{12} f(x, y)=D_{21} f(x, y) .\) $$ f(x, y)=4 x \sinh y+3 y \cosh x $$

In Exercises 5 through 10, find the indicated partial derivative by using the chain rule. $$ u=\cosh \frac{y}{x^{\prime}} x=3 r^{2} s ; y=6 s e^{r} ; \frac{\partial u}{\partial r} ; \frac{\partial u}{\partial s} $$

In Exercises 5 through 8 , prove that \(f\) is differentiable at all points in its domain by doing each of the following: (a) Find \(\Delta f\left(x_{0}, y_{0}\right)\) for the given function; (b) find an \(\epsilon_{1}\) and an \(\epsilon_{2}\) so that Eq. (3) holds; (c) show that the \(\epsilon_{1}\) and the \(\epsilon_{2}\) found in part (b) both approach zero as \((\Delta x, \Delta y) \rightarrow(0,0)\). $$ f(x, y)=\frac{x^{2}}{y} $$

In Exercises 7 through 12, prove that for the given function \(f, \lim _{(x, y) \rightarrow(0,0)} f(x, y)\) does not exist. \(f(x, y)=\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\)

In Exercises 1 through 8 , do each of the following: (a) Find \(D_{11} f(x, y)\); (b) find \(D_{22} f(x, y) ;\) (c) show that \(D_{12} f(x, y)=D_{21} f(x, y) .\) $$ f(x, y)=x \cos y-y e^{x} $$

In Exercises 5 through 10, find the indicated partial derivative by using the chain rule. $$ u=x y+x z+y z ; x=r s ; y=r^{2}-s^{2} ; z=(r-s)^{2} ; \frac{\partial u}{\partial r} ; \frac{\partial u}{\partial s} $$

In Exercises 7 through 12, prove that for the given function \(f, \lim _{(x, y) \rightarrow(0,0)} f(x, y)\) does not exist. \(f(x, y)=\frac{x^{2}}{x^{2}+y^{2}}\)

Given \(f(x, y)= \begin{cases}x+y-2 & \text { if } x=1 \text { or } y=1 \\ 2 & \text { if } x \neq 1 \text { and } y \neq 1\end{cases}\) Prove that \(D_{1} f(1,1)\) and \(D_{2} f(1,1)\) exist, but \(f\) is not differentiable at \((1,1)\).

In Exercises 7 through 12, prove that for the given function \(f, \lim _{(x, y) \rightarrow(0,0)} f(x, y)\) does not exist. \(f(x, y)=\frac{x^{3}+y^{3}}{x^{2}+y}\)

In Exercises 5 through 10, find the indicated partial derivative by using the chain rule. $$ u=x^{2} y z ; x=\frac{r}{s^{\prime}} y=r e^{s} ; z=r e^{-s} ; \frac{\partial u}{\partial r} ; \frac{\partial u}{\partial s} $$

Given \(f(x, y)= \begin{cases}\frac{3 x^{2} y^{2}}{x^{4}+y^{4}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases}\) Prove that \(D_{1} f(0,0)\) and \(D_{2} f(0,0)\) exist, but \(f\) is not differentiable at \((0,0)\).

In Exercises 8 through 17, determine the region of continuity of \(f\) and draw a sketch showing as a shaded region in \(R^{2}\) the region of continuity of \(f\). f(x, y)=\frac{x^{2}+y^{2}}{\sqrt{9-x^{2}-y^{2}}}

In Exercises 7 through 12, prove that for the given function \(f, \lim _{(x, y) \rightarrow(0,0)} f(x, y)\) does not exist. \(f(x, y)=\frac{x^{4}+3 x^{2} y^{2}+2 x y^{3}}{\left(x^{2}+y^{2}\right)^{2}}\)

In Exercises 3 through 11 , find the domain and range of the function \(f\) and draw a sketch showing as a shaded region in \(R^{2}\) the set of points in the domain of \(f\). $$ f(x, y)=\sin ^{-1}(x+y) $$

In Exercises 11 through 14, find the total derivative \(d u / d t\) by two methods: (a) Use the chain rule; (b) make the substitutions for \(x\) and \(y\) or for \(x, y\), and \(z\) before differentiating. $$ u=y e^{x}+x e^{y} ; x=\cos t ; y=\sin t $$

In Exercises 8 through 17, determine the region of continuity of \(f\) and draw a sketch showing as a shaded region in \(R^{2}\) the region of continuity of \(f\). f(x, y)=\frac{x}{\sqrt{4 x^{2}+9 y^{2}-36}}

In Exercises 7 through 12, prove that for the given function \(f, \lim _{(x, y) \rightarrow(0,0)} f(x, y)\) does not exist. \(f(x, y)=\frac{x^{4} y^{4}}{\left(x^{2}+y^{4}\right)^{3}}\)

In Exercises 3 through 11 , find the domain and range of the function \(f\) and draw a sketch showing as a shaded region in \(R^{2}\) the set of points in the domain of \(f\). $$ f(x, y)=\ln (x y-1) $$

In Exercises 9 through 14, find the indicated partial derivatives. $$ f(u, v)=\ln \cos (u-v) ;\left(\text { a) } f_{\text {uuv }}(u, v) ;\left(\text { b) } f_{\text {vuv }}(u, v)\right.\right. $$

In Exercises 11 through 14 , find the total derivative \(d u / d t\) by two methods: (a) Use the chain rule; (b) make the substitutions for \(x\) and \(y\) or for \(x, y\), and \(z\) before differentiating. $$ u=\ln x y+y^{2} ; x=e^{t} ; y=e^{-t} $$

In Exercises 8 through 17, determine the region of continuity of \(f\) and draw a sketch showing as a shaded region in \(R^{2}\) the region of continuity of \(f\). f(x, y)=\ln \left(x^{2}+y^{2}-9\right)-\ln \left(1-x^{2}-y^{2}\right)

In Exercises 7 through 12, prove that for the given function \(f, \lim _{(x, y) \rightarrow(0,0)} f(x, y)\) does not exist. \(f(x, y)=\frac{x^{2} y^{2}}{x^{3}+y^{3}}\)

In Exercises 11 through 14 , find the total derivative \(d u / d t\) by two methods: (a) Use the chain rule; (b) make the substitutions for \(x\) and \(y\) or for \(x, y\), and \(z\) before differentiating. $$ u=\sqrt{x^{2}+y^{2}+z^{2}} ; x=\tan t ; y=\cos t ; z=\sin t ; 0

In Exercises 13 through 24 , find the indicated partial derivatives by holding all but one of the variables constant and applying theorems for ordinary differentiation. $$ f(x, y)=4 y^{3}+\sqrt{x^{2}+y^{2}} ; D_{1} f(x, y) $$

In Exercises 13 through 16, prove that \(\lim _{(x, y) \rightarrow(0,0)} f(x, y)\) exists. \(f(x, y)=\frac{x y}{\sqrt{x^{2}+y^{2}}}\)

In Exercises 11 through 14 , find the total derivative \(d u / d t\) by two methods: (a) Use the chain rule; (b) make the substitutions for \(x\) and \(y\) or for \(x, y\), and \(z\) before differentiating. $$ u=\frac{t+e^{x}}{y-e^{t}} ; x=3 \sin t ; y=\ln t $$

In Exercises 13 through 24 , find the indicated partial derivatives by holding all but one of the variables constant and applying theorems for ordinary differentiation. $$ f(x, y)=\frac{x+y}{\sqrt{y^{2}-x^{2}}} ; D_{2} f(x, y) $$