1. True or False  To find the partial derivative $f_{y}(x,y) ,$ differentiate $f$ with respect to $x$ while treating $y$ as if it were a constant.

2. Multiple Choice  The partial derivative $f_{x}(x_{0},y_{0})$ equals the slope of the tangent line to the curve of intersection of the surface $z=f(x,y)$ and the plane [(a) $x=x_{0},$ (b) $y=y_{0},$ (c) $z=z_{0},$ (d) $x+y+z=0$] at the point $(x_{0},y_{0},f(x_{0},y_{0}) )$ on the surface.

3. True or False  $f_{x}(x,y)$ equals the rate of change of $f$ in the direction of the positive $x$-axis.

4. The two second-order partial derivatives $\dfrac{\partial ^{2}z}{\partial x\partial y}$ and $\dfrac{\partial ^{2}z}{\partial y\partial x}$ are called ____ partials.

5. True or False  If $f(x,y)=x\;\cos\;y,$ then $f_{x}(x,y)=$ $f_{y}(x,y)$.

6. For a function $w=f(x,y,z)$ of three variables, to find the partial derivative $f_{y}(x,y,z)$, treat the variables ____ and ____ as constants, and differentiate $f$ with respect to ____.

7. $f(x,y)=x^{2}y+6y^{2}$

8. $f(x,y)=3x^{2}+6xy^{3}$

9. $f(x,y)=\dfrac{x-y}{x+y}$

10. $f(x,y)=\dfrac{x+y}{y^{2}}$

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

12. $f(x,y)=x^{2}\cos y+y^{2}\sin x$

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

14. $f(x,y)=\cos (x^{2}y^{3})$

15. $z= f(x,y)=\tan ^{-1}\dfrac{y}{x}$

16. $z=f(x,y)=\sin ^{2}(2xy)$

17. $z=f(x,y) =\sin ( e^{x^{2}y})$

18. $z=f(x, y)=\sin [ \ln\;(x^{2}+y^{2})]$

19. $z=f(x, y)=e^{( x^{2}+y^{2})^{1.2}}$

20. $z=f(x,y)=\ln\;\sqrt{x^{2}+y^{2}}$

21. $f(x,y)=6x^{2}-8xy+9y^{2}$

22. $f(x,y)=(2x+3y) ( 3x-2y)$

23. $f(x,y)=\ln\;(x^{3}+y^{2})$

24. $f(x,y)=e^{2x+3y}$

25. $f(x,y)=\cos (x^{2}y^{3})$

26. $f(x,y)=\sin ^{2}(xy)$

27. $f(x,y,z)=xy+yz+xz$

28. $f(x,y,z)=xe^{y}+ye^{z}+ze^{x}$

29. $f(x,y,z)=xy\;\sin\;z-yz\sin\;x$

30. $f(x,y,z)=\dfrac{1}{\sqrt{x^{2}+y^{2}+z^{2}}}$

31. $f(x,y,z)=z\;\tan ^{-1}\dfrac{y}{x}$

32. $f(x,y,z)=\tan ^{-1}\dfrac{xy}{z}$

33. $f(x,y,z)=\sin [\ln\;(x^{2}+y^{2}+z^{2})]$

34. $f(x,y,z)=e^{x^{2}+y^{2}}\ln\;z$

35. $f(x,y)=\left\{ \begin{array}{@{}c@{\quad}c@{\quad}c} \dfrac{x^{3}+y^{3}}{x^{2}+y^{2}} & \hbox{if} & (x,y)\neq (0,0)\\ 0 & \hbox{if} & (x,y)=(0,0) \end{array} \right.$

36. $f(x,y)=\left\{ \begin{array}{@{}c@{\quad}c@{\quad}c} \dfrac{x^{2}y^{2}}{x^{2}+4y^{3}} & \hbox{if} & (x,y)\neq (0,0)\\ 0 & \hbox{if} & (x,y)=(0,0) \end{array} \right.$

37. $z=x^{2}+y^{2}$ and $y=2$ at $(1,2,5)$

38. $z=x^{2}-y^{2}$ and $x=3$ at $(3,1,8)$

39. $z=\sqrt{1-x^{2}-y^{2}}$ and $x=0$ at $\left( 0,\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right)$

40. $z=\sqrt{16-x^{2}-y^{2}}$ and $y=2$ at $( \sqrt{3},2,3)$

41. $z=\sqrt{x^{2}+y^{2}}$ and $x=4$ at $( 4,2,2\sqrt{5})$

42. $z=e^{x}\ln\;y$ and $y=e$ at $(0,e,1)$

43. Find the rate of change of $z=\ln\;\sqrt{x^{2}+y^{2}}$ at $(3,4,\ln\;5)$,

    1. (a) In the direction of the positive $x$-axis.
    2. (b) In the direction of the positive $y$-axis.

44. Find the rate of change of $z=e^{y}\sin x$ at $\left( \dfrac{\pi }{3},0,\dfrac{\sqrt{3}}{2}\right)\! ,$

    1. (a) In the direction of the positive $x$-axis.
    2. (b) In the direction of the positive $y$-axis.

45. Temperature Distribution  The temperature distribution $T$ (in degrees Celsius) of a heated plate at a point $(x,y)$ in the $xy$-plane is modeled by $$T=T(x,y)=\left( \frac{100}{\ln\;2}\right)\;\ln\;(x^{2}+y^{2}) \qquad 1\leq x^{2}+y^{2}\leq 9$$

    1. (a) Show that $T=0\ ^{\circ}{\rm C}$ on the circle $x^{2}+y^{2}=1$, and $T=200\ ^{\circ}{\rm C}$ on the circle $x^{2}+y^{2}=4$.
    2. (b) Find the rate of change of $T$ in the direction of the positive $x$-axis at the point $(1,2)$ and at the point $(2,1)$. Describe the rate of change.
    3. (c) Find the rate of change of $T$ in the direction of the positive $y$-axis at the point $(2,0)$ and at the point $(0,2)$. Describe the rate of change.

46. Temperature Distribution  The temperature distribution (in degrees Celsius) of a heated plate at a point $(x,y)$ in the $xy$-plane is modeled by $T=T(x,y)=\dfrac{100}{\sqrt{x^{2}+y^{2}}}$, $1\leq x^2 + y^2\leq 9$.

    1. (a) Show that $T=100{{}^{\circ}{\rm C}}$ on the circle $x^{2}+y^{2}=1$, and $T=50{ {}^{\circ}{\rm C}}$ on the circle $x^{2}+y^{2}=4$.
    2. (b) Find the rate of change of $T$ in the direction of the positive $x$ -axis at the point $(1,0)$ and at the point $(0,1)$. Describe the rate of change.
    3. (c) Find the rate of change of $T$ in the direction of the positive $y$-axis at the point $(2,0)$ and at the point $(0,2)$. Describe the rate of change.

47. Thermodynamics  The Ideal Gas Law $PV=nrT$ is used to describe the relationship between pressure $P$, volume $V$, and temperature $T$ of a confined gas, where $n$ is the number of moles of the gas and $r$ is the universal gas constant. Show that $$\dfrac{\partial V}{\partial T}\cdot \dfrac{\partial T}{\partial P}\cdot \dfrac{\partial P}{\partial V}=-1$$

48. Thermodynamics  The volume $V$ of a fixed amount of gas varies directly with the temperature $T$ and inversely with the pressure $P$. That is, $V=k\dfrac{T}{P},$ where $k>0$ is a constant.

    1. (a) Find $\dfrac{\partial V}{\partial T}$ and $\dfrac{\partial V}{\partial P}$.
    2. (b) Show that $T\,\dfrac{\partial V}{\partial T}+P\,\dfrac{\partial V}{\partial P}=0$.

49. Economics  The data used to develop the Cobb–Douglas productivity model included capital input $K$ and labor input $L$ for each year during the period 1899–1922. Using the model $P=aK^{b}L^{1-b}$ and multiple regression techniques, Cobb and Douglas determined that manufacturing productivity was represented by the function $$P=1.014651K^{0.254124}L^{0.745876}\approx 1.01K^{0.25}L^{0.75}$$

    1. (a) Find the marginal productivity with respect to capital input and the marginal productivity with respect to labor input in 1900 when $K=107$ and $L=105.$
    2. (b) Find the marginal productivity with respect to capital input and the marginal productivity with respect to labor input in 1920 when $K=407$ and $L=193.$
    3. (c) Compare the answers. What do you conclude about the change in manufacturing productivity in the United States during the 20-year period?

50. Economics  The function $$z=f(x,y,r)=\frac{1+(1-x)y}{1+r}-1$$

    describes the net gain or loss of money invested, where $x=$ annual marginal tax rate, $y=$ annual effective yield on an investment, and $r=$ annual inflation rate.

    1. (a) Find the annual net gain or loss if money is invested at an effective yield of $4\%$ when the marginal tax rate is $25\%$ and the inflation rate is $5\%$; that is, find $f(0.25,0.04,0.05)$.
    2. (b) Find the rate of change of gain (or loss) of money with respect to the marginal tax rate when the effective yield is $4\%$ and the inflation rate is $5\%$.
    3. (c) Find the rate of change of gain (or loss) of money with respect to the effective yield when the marginal tax rate is $25\%$ and the inflation rate is $5\%$.
    4. (d) Find the rate of change of gain (or loss) of money with respect to the inflation rate when the marginal tax rate is $25\%$ and the effective yield is $4\%$. Use $r = 5\%$

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51. Economics  Let $w=2x^{1/2}y^{1/3}z^{1/6}$ be a production function that depends on three inputs: $x,$ $y,$ and $z$. Find the marginal productivity with respect to $x,$ the marginal productivity with respect to $y,$ and the marginal productivity with respect to $z.$

52. Speed of Sound  The speed $v$ of sound in a gas depends on the pressure $p$ and density $d$ of the gas and is modeled by the formula $v(p,d)=k\sqrt{\dfrac{p}{d}}$, where $k$ is some constant. Find the rate of change of speed with respect to $p$ and with respect to $d$.

53. Vibrating Strings  Suppose a vibrating string is governed by the equation $f(x,t)=2\cos ( 5t) \sin x$, where $x$ is the horizontal distance of a point on the string, $t$ is time, and $f(x,t)$ is the displacement. Show that $\dfrac{\partial ^{2}f}{\partial t^{2}}=25\dfrac{ \partial ^{2}f}{\partial x^{2}}$ at all points $(x,t)$.

54. Temperature Distribution  Suppose a thin metal rod extends along the $x$-axis from $x=0$ to $x=20$, and for each $x$, where $0\leq x\leq 20$, the temperature $T$ of the rod at time $t\geq 0$ and position $x$ is $T(t,x)=40e^{-\lambda t}\sin \dfrac{\pi x}{20}$, where $\lambda >0$ is a constant.

    1. (a) Show that $T_{t}=-\lambda T$, $T_{xx}=-\dfrac{\pi ^{2}}{400}T$, and $T_{t}=\dfrac{1}{k^{2}}T_{xx}$ for some $k$.
    2. (b) Graph the initial temperature distribution, $y=T(0,x),$ where $0\leq x\leq 20$.

55. Find $\dfrac{\partial x}{\partial r}, \dfrac{\partial x}{\partial \theta }, \dfrac{\partial y}{\partial r}$, and $\dfrac{\partial y}{\partial \theta }$ if $x=r\cos \theta$ and $y=r\;\sin \theta$.

56. Find $\dfrac{\partial r}{\partial x}, \dfrac{\partial \theta }{\partial x}, \dfrac{\partial r}{\partial y}$, and $\dfrac{\partial \theta }{\partial y}$

    if $r=\sqrt{x^{2}+y^{2}}$ and $\theta =\tan^{-1}\dfrac{y}{x},$ $x\neq 0.$

57. 

    1. (a) Graph $f(x,y) =\dfrac{1}{2}x^{2}+\dfrac{1}{3}y^{2}$ and the plane $y=1.$
    2. (b) Find symmetric equations of the tangent line to the curve of intersection of the surface and the plane at the point $\left( 2,1,\dfrac{7}{3}\right)$.

58. 

    1. (a) Graph $f(x,y) =\dfrac{5}{\sqrt{x^{2}+y^{2}+1}}$ and the plane $x=1.$
    2. (b) Find symmetric equations of the tangent line to the curve of intersection of the surface and the plane at the point $\left( 1,-2,\dfrac{5\sqrt{6}}{6}\right)$.

59. Show that $\dfrac{\partial u}{\partial x}=\dfrac{\partial v}{\partial y}$ and $\dfrac{\partial u}{\partial y}=-\dfrac{\partial v}{\partial x}$ for $u=e^{x}\cos y$ and $v=e^{x}\sin y$.

60. Show that $\dfrac{\partial u}{\partial x}=\dfrac{\partial v}{\partial y}$ and $\dfrac{\partial u}{\partial y}=-\dfrac{\partial v}{\partial x}$ for $u=\ln\;\sqrt{x^{2}+y^{2}}$ and $v=\tan ^{-1}\dfrac{y}{x}$.

61. If $u=x^{2}+4y^{2}$, show that $x\dfrac{\partial u}{\partial x}+y\dfrac{\partial u}{\partial y}=2u$.

62. If $u=xy^{2}$, show that $x\dfrac{\partial u}{\partial x}+y \dfrac{\partial u}{\partial y}=3u$.

63. If $w=x^{2}+y^{2}-3yz$, show that $x\dfrac{\partial w}{\partial x}+y\dfrac{\partial w}{\partial y}+z\dfrac{\partial w}{\partial z}=2w$.

64. If $w=\dfrac{xz+y^{2}}{yz}$, show that $x\dfrac{\partial w}{\partial x}+y\dfrac{\partial w}{\partial y}+z\dfrac{\partial w}{\partial z}=0$.

65. If $z=\cos (x+y)+\cos (x-y)$, show that $\dfrac{\partial ^{2}z}{\partial x^{2}}-\dfrac{\partial ^{2}z}{\partial y^{2}}=0$.

66. If $z=\sin (x-y)+\ln\;(x+y)$, show that $\dfrac{\partial ^{2}z}{\partial x^{2}}=\dfrac{\partial ^{2}z}{\partial y^{2}}$.

67. Show that $u=e^{-\alpha ^{2}t}\sin \left( \alpha x\right)$ satisfies the equation $\dfrac{\partial u}{\partial t}=\dfrac{\partial ^{2}u}{\partial x^{2}}$ for all values of the constant $\alpha$.

68. $z=\ln\;\sqrt{x^{2}+y^{2}}$

69. $z=e^{ax}\sin ( ay)$

70. $z=\tan ^{-1}\dfrac{y}{x}$

71. $z=e^{ax}\cos ( ay)$

72. Harmonic Functions  Suppose $u(x,y)$ and $v(x,y)$ have continuous second-order partial derivatives, $u_{x}=v_{y}$ and $u_{y}=-v_{x}$. Show that $u$ and $v$ are harmonic functions.

73. Harmonic Functions  If $u=z\tan ^{-1}\dfrac{x}{y}$, show that $$\dfrac{\partial ^{2}u}{\partial x^{2}}+\dfrac{\partial ^{2}u}{\partial y^{2}} +\dfrac{\partial ^{2}u}{\partial z^{2}}=0.$$

74. Harmonic Functions  Show that $f(x,y,z)=(x^{2}+y^{2}+z^{2})^{-1/2}$ satisfies the three-dimensional Laplace equation $$f_{xx}+f_{yy}+f_{zz}=0$$

75. Let $f(x,y)=\left\{ \begin{array}{@{}c@{\quad}c@{\quad}c} \dfrac{xy^{3}}{x^{2}+y^{2}} & \hbox{if} & (x,y)\neq (0,0) \\ 0 & \hbox{if} & (x,y)=(0,0) \end{array} \right.$

    1. (a) Find $f_{x}$ and $f_{y}$. [Hint: $f_{x}$ and $f_{y}$, where $(x,y)\neq (0,0)$, can be found using derivative formulas. To find $f_{x}(0,0)$ and $f_{y}(0,0)$, use the definition of a partial derivative.]
    2. (b) Show that $f_{xy}(0,0)\neq f_{yx}(0,0)$.

76. Let $f(x,y)=\left\{ \begin{array}{@{}c@{\quad}c@{\quad}c} \dfrac{xy(x^{2}-y^{2})}{x^{2}+y^{2}} & \hbox{if} & (x,y)\neq (0,0) \\ 0 & \hbox{if} & (x,y)=(0,0) \end{array} \right.$

    Show that:

    1. (a) $f_{x}(0,y)=-y$
    2. (b) $f_{y}(x,0)=x$
    3. (c) $f_{xy}(0,0)=-1$
    4. (d) $f_{yx}(0,0)=1$

77. If you are told that $f$ is a function of two variables whose partial derivatives are $f_{x}(x,y)=3x-y$ and $f_{y}(x,y)=x-3y$, should you believe it? Explain.

*Laplace’s equation is important in many applications including fluid dynamics, heat, elasticity, and electricity.

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78. Show that there is no function $z=f(x,y)$ for which $f_{x}(x,y)=2x-y$ and $f_{y}(x, y)=x-2y$.

79. Use the definition of a partial derivative to show that the function $z=\sqrt{x^{2}+y^{2}}$ does not have partial derivatives at $(0,0)$. By discussing the graph of the function, give a geometric reason why this should be so.

80. If $z=f(x,y)=4x^{2}+9y^{2}-12$, interpret $f_{x}\left( 1,-\dfrac{1}{3}\right)$ and $f_{y}\left( 1,-\dfrac{1}{3}\right)$ geometrically.

81. Show that $xf_{x}+yf_{y}+zf_{z}=0$ for $f(x,y,z)= e^{x/y}+e^{y/z}+e^{z/x}$.

82. Find $a$ in terms of $b$ and $c$ so that $f(t,x,y)= e^{at}\sin\;(bx)\;\cos\;(cy)$ satisfies $f_{t}=f_{xx}+f_{yy}$.

83. Wave Equation  Show that $f(x,t)=\cos (x+ct)$ satisfies the one-dimensional wave equation $f_{tt}=c^{2}f_{xx}$, where $c$ is a constant.

84. Find $f_{x}$ and $f_{y}$ if $f(x, y)=\int_{x}^{y}\ln\;(\cos\;\sqrt{t})\ dt$.

85. Find $\dfrac{\partial a}{\partial b}$, $\dfrac{\partial a}{\partial c}$, and $\dfrac{\partial a}{\partial A}$ for the Law of Cosines: $a^{2}=b^{2}+c^{2}-2bc\;\cos\;A$.

86. If $x=r\cos\;\theta$ and $y=r\sin\;\theta$, show that $$\begin{equation*} \left\vert \begin{array}{c@{\quad}c} \dfrac{\partial x}{\partial r} & \dfrac{\partial x}{\partial \theta } \\ \dfrac{\partial y}{\partial r} & \dfrac{\partial y}{\partial \theta } \end{array} \right\vert =r \qquad\hbox{and}\qquad \left\vert \begin{array}{c@{\quad}c} \dfrac{\partial r}{\partial x} & \dfrac{\partial r}{\partial y} \\ \dfrac{\partial \theta }{\partial x} & \dfrac{\partial \theta }{\partial y} \end{array} \right\vert =\frac{1}{r} \end{equation*}$$

87. $f(x,y) =x^{y}$

88. $f(x,y)=x^{2x+3y}$

89. $f(x,y,z)=x^{y+z}$

90. $f(x,y,z)=x^{yz}$

91. $f(x,y,z)=(x+y)^{z}$

92. $f(x,y,z)=(xy) ^{z}$

93. Find $f_{x}$ and $f_{y}$ at $(0,0)$ if $$f(x, y)=\left\{ \begin{array}{@{}c@{\quad}c@{\quad}c} e^{-1/(x^{2}+y^{2})} & \hbox{if} & (x,y)\neq (0,0)\\ 0 & \hbox{if} & (x,y)=(0,0) \end{array} \right.$$

94. Find $f_{x},$ $f_{y},$ $f_{xx},$ $f_{yy}$, and $f_{xy}$ for $f(x,y)=(xy)^{xy}$. What is the domain of $f$?

95. Show that the following function has first partial derivatives at all points in the plane: $$f(x,y)=\left\{ \begin{array}{@{}c@{\quad}c@{\quad}c} \dfrac{x^{3}-y^{3}}{x^{2}+y^{2}} & \hbox{if} & (x,y)\neq (0,0)\\ 0 & \hbox{if} & (x,y)=(0,0) \end{array} \right.$$

96. Laplace’s Equation in Polar Coordinates  Show that the function $f(r,\theta )=r^{n}\sin ( n\theta )$ satisfies the Laplace equation $$f_{r\,r}+\dfrac{1}{r}\,f_{r}+\dfrac{1}{r^{2}}\,f_{\theta \,\theta }=0$$

97. Let $u=r^{m}\cos ( m\theta )$. Show that $$\dfrac{\partial ^{2}u}{\partial r^{2}}+\dfrac{1}{r^{2}}\left( \dfrac{ \partial ^{2}u}{\partial \theta ^{2}}\right) +\dfrac{1}{r}\left(\dfrac{ \partial u}{\partial r}\right) = 0\;{\rm for\;all}\;m.$$

98. 1. (a) Find symmetric equations of the tangent lines at $( x_{0},y_{0}, f(x_{0},y_{0}))$ to the curve of intersection of $z=f(x,y)$ and $y=y_{0}$, and the curve of intersection of $z=f(x, y)$ and $x=x_{0}$.
    2. (b) Write an equation of the plane determined by these two lines.
    3. (c) What is the geometric relationship of this plane to the surface $z=f(x, y)$?

99. Consider two coordinate systems as given in the figure.

    1. (a) Let $f(x,y)=3x^{2}+4y^{3}$. Find $f_{x}(1,6)$ and $f_{y}(1,6)$.
    2. (b) Let $(a,b)$ be the $x^{\prime }y^{\prime }$-coordinates of $(1,6)$. Let $\bar{f}(x^{\prime },y^{\prime })=f(x,y)$. Find $\bar{f}_{x^{\prime }}(a,b)$ and $\bar{f}_{y^{\prime }}(a,b)$.