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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.

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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.

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

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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.

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

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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 ____.

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\(f(x,y)=x^{2}y+6y^{2}\)

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\(f(x,y)=3x^{2}+6xy^{3}\)

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

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

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

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

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\(f(x,y) =x^{2}e^{xy}\)

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

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

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

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

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

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\(z=f(x, y)=e^{( x^{2}+y^{2})^{1.2}}\)

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

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

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\(f(x,y)=(2x+3y) ( 3x-2y)\)

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

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\(f(x,y)=e^{2x+3y}\)

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

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

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\(f(x,y,z)=xy+yz+xz\)

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\(f(x,y,z)=xe^{y}+ye^{z}+ze^{x}\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

1. In the direction of the positive \(x\)-axis.
2. In the direction of the positive \(y\)-axis.

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

1. In the direction of the positive \(x\)-axis.
2. In the direction of the positive \(y\)-axis.

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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. 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. 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. 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.

Question

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. 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. 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. 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.

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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 \[ \dfrac{\partial V}{\partial T}\cdot \dfrac{\partial T}{\partial P}\cdot \dfrac{\partial P}{\partial V}=-1 \]

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

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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. 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. 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. Compare the answers. What do you conclude about the change in manufacturing productivity in the United States during the 20-year period?

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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. 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. 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. 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. 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 \(r = 5\%\)

839

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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 \(z.\)

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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 \(d\).

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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 \((x,t)\).

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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. 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. Graph the initial temperature distribution, \(y=T(0,x),\) where \(0\leq x\leq 20\).

Question

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 \(y=r\;\sin \theta\).

Question

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 \(\theta =\tan^{-1}\dfrac{y}{x},\) \(x\neq 0.\)

Question

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

Question

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

Question

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 \(v=e^{x}\sin y\).

Question

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 \(v=\tan ^{-1}\dfrac{y}{x}\).

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

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

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

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

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

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

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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 \(\alpha\).

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\(z=\ln\;\sqrt{x^{2}+y^{2}}\)

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

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\(z=\tan ^{-1}\dfrac{y}{x}\)

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

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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.

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

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

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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. 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. Show that \(f_{xy}(0,0)\neq f_{yx}(0,0)\).

Question

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. \(f_{x}(0,y)=-y\)
2. \(f_{y}(x,0)=x\)
3. \(f_{xy}(0,0)=-1\)
4. \(f_{yx}(0,0)=1\)

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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.

840

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

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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.

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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.

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

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

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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.

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

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

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

Question

\(f(x,y) =x^{y}\)

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\(f(x,y)=x^{2x+3y}\)

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\(f(x,y,z)=x^{y+z}\)

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\(f(x,y,z)=x^{yz}\)

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\(f(x,y,z)=(x+y)^{z}\)

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\(f(x,y,z)=(xy) ^{z}\)

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

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

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

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

Question

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

Question

1. 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. Write an equation of the plane determined by these two lines.
3. What is the geometric relationship of this plane to the surface \(z=f(x, y)\)?

Question

Consider two coordinate systems as given in the figure.

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