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True or False If a differentiable function \(z\) is defined implicitly by the equation \(F(x,y,z)=0\), then \(\dfrac{\partial z}{ \partial x}=\dfrac{F_{x}(x,y,z)}{F_{z}(x,y,z)},\) provided \(F_{z}(x,y,z)\neq 0.\)

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True or False If \(x=x( t)\) and \(y=y( t)\) are differentiable functions of \(t\) and if \(z=f(x,y)\) is a differentiable function of \(x\) and \(y,\) then \(\dfrac{dz}{dt}=\dfrac{ \partial z}{\partial x}+\dfrac{\partial z}{\partial y}.\)

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

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\(z=x^{2}-y^{2}\), \(x=\sin ( 2t)\), \(y=\cos t\)

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\(z=x^{2}+y^{2}\), \(x=te^{t}\), \( y=te^{-t}\)

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\(z=x^{2}-y^{2}\), \(x=te^{-t}\), \(y=t^{2}e^{-t}\)

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\(z=e^{u}\sin v\), \(u=\sqrt{t}\), \(v=\pi t\)

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\(z=e^{u/v}\), \(u=\sqrt{t}\), \(v=t^{3}+1\)

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\(z=e^{u/v}\), \(u=te^{t}\), \(v=e^{t^{2}}\)

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\(z=\ln (uv)\), \(u=t^{5}\), \(v=\sqrt{t+1}\)

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\(z=e^{x^{2}+y^{2}}\), \(x=\sin (2t) \), \(y=\cos t\)

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\(z=e^{x^{2}-y^{2}}\), \(x=\sin (2t) \), \(y=\cos (2t) \)

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\(z=\dfrac{xy}{x^{2}+y^{2}}\), \(x=\sin t\), \(y=\cos t\)

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\(z=y\ln x+xy+\tan y\), \(x=\dfrac{t}{t+1}\), \(y=t^{3}-t\)

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\(p=x^{2}+y^{2}-z^{2},\) \(x=te^{t}\), \(y=te^{-t},\) \(z=e^{2t}\)

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\(p=x^{2}-y^{2}-z^{2}\), \(x=te^{-t}\), \(y=t^{2}e^{-t}\), \(z=e^{-t}\)

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\(p=e^{x}\sin y\cos z,\) \(x=\sqrt{t}\), \(y=\pi t,\) \(z=\dfrac{t}{2}\)

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\(p=\ln (xyz)\), \(x=t^{5}\), \(y=\sqrt{t+1},\) \(z=t^{2}\)

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\(p=w \ln \left( \dfrac{u}{v}\right) \), \(u=te^{t}\), \(v=e^{t^{2}},\) \(w=e^{2t}\)

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\(p=we^{u/v},\) \(u=\sqrt{t}\), \(v=t^{3}+1\), \(w=e^{t}\)

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\(p=u^{2}vw,\) \(u=\sin t\), \(v=\cos t\), \(w=e^{t}\)

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\(p=\sqrt{uvw}\), \(u=e^{t}\), \(v=te^{t}\), \(w=t^{2}e^{2t}\)

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\(z=x^{2}+y^{2},\) \(x=ue^{v},\) \(y=ve^{u}\)

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\(z=x^{2}-y^{2},\) \(x=u\ln v,\) \(y=v\ln u\)

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\(z=e^{x}\sin y,\) \(x=u^{2}v,\) \(y=\ln (uv)\)

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\(z=\dfrac{1}{y}\ln x,\) \(x=\sqrt{uv},\) \(y=\dfrac{v}{u}\)

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\(z=\ln (x^{2}+y^{2}),\) \(x=\dfrac{v^{2}}{u},\) \(y=\dfrac{u}{v^{2}}\)

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\(z=x\sin y-y\sin x,\) \(x=u^{2}v,\) \(y=uv^{2}\)

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\(z=x^{2}+y^{2},\) \(x=\sin (u-v),\) \(y=\cos (u+v)\)

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\(z=e^{x}+y,\) \(x=\tan ^{-1}\left( \dfrac{u}{v}\right) ,\) \(y=\ln (u+v)\)

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\(z=se^{r},\) \(r=u^{2}+v^{2},\) \(s=\dfrac{v}{u}\)

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\(z=\sqrt{s^{2}+r^{2}}\), \(s=\ln (uv),\) \(r=\sqrt{uv}\)

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\(z=xy^{2}w^{3},\) \(x=2u+v,\) \(y=5u-3v,\) \(w=2u+3v\)

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\(z=x^{2}-y^{2}+w,\) \(x=e^{u+v},\) \(y=uv,\) \(w=\dfrac{v}{u}\)

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Find \(\dfrac{\partial f}{\partial u},\) \(\dfrac{\partial f}{ \partial v},\) \(\dfrac{\partial f}{\partial w}\) if \( f(x,y,z)=x^{2}+y^{2}+z^{2},\) \(x=uv\), \(y=e^{u+2v+3w}\), \(\ z=2v+3w\).

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Find \(\dfrac{\partial f}{\partial u},\) \(\dfrac{\partial f}{ \partial v},\) \(\dfrac{\partial f}{\partial w}\) if \(f(x,y,z)=x-y^{2}+z^{2},\) \(x=\sqrt{u+v}\), \(y=(u+w) \ln v\), \(\ z=2-v+3w\).

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Find \(\dfrac{\partial f}{\partial u},\) \(\dfrac{ \partial f}{\partial v},\) \(\dfrac{\partial f}{\partial w}\) if \( f(x,y,z)=x\cos y-z\cos x+x^{2}yz,\) \(x=uvw,\) \(y=u^{2}+v^{2}+w^{2},\) \(\ z=w\).

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Find \(\dfrac{\partial f}{\partial u},\) \(\dfrac{\partial f}{ \partial v},\) \(\dfrac{\partial f}{\partial w}\) if \(\ f(x,y,z)=x^{2}+y^{2}\), \(x=\sin (u-v),\) \(y=\cos (u+v)\), \(\ z=uw^{2}\).

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Find \(\dfrac{\partial f}{\partial u},\) \(\dfrac{\partial f}{ \partial v},\) \(\dfrac{\partial f}{\partial w},\) \(\dfrac{\partial f}{ \partial t}\) if \(\ f(x,y,z)=x+2y^{2}-z^{2},\) \(x=ut\), \(\ y=e^{u+2v+3w+4t}\), \(\ z=u+\dfrac{1}{2}v+4t\).

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Find \(\dfrac{\partial f}{\partial u},\) \(\dfrac{\partial f}{ \partial v},\) \(\dfrac{\partial f}{\partial w},\) \(\dfrac{\partial f}{ \partial t}\) if \(f(x,y,z)=x^{2}+y^{2}+z,\) \(x=\sin ( u+t)\), \( \ y=\cos (v-t) \), \(\ z=uw^{2}\).

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\(F(x, y)=x^{2}y-y^{2}x+xy-5=0\)

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\(F(x, y)=x^{3}y^{2}-xy+x^{2}y-10=0\)

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\(F(x, y)=x\sin y+y\sin x-2=0\)

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\(F(x, y)=xe^{y}+ye^{x}-xy=0\)

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\(F(x, y)=x^{1/3}+y^{1/3}-1=0\)

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\(F(x, y)=x^{2/3}+y^{2/3}-1=0\)

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\(F(x,y,z)=xz+3yz^{2}+x^{2}y^{3}-5z=0\)

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\(F(x,y,z)=x^{2}z+y^{2}z+x^{3}y-10z=0\)

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\(F(x,y,z)=\sin z+y\cos z+xyz-10=0\)

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\(F(x,y,z)=x\sin y-\cos z+x^{2}z=0\)

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\(F(x,y,z)=xe^{yz}+ye^{xz}+xyz=0\)

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\(F(x,y,z)=e^{yz}\ln x+ye^{xz}-yz=0\)

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\(w=(2x+3y)^{4z}\)

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\(w=(2x)^{3y+4z}\)

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Ideal Gas Law One mole of a gas obeys the Ideal Gas Law \( PV=20T\), where \(P\) is pressure, \(V\) is volume, and \(T\) is temperature. If the temperature \(T\) of the gas is increasing at the rate of 5 \(^{\circ}{\rm C}/{\rm s}\) and if, when the temperature is 80 \(^{\circ}{\rm C}\), the pressure \(P\) is \(10{\rm N}/{\rm m}^{2}\) and is decreasing at the rate of \(2\dfrac{{\rm N}}{{\rm m}^{2}\cdot {\rm s}}\), find the rate of change of the volume \(V\) with respect to time.

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Melting Ice A block of ice of dimensions \(l,\) \(w,\) and \(h\) is melting. When \(l=3{\rm m}\), \(w=2{\rm m},\) \(h=1{\rm m},\) these variables are changing so that \(\dfrac{dl}{dt}=-1{\rm m}/{\rm h},\) \(\dfrac{dw}{dt}=-1 {\rm m}/{\rm h}\), and \(\dfrac{dh}{dt}=-0.5{\rm m}/{\rm h}\).

1. What is the rate of change in the surface area of the block of ice?
2. What is the rate of change in the volume of the block of ice?

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Wave Equation The one-dimensional wave equation \(\dfrac{ \partial ^{2}f}{\partial x^{2}}=\dfrac{1}{v^{2}}\dfrac{\partial ^{2}f}{ \partial t^{2}}\) describes a wave traveling with speed \(v\) along the \(x\) -axis. The function \(f\) represents the displacement \(x\) from the equilibrium of the wave at time \(t\).

1. Show that \(z=f(x,t) =\sin (x+vt)\) satisfies the wave equation.
2. Show that \(z=f(x,t) =e^{x-vt}\) satisfies the wave equation.
3. Show that \(z=f(x,t) =\sin x+\sin (vt) \) does not satisfy the wave equation.
4. Show that any twice-differentiable function of the form \(f(x+vt)\) is a solution of the wave equation.

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Economics A toy manufacturer’s production function satisfies a Cobb–Douglas model, \(Q( L,M) =400L^{0.3}M^{0.7}\), where \(Q\) is the output in thousands of units, \(L\) is the labor in thousands of hours, and \(M\) is the machine hours (in thousands). Suppose the labor hours are decreasing at a rate of \(4000{\rm h}/{\rm yr}\) and the machine hours are increasing at a rate of \(2000{\rm h}/{\rm yr}.\) Find the rate of change of production when

1. \(L=19\) and \(M=21\)
2. \(L=21\) and \(M=20\)

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Related Rates Let \(y=f(a,b),\) where \(a=h(s,t)\) and \(\ b=k(s,t) \). Suppose when \(s=1\) and \(t=3\), we know that \[ \frac{\partial h}{\partial s}=4\qquad \frac{\partial k}{\partial s} =-3\qquad \frac{\partial h}{\partial t}=1\qquad \frac{\partial k}{ \partial t}=-5 \]

Also, suppose \[ h(1,3)=6 \qquad k(1,3)=2 \qquad f_{a}( 6,2) =7 \qquad f_{b}( 6,2) =2 \]

What are \(\dfrac{\partial y}{\partial s}\) and \(\dfrac{\partial y}{\partial t} \) at the point \((1,3)\)?

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Related Rates Let \(z=f(x,y),\) where \(x=g(s,t)\) and \(\ y=h(s,t)\) . Suppose when \(s=2\) and \(t=-1\), we know that \[ \frac{\partial x}{\partial s}=5\qquad \frac{\partial y}{\partial s} =2\qquad \frac{\partial x}{\partial t}=-3\qquad \frac{\partial y}{ \partial t}=-2 \]

Also, suppose \[ g(2,-1)=3 \quad\hspace{6pt} h(2,-1)=4\quad\hspace{6pt} f_{x}( 3,4) =12\quad\hspace{6pt} f_{y}( 3,4) =7 \]

What are \(\dfrac{\partial z}{\partial s}\) and \(\dfrac{\partial z}{\partial t} \) at the point \((2,-1)\)?

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If \(z=f(x, y),\) where \(x=g(u,v)\) and \(y=h(u,v)\), find expressions for \(\dfrac{\partial ^{2}z}{\partial u^{2}}\), \(\dfrac{\partial ^{2}z}{\partial u~\partial v}\), and \( \dfrac{\partial ^{2}z}{\partial v^{2} }.\)

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If \(z=f(x, y)\) and \(x=r\cos \theta \), \(y=r\sin \theta \), show that \[ \left( \frac{\partial z}{\partial r}\right) ^{2}+\frac{1}{r^{2}}\left( \frac{ \partial z}{\partial \theta }\right) ^{2}=\left( \frac{\partial z}{\partial x }\right) ^{2}+\left( \frac{\partial z}{\partial y}\right) ^{2} \]

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If \(z=f(x, y),\) where \(x=u\cos \theta -v\sin \theta \) and \( y=u\sin \theta +v\cos \theta \), with \(\theta \) a constant, show that \(\left( \dfrac{\partial f}{\partial u}\right) ^{2}+\left( \dfrac{\partial f}{ \partial v}\right) ^{2}=\left( \dfrac{\partial f}{\partial x}\right) ^{2}+\left( \dfrac{\partial f}{\partial y}\right) ^{2}\).

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If \(z=f(u-v,v-u)\), show that \(\dfrac{\partial z}{ \partial u}+\dfrac{\partial z}{\partial v}=0\).

(Hint: Let \(x=u-v\) and \(y=v-u\).)

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If \(z=vf(u^{2}-v^{2})\), show that \(v\dfrac{\partial z}{ \partial u}+u\dfrac{\partial z}{\partial v}=\dfrac{uz}{v}.\)

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If \(w=f(u)\) and \(u=\sqrt{x^{2}+y^{2}+z^{2}}\), show that \( \left( \dfrac{\partial w}{\partial x}\right) ^{2}+\left( \dfrac{\partial w}{ \partial y}\right) ^{2}+\left( \dfrac{\partial w}{\partial z}\right) ^{2}=\left( \dfrac{d w}{d u}\right) ^{2}\).

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If \(z=f\left( \dfrac{x}{y}\right) \), show that \(x\dfrac{ \partial z}{\partial x}+y\dfrac{\partial z}{\partial y}=0\).

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Show that if \(z=f\left( \dfrac{u}{v}, \dfrac{v}{w}\right) \), then \(u\dfrac{\partial z}{\partial u}+v\dfrac{\partial z}{\partial v}+w \dfrac{\partial z}{\partial w}=0\).

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Suppose we denote the expression \(\dfrac{\partial ^{2}}{ \partial x^{2}}+\dfrac{\partial ^{2}}{\partial y^{2}}\) by \(\mathbf{\Delta }\) . If \(z=f(x,y)\), where \(x=r\cos \theta \) and \(y=r\sin \theta ,\) show that \( \mathbf{\Delta }f=\dfrac{\partial ^{2}f}{\partial r^{2}}+\dfrac{1}{r}\left( \dfrac{\partial f}{\partial r}\right) +\dfrac{1}{r^{2}}\left( \dfrac{ \partial ^{2}f}{\partial \theta ^{2}}\right) \).

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Prove that if \(F(x,y,z)=0\) is differentiable, then \[ \dfrac{\partial z}{\partial x}\cdot \dfrac{\partial x}{\partial y}\cdot \dfrac{\partial y}{\partial z}=-1. \]

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1. Suppose that \(F(x,y)\) has continuous second-order partial derivatives and \(F(x,y)=0\) defines \(y\) as a function of \(x\) implicitly. Show that \[ \frac{d^{2}y}{dx^{2}}=-\frac{F_{y}^{2}F_{xx}-2F_{x}F_{y}F_{xy}+F_{x}^{2}F_{yy}}{F_{y}^{3}}\qquad F_{y}\neq 0 \]
2. Use the result from (a) to find \(\dfrac{d^{2}y}{dx^{2}}\) for \( x^{3}+3xy-y^{3}=6\).

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\(f(t,x)=\int_{0}^{x/(2\sqrt{\lambda t})}e^{-u^{2}}du\). Show that \(f_{t}=\lambda f_{xx}\).