Question

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

1. \(f( 1,1) \)
2. \(f(x+\Delta x, y)\)
3. \(f(x, y+\Delta y)\)

Question

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

1. \(f ( 1,1 ) \)
2. \(f(x+\Delta x, y)\)
3. \(f(x, y+\Delta y)\)

Question

\(f(x,y,z) =e^{x}\sin ^{-1}(y+2z) \)

1. \(f\left( \ln 3,\dfrac{1}{2},\dfrac{1}{4}\right) \)
2. \(f\left( 1,0,\dfrac{1}{4}\right) \)
3. \(f( 0,0,0) \)

Question

\(z=f(x,y)=\ln ( x^{2}-3y) \)

Question

\(z=f(x,y)=\sqrt{9-x^{2}-4y^{2}}\)

Question

\(z=f(x,y) =\dfrac{25xy}{\sqrt{5-y^{2}}}\)

Question

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

Question

Find the domain of \(w=f(x,y,z)=\dfrac{y^{2}+z^{2}}{x^{2}}.\)

Question

Find the domain of \(w=f(x,y,z)=e^{x+y}\ln z\).

Question

\(z=f(x,y)=x-y+5\)

Question

\(z=f(x, y)=\sin x\)

Question

\(z=f(x,y)=\ln x\)

Question

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

Question

For \(z=f(x, y)=x^{2}-2y,\) graph the level curves corresponding to \(c=-4,\) \(-1,\) \(0,\) \(1,\) \(4.\)

Question

For \(z=f(x, y)=\sqrt{x^{2}+y^{2}},\) graph the level curves corresponding to \(c=0, 1, 4, 9.\)

Question

For \(z=f(x,y) =e^{4x^{2}+y^{2}},\) graph the level curves corresponding to \(c=1,\) \(e,\) \(e^{4},\) \(e^{16}\).

Question

Describe the level surfaces associated with the function \( w=f(x, y, z)=4x^{2}+y^{2}+z^{2}\).

Question

Describe the level surfaces associated with the function \(w=f(x, y, z)=x+y+2z\).

Question

Find \(\lim\limits_{(x,y) \rightarrow \left( {\pi }/{2},0\right) }\dfrac{\sin x\cos y}{x}\).

Question

Find \(\lim\limits_{(x, y)\rightarrow (1, 2)}\dfrac{4x^{2}+y^{2} }{2x+y}\).

Question

Let \(f(x, y)=\dfrac{3xy^{2}}{x^{2}+y^{4}}\).

1. Show that \(\lim\limits_{(x, y)\rightarrow (0, 0)}f(x, y)=0\) along the lines \(y=mx\).
2. Find \(\lim\limits_{(x, y)\rightarrow (0, 0)}f(x, y)\) along the parabola \( x=y^{2}\).
3. What can you conclude?

860

Question

Show that \(\lim\limits_{(x, y, z)\rightarrow (0, 0, 0)}\dfrac{ 4xy}{x^{2}+y^{2}+z^{2}}\) does not exist.

Question

Determine where the function \(f(x,y) =2x^{2}y-y^{2}+3\) is continuous.

Question

Determine where the function \(R(x,y) =\dfrac{xy}{ x^{2}-y^{2}}\) is continuous.

Question

1. Determine where the function \(f(x,y) =\tan ^{-1}\dfrac{1}{x^{2}+y^{2}}\) is continuous.
2. Find \(\lim\limits_{(x,y) \rightarrow ( 0,1) }\tan ^{-1}\dfrac{1}{x^{2}+y^{2}}.\)

Question

Find \(f_{x}\) and \(f_{y}\) for \(z=f(x, y)=e^{x^{2}+y^{2}}\sin (xy)\).

Question

Find \(f_{x}\) and \(f_{y}\) for \(z=f(x, y)=\dfrac{x+y}{y}\).

Question

Find \(\dfrac{\partial z}{\partial x}\) and \(\dfrac{\partial z}{ \partial y}\) for \(z=f(x, y)=\sqrt{x-2y^{2}}\).

Question

Find \(\dfrac{\partial z}{\partial x}\) and \(\dfrac{\partial z}{ \partial y}\) for \(z=f(x, y)=e^{x}\ln ( 5x+2y) \).

Question

For \(f(x, y)=\sqrt{x^{2}-y^{2}}\), find \(f_{x}(2,1)\) and \( f_{y}(2,-1)\).

Question

For \(F(x, y)=e^{x}\sin y\), find \(F_{x}\!\left( 0,\dfrac{\pi }{6} \right) \) and \(F_{y}\!\left( 0,\dfrac{\pi }{6}\right) \).

Question

Find symmetric equations for the tangent line to the curve of intersection of the ellipsoid \(\dfrac{x^{2}}{24}+\dfrac{y^{2}}{12}+\dfrac{ z^{2}}{6}=1\), and the plane \(y=1\), where \(x=4\) and \(z\) is positive.

Question

Find symmetric equations for the tangent line to the curve of intersection of the surface \(z=4x^{2}-y^{2}+7\):

1. With the plane \(y=-2\) at the point \((1,-2,7)\).
2. With the plane \(x=1\) at the point \((1,-2,7)\).

Question

Boyle’s Law The volume \(V\) of a gas varies directly with the temperature \(T\) and inversely with the pressure \(P\).

1. Find the rate of change of the volume \(V\) with respect to the temperature \(T.\)
2. Find the rate of change of the volume \(V\) with respect to the pressure \(P.\)
3. If \(T\) and \(P\) are functions of \(t\), what is \(\dfrac{\partial V}{\partial t}\)?

Question

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

Question

\(z=f(x,y) =\sec (xy) \)

Question

\(w=f(x,y,z)=e^{xyz}\)

Question

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

Question

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

Question

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

Question

For the function \(z=x^{3}y^{2}-2xy^{4}+3x^{2}y^{3},\) show that \(x\dfrac{\partial z}{\partial x}+y\dfrac{\partial z}{\partial y}=5z\).

Question

Find the change \(\Delta z\) in \(z=f(x,y)=xy^{2}+2\) from \( (x_{0},y_{0})\) to \((x_{0}+\Delta x, y_{0}+\Delta y)\). Use the answer to calculate the change in \(z\) from \((1,0)\) to \((0.9,0.2)\).

Question

Show that the function \(z=f(x,y)=xy-5y^{2}\) is differentiable at any point \((x,y)\) in its domain by:

1. Finding \(\Delta z\).
2. Finding \(\eta _{1}\) and \(\eta _{2}\) so that \(\Delta z=f_{x}(x_{0},y_{0})\Delta x+ f_{y}(x_{0},y_{0})\Delta y+\eta _{1}\Delta x+\eta _{2}\Delta y\).
3. Show that \(\lim\limits_{(\Delta x, \Delta y)\rightarrow (0, 0)}\eta_{1}=0\) and \(\lim\limits_{(\Delta x, \Delta y)\rightarrow (0, 0)}\eta _{2}=0\).

Question

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

Question

\(z=\sin ^{-1}\dfrac{x}{y}\), \(y>0\)

Question

\(w=ze^{xy}\)

Question

\(w=\ln (xyz)\)

Question

Use differentials to estimate the change in \(z=x\sqrt{1+y^{2}}\) from \(( 4,0) \) to \(( 4.1,0.1) .\)

Question

Use differentials to estimate the change in \(z=\sin ^{-1}\dfrac{x}{y}\) from \(( 0,1)\) to \(( 0.1,1.1) .\)

Question

Use the differential of \(f(x,y)=y^{2}\cos x\) to find an approximate value of \(f(0.05,1.98)\). (Compare your answer with a calculator result.)

Question

Electricity The electrical resistance \(R\) of a wire is \(R=k \dfrac{L}{D^{2}}\) where \(L\) is the length of the wire, \(D\) is the diameter of the wire, and \(k\) is a constant. If \(L\) has a \(1\%\) error and \(D\) has a \(2\%\) error, what is the approximate maximum percentage error in the computation of \(R\)?

Question

Find \(\dfrac{dz}{dt}\) if \(z=\sin (xy)-x\sin y,\) where \( x=e^{t} \) and \(y=te^{t}\).

Question

Find \(\dfrac{dw}{dt}\) if \(w=\dfrac{x}{y}+\dfrac{y}{z}+\dfrac{z}{ x},\) where \(x=\dfrac{1}{t}\), \(y=\dfrac{1}{t^{2}}\), and \(z=\dfrac{1}{t^{3}}\).

Question

Find \(\dfrac{\partial w}{\partial u}\) and \(\dfrac{\partial w}{ \partial v}\) if \(w=xy+yz-xz,\) where \(x=u+v\), \(y=uv\), and \(z=v\). Express each answer in terms of \(u\) and \(v.\)

Question

Find \(\dfrac{\partial u}{\partial r}\) and \(\dfrac{\partial u}{ \partial s}\) if \(u=\sqrt{x^{2}+y^{2}+z^{2}},\) where \(x=r\cos s\), \(y=r\sin s\) , and \(z=\sqrt{r^{2}+s^{2}}\). Express each answer in terms of \(r\) and \(s.\)

Question

Find \(\dfrac{\partial z}{\partial x}\) and \(\dfrac{\partial z}{ \partial y}\) if \(z=f(x,y) \) is defined implicitly by \( F(x,y,z)=x^{2}+y^{2}-2xyz=0.\)

Question

Find \(\dfrac{\partial z}{\partial x}\) and \(\dfrac{\partial z}{ \partial y}\) if \(z=f(x,y) \) is defined implicitly by \( F(x,y,z)=2x\sin y+2y\sin x+2xyz=0.\)

Question

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