Use the linear approximation to find approximate values of \({ (123) }^{ \frac { 2 }{ 3 } }\)

Use the linear approximation to find approximate values of \(\sqrt [ 4 ]{ 15 } \)

Use the linear approximation to find approximate values of \(\sqrt [ 3 ]{ 26 } \)

Let g(x) = x² + sin x. Calculate the differential dg.

Find differential dy for each of the following function \(y=\frac { { \left( 1-2x \right) }^{ 3 } }{ 3-4x } \)

Find differential dy for each of the following function
y = (3 + sin(2x)) ^2/3 

Find differential dy for each of the following function
y = _ex^2_-5x+7 cos (x² - 1)

Consider g(x,y) = \(\frac { 2{ x }^{ 2 }y }{ { x }^{ 2 }+{ y }^{ 2 } } \), if (x, y) ≠ (0, 0) and g(0, 0) = 0 Show that g is continuous on R²

Let f (x, y) = 0 if xy ≠ 0 and f (x, y) = 1 if xy = 0.
Calculate: \(\frac { \partial f }{ \partial x } (0,0),\frac { \partial f }{ \partial y } (0,0).\)

Let w(x, y) = xy+\(\frac { { e }^{ y } }{ { y }^{ 2 }+1 } \) for all (x, y) ∈ R². Calculate \(\frac { { \partial }^{ 2 }w }{ { \partial y\partial x } } \) and \(\frac { { \partial }^{ 2 }w }{ { \partial x\partial y } } \)

Let (x, y) = e^-2y cos(2x) for all (x, y) ∈ R². Prove that u is a harmonic function in R².

Let \(f(x,y)=\frac { { y }^{ 2 }-xy }{ \sqrt { x } -\sqrt { y } } \) for (x, y) ≠ (0, 0). Show that \(\begin{matrix} lim \\ (x,y)\rightarrow (0,0) \end{matrix}\) f(x, y) = 0

Evaluate \(\begin{matrix} lim \\ (x,y)\rightarrow (0,0) \end{matrix}cos=\left( \frac { { e }^{ x }siny }{ y } \right) \), if the limit exists.

Let g(x, y) = \(\frac { { x }^{ 2 }y }{ { x }^{ 4 }+{ y }^{ 2 } } \) for (x, y) ≠ (0, 0) and f(0, 0) = 0
Show that \(\begin{matrix} lim \\ (x,y)\rightarrow (0,0) \end{matrix}\) g(x, y) = 0 along every line y = mx, m ∈ R

Let g(x, y) = \(\frac { { x }^{ 2 }y }{ { x }^{ 4 }+{ y }^{ 2 } } \) for (x, y) ≠ (0, 0) and f(0, 0) = 0
Show that \(\begin{matrix} lim \\ (x,y)\rightarrow (0,0) \end{matrix}\) g(x, y) = \(\frac { k }{ 1+{ k }^{ 2 } } \) along every parabola y = kx², k ∈ R \ {0}.

Find the partial derivatives of the following functions at the indicated point.
f(x, y) = 3x² - 2xy + y² + 5x + 2, (2,-5)

Find the partial derivatives of the following functions at the indicated point
g(x, y) = 3x² + y² + 5x + 2, (1, -2)

Find the partial derivatives of the following functions at the indicated point
h (x, y, z) = x sin (xy) + z²x, \(\left( 2,\frac { \pi }{ 4 }, 1\right) \) 

Find the partial derivatives of the following functions at the indicated point
G(x, y) = e^x+3y log (x² + y²), (-1, 1)

If U(x, y, z) = log (x³ + y³ + z³), find \(\frac { \partial U }{ \partial x } +\frac { \partial U }{ \partial y } +\frac { \partial U }{ \partial z } \)

If v(x, y, z) = x³ + y³ + z³ + 3xyz, show that \(\frac { { \partial }^{ 2 }v }{ \partial y\partial z } =\frac { { \partial }^{ 2 }v }{ \partial z\partial y } \)

If w(x, y, z) = x² y + y²z + z²x, x, y, z∈R, find the differential dw .

If v(x, y) = x² - xy + \(\frac14\) y + 7, x, y ∈ R, find the differential dv.

If u (x, y) = x²y + 3xy⁴, x = e^t and y = sin t, find \(\frac{du}{dt}\) and evaluate If at t = 0. 

If u(x, y, z) = xy²z³, x = sin t, y = cos t, z = 1+ e^2t, find \(\frac{du}{dt}\)

If w (x, y, z) = x² + y² + y², x = e^t, y = e^t sin t, z = e^t cos t, find \(\frac{dw}{dt}\)

If w(x, y) = 6x² - 3xy + 2y², x = e^x, y = cos s, s ∈ R find \(\frac{dw}{ds}\), and evaluate at s = 0

In each of the following cases, determine whether the following function is homogeneous or not. If it is so, find the degree.
f(x, y) = x²y + 6x³ + 7

Determine whether the following function is homogeneous or not. If it is so, find the degree.
\(h(x,y)=\frac { 6{ x }^{ 2 }{ y }^{ 3 }-\pi { y }^{ 5 }+9{ x }^{ 4 }y }{ 2020{ x }^{ 2 }+2019{ y }^{ 2 } } \)

In each of the following cases, determine whether the following function is homogeneous or not. If it is so, find the degree.
\(g(x,y,z)=\frac { \sqrt { { 3 }x^{ 2 }+5{ y }^{ 2 }+{ z }^{ 2 } } }{ 4x+7y } \)

In each of the following cases, determine whether the following function is homogeneous or not. If it is so, find the degree.
\(U(x,y,z)=xy+sin\left( \frac { { y }^{ 2 }-2{ x }^{ 2 } }{ xy } \right) \)

If u(x, y) = \(\frac { { x }^{ 2 }+{ y }^{ 2 } }{ \sqrt { x+y } } \), prove that \(x\frac { \partial u }{ \partial x } +y\frac { \partial u }{ \partial y } =\frac { 3 }{ 2 } u\)

Show that F(x,y) = \(\frac { { x }^{ 2 }+5xy-10{ y }^{ 2 } }{ 3x+7y } \) is a homogeneous function of degree 1.

Let f (x, y) = 0 if xy ≠ 0 and f (x, y) =1 if xy = 0.
Show that f is not continuous at (0,0)