If w (x, y) = x^y, x > 0, then \(\frac { \partial w }{ \partial x } \) is equal to

If f (x, y) = e^xy then \(\frac { { \partial }^{ 2 }f }{ \partial x\partial y } \) is equal to

The change in the surface area S = 6x² of a cube when the edge length varies from x_o to x_o+ dx is

If u = log \(\sqrt { { x }^{ 2 }+{ y }^{ 2 } } \), then \(\frac { { \partial }^{ 2 }u }{ \partial { x }^{ 2 } } +\frac { { \partial }^{ 2 }u }{ { \partial y }^{ 2 } } \) is _____________

If u = x^y + y^x then u_x + u_y at x = y = 1 is _____________

Find a linear approximation for the following functions at the indicated points.
g(x) = \(g(x)=\sqrt { { x }^{ 2 }+9 } ,{ x }_{ 0 }=-4\)

Find a linear approximation for the following functions at the indicated points.
\(h(x)=\frac{x}{x+1}, x_{0}=1\)

The time T, taken for a complete oscillation of a single pendulum with length l, is given by the equation T = 2ㅠ\(\sqrt { \frac { 1 }{ g } } \), where g is a constant. Find the approximate percentage error in the calculated value of T corresponding to an error of 2 percent in the value of l.

If f (x, y) = 2x³ - 11x²y + 3y³, prove that \(x\frac { \partial f }{ \partial x } +y\frac { \partial f }{ \partial y } =3f\)

If of f(x, y) = x² + y³ + 2xy² find f_xx, f_yy, f_xy and f_yx.

f(x,y) = \(\frac { xy }{ { x }^{ 2 }+{ y }^{ 2 } } \), if (x,y) ≠ (0, 0) and f (0, 0) = 0. Show that f is not continuous at (0, 0) and continuous at all other points of R²

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 U(x, y, z) = x² − xy + 3 sin z, x, y, z ∈ R Find the linear approximation for U at (2,−1,0).

Find the approximate value of f (3.02) where f(x) = 3x² + 5x +3.

Using differentials find the approximate value of tan 46° if it is given that 1⁰ = 0.01745 radians

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

If V = log r and r² = x² +y² + z², then prove that \(\frac { { \partial }^{ 2 }V }{ \partial { x }^{ 2 } } +\frac { { \partial }^{ 2 }V }{ \partial { y }^{ 2 } } +\frac { { \partial }^{ 2 } }{ \partial { z }^{ 2 } } =\frac { 1 }{ { r }^{ 2 } } \)