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

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

Find ∆f and df for the function f for the indicated values of x, ∆x and compare

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

If w(x, y) = x³ − 3xy + 2y², x, y ∊ R, find the linear approximation for w at (1,−1)

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

A circular metal plate expands under heating so that its radius increases by 2%. Find the approximate increase in the area of the plate if the radius of the plate before heating is 10cm.

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

If w=log(x²+y²),x=cosθ,y=sinθ, find \(\frac { dw }{ d\theta } \)

Find the linear approximation to \(g(z)=\sqrt [ 4 ]{ zat } z=2\)