Find the linear approximation for f(x) = \(\sqrt { 1+x } ,x\ge -1\) at x₀ = 3. Use the linear approximation to estimate f(3.2) 

Use linear approximation to find an approximate value of \(\sqrt { 9.2 } \) without using a calculator.

Let us assume that the shape of a soap bubble is a sphere. Use linear approximation to approximate the increase in the surface area of a soap bubble as its radius increases from 5 cm to 5.2 cm. Also, calculate the percentage error.

A right circular cylinder has radius r =10 cm. and height h = 20 cm. Suppose that the radius of the cylinder is increased from 10 cm to 10. 1 cm and the height does not change. Estimate the change in the volume of the cylinder. Also, calculate the relative error and percentage error.

Let f (x,y) = \(\frac { 3x-5y+8 }{ { x }^{ 2 }+{ y }^{ 2 }+1 } \) for all (x, y) ∈ R² Show that f is continuous on R² 

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

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

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

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

If u = sin^-1 \(\left( \frac { x+y }{ \sqrt { x } +\sqrt { y } } \right) \), Show that  \(x\frac { \partial u }{ \partial x } +y\frac { \partial u }{ \partial y } =\frac { 1 }{ 2 } tanu\)