A circular template has a radius of 10 cm. The measurement of radius has an approximate error of 0.02 cm. Then the percentage error in calculating area of this template is

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

If we measure the side of a cube to be 4 cm with an error of 0.1 cm, then the error in our calculation of the volume is

If \(g(x, y)=3 x^{2}-5 y+2 y^{2}, x(t)=e^{t}\) and y(t) = cos t, then \(\frac{dg}{dt}\) is equal to

If \(f(x)=\frac{x}{x+1}\), then its differential is given by

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

A sphere is made of ice having radius 10 cm. Its radius decreases from 10 cm to 9.8 cm. Find approximations for the following:
(i) change in the volume
(ii) change in the surface area

Find df for f(x) = x² + 3x and evaluate it for
x = 2 and dx = 0.1

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

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

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

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

Verify the above theorem for F(x, y) = x² - 2y² + 2xy and x(t) = cos t, y(t) = sin t, t ∈ [0, 2\(\pi\)]