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

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

The approximate change in the volume V of a cube of side x metres caused by increasing the side by 1% is

If f(x,y, z) = xy +yz +zx, then f_x - f_z is equal to

If log_e4 = 1.3868, then log_e4.01 = _____________

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 _____________

lf u = (x-y)⁴+(y-z)⁴ +(z-x)⁴ then \(\sum { \frac { \partial u }{ \partial x } } \) = _____________

The cube root of 127 is ............

If y = sin x and x changes from \(\frac{\pi}{2}\) to ㅠ the approximate change in y is ..............

If u = y^x then \(\frac { \partial u }{ \partial y } \) = ............

If u = sin-1 \(\left( \frac { { x }^{ 4 }+{ y }^{ 4 } }{ { x }^{ 2 }+{ y }^{ 2 } } \right) \) and f = sin u then f is a homogeneous function of degree ..................

If u = y sin x then \(\frac { { \partial }^{ 2 }u }{ \partial x\partial y } \) = ..........

If is a homogeneous function of x and y of degree n, then \(x\frac { { \partial }^{ 2 }u }{ \partial { x }^{ 2 } } +y\frac { { \partial }^{ 2 }u }{ \partial x\partial y } \) = ...... \(\frac { { \partial }u }{ \partial { x } } \)

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.

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

Evaluate \(\begin{gathered} \text { lim } \\ (x, y) \rightarrow(1,2) \end{gathered}\)g(x, y), if the limit exists, where g\((x,y)=\frac { { 3x }^{ 2 }-xy }{ { x }^{ 2 }+{ y }^{ 2 }+3 } \)

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 \(x\frac { \partial f }{ \partial x } +y\frac { \partial f }{ \partial y } =3f\)

Find a linear approximation to f(x)=3xe^2x-10 at x=5

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) 

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.

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

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

A circular plate expands uniformly under the influence of heat. If it’s radius increases from 10.5 cm to 10.75 cm, then find an approximate change in the area and the approximate percentage change in the area.

A firm produces two types of calculators each week, x number of type A and y number of type B. The weekly revenue and cost functions (in rupees) are R(x, y) = 80x + 90y + 0.04xy − 0.05x² − 0.05y² and C(x, y) = 8x + 6y + 2000 respectively
(i) Find the profit function P(x, y) 
(ii) Find \(\frac { { \partial P } }{ \partial { x } } \) (1200, 1800) and \(\frac { \partial v }{ \partial y} \) (1200, 1800)

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

Find the approximate value of \(\left( \frac { 17 }{ 81 } \right) ^{ \frac { 1 }{ 4 } }\) using linear approximation.

If w=x+2y+z² and x=cos t,y=sint,z=t then find \(\frac { dw }{ dt } \) 

Let f(x, y) = sin(xy²) + \(e^{{x^3}+5y}\) for all ∈ R². Calculate \(\frac { \partial f }{ \partial x } ,\frac { \partial f }{ \partial y } ,\frac { { \partial }^{ 2 }f }{ { \partial y\partial x } } \)and \(\frac { { \partial }^{ 2 }f }{ { \partial x\partial y } } \)

If w(x, y) = xy + sin (xy), then prove that \(\frac { { \partial }^{ 2 }w }{ \partial y\partial x } =\frac { { \partial }^{ 2 }w }{ \partial x\partial y } \)

Let U(x, y) = e^x sin y, where x = st², y = s² t, s, t ∈ R. Find \(\frac { \partial U }{ \partial s } ,\frac { \partial U }{ \partial t } \) and evaluate them at s = t = 1.

If v(x, y) = log \(\left( \frac { { x }^{ 2 }+{ y }^{ 2 } }{ x+y } \right) \), prove that \(x\frac { \partial v }{ \partial x } +y\frac { \partial u }{ \partial y } \) = 1

If w(x,y, z) = log \(\left( \frac { { 5x }^{ 3 }{ y }^{ 4 }+7{ y }^{ 2 }{ xz }^{ 4 }-{ 75y }^{ 3 }{ z }^{ 4 } }{ { x }^{ 2 }+{ y }^{ 2 } } \right) \) find \(x\frac { \partial w }{ \partial x } +y\frac { \partial w }{ \partial y } +z\frac { \partial w }{ \partial z } \)

If u = tan ^-1 \(\left( \frac { { x }^{ 3 }+{ y }^{ 3 } }{ x-y } \right) \) Prove that \(x\frac { \partial u }{ \partial x } +y\frac { \partial u }{ \partial y } \) sin 2u.

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