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.

The radius of a circular plate is measured as 12.65 cm instead of the actual length 12.5 cm. find the following in calculating the area of the circular plate:
Absolute error

The radius of a circular plate is measured as 12.65 cm instead of the actual length 12.5 cm. Find the following in calculating the area of the circular plate:
Relative error

The radius of a circular plate is measured as 12.65 cm instead of the actual length 12.5 cm.find the following in calculating the area of the circular plate:
Percentage error

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

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:
change in the surface area

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.

Show that the percentage error in the n^th root of a number is approximately \(\frac1n\) times the percentage error in the number.

The trunk of a tree has diameter 30 cm. During the following year, the circumference grew 6cm.
(i) Approximately, how much did the tree's diameter grow?
(ii) What is the percentage increase in area of the tree's cross-section?

The trunk of a tree has diameter 30 cm. During the following year, the circumference grew 6cm.

In a newly developed city, it is estimated that the voting population (in thousands) will increase according to V(t) = 30 + 12t² - t³, 0 ≤ t ≤ 8 where t is the time in years. Find the approximate change in voters for the time change from 4 to 4\(\frac16\) year

The relation between the number of words y a person learns in x hours is given by y = 52 \(\sqrt { x } \), 0, ≤ x ≤ 9. What is the approximate number of words learned when x changes from
(i) 1 to 1.1 hour?
(ii) 4 to 4.1 hour?

The relation between the number of words y a person learns in x hours is given by y = 52 \(\sqrt { x } \), 0, ≤ x ≤ 9. What is the approximate number of words learned when x changes from

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 coat of paint of thickness 0.2 cm is applied to the faces of a cube whose edge is 10 cm. Use the differentials to find approximately how many cubic centimeters of paint is used to paint this cube. Also calculate the exact amount of paint used to paint this cube.

Let F(x, y) = x³ y + y²x + 7 for all (x, y)∈ R². Calculate \(\frac { \partial F }{ \partial x } \)(-1, 3) and \(\frac { \partial F }{ \partial y } \)(-2, 1).

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 \(\frac { { \partial }^{ 2 }f }{ { \partial x\partial y } } \)

Let w(x, y, z) = \(\frac { 1 }{ \sqrt { { x }^{ 2 }+{ y }^{ 2 }+{ z }^{ 2 } } } ,(x,y,z)\neq (0,0,0)\). Show that \(\frac { { \partial }^{ 2 }w }{ \partial { x }^{ 2 } } +\frac { { \partial }^{ 2 }w }{ \partial { y }^{ 2 } } +\frac { { \partial }^{ 2 }w }{ \partial { z }^{ 2 } } =0\)

If V(x,y) = e^x(x cos y - y siny), then prove that \(\frac { { \partial }^{ 2 }V }{ \partial { x }^{ 2 } } =\frac { { \partial }^{ 2 }V }{ \partial { y }^{ 2 } } \) = 0

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

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)

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

W(x, y, z) = xy + yz + zx, x = u - v, y = uv, z = u + v, u ∈ R. Find \(\frac { \partial W }{ \partial u } ,\frac { \partial W }{ \partial v } \), and evaluate them at \(\left( \frac { 1 }{ 2 } ,1 \right) \)