Samacheer Kalvi solutions for Mathematics - Volume 1 and 2 [English] Class 12 TN Board chapter 8 - Differentials and Partial Derivatives [Latest edition]

Let f(x) = `root(3)(x)`. Find the linear approximation at x = 27. Use the linear approximation to approximate `root(3)(27.2)`

Use the linear approximation to find approximate values of `(123)^(2/3)`

Use the linear approximation to find approximate values of `root(4)(15)`

Use the linear approximation to find approximate values of `root(3)(26)`

Find a linear approximation for the following functions at the indicated points.

f(x) = x³ – 5x + 12, x₀ = 2

Find a linear approximation for the following functions at the indicated points.

g(x) = `sqrt(x^2 + 9)`,  x₀ = – 4

Find a linear approximation for the following functions at the indicated points.

h(x) = `x/(x + 1), x_0` = 1

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:

Change in the volume

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 = `2pi sqrt(l/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 `1/"n"` times the percentage error in the number

Find the differential dy for the following functions:

y = `(1 - 2x)^3/(3 - 4x)`

Find the differential dy for the following functions:

y = `(3 + sin(2x))^(2/3)`

Find the differential dy for the following functions:

y = `"e"^(x^2 - 5x + 7) cos(x^2 - 1)`

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

Find df for f(x) = x² + 3x and evaluate it for x = 3 and dx = 0.02

Find Δf and df for the function f for the indicated values of x, Δx and compare:

f(x) = x³ – 2x², x = 2, Δx = dx = 0.5

Find Δf and df for the function f for the indicated values of x, Δx and compare:

f(x) = x² + 2x + 3, x = – 0.5, Δx = dx = 0.1

Assuming log₁₀ e = 0.4343, find an approximate value of Iog₁₀ 1003

The trunk of a tree has a diameter of 30 cm. During the following year, the circumference grew 6 cm. Approximately how much did the tree diameter grow?

The trunk of a tree has a diameter of 30 cm. During the following year, the circumference grew 6 cm. What is the percentage increase in the area of the cross-section of the tree?

An egg of a particular bird is very nearly spherical. If the radius to the inside of the shell is 5 mm and the radius to the outside of the shell is 5.3 mm, find the volume of the shell approximately

Assume that the cross-section of the artery of human is circular. A drug is given to a patient to dilate his arteries. If the radius of an artery is increased from 2 mm to 2.1 mm, how much is cross-sectional area increased approximately?

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 1/6` years

The relation between the number of words y a person learns in x hours is given by y = `sqrt(x), 0 ≤ x ≤ 9`. What is the approximate number of words learned when x changes from 1 to 1.1 hours?

The relation between the number of words y a person learns in x hours is given by y = `sqrt(x), 0 ≤ x ≤ 9`. What is the approximate number of words learned when x changes from 4 to 4.1 hours?

A circular plate expands uniformly under the influence of heat. If its 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 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

Evaluate `lim_((x,  y) -> (1,  2))  "g"(x, y)`, if the limit exists, where `"g"(x, y) = (3x2 - xy)/(x^2 + y^2 + 3)`

Evaluate `lim_((x,  y) -> (0,  0)) cos((x^3 + y^2)/(x + y + 2))` If the limits exists

Let f(x, y) = `(y^2 - xy)/(sqrt(x) - sqrt(y))` for (x, y) ≠ (0, 0). Show that `lim_((x,  y) -> (0,  0)) "f"(x,  y)` = 0

Evaluate `lim_((x,  y) -> (0,  0)) cos(("e"^x sin y)/y)`, if the limit exists

Let g(x, y) = `(x^2y)/(x^4 + y^2)` for (x, y) ≠ (0, 0) = 0. Show that `lim_((x,  y) -> (0,  0)) "g"(x,  y)` = 0 along every line y = mx, m ∈ R

Let g(x, y) = `(x^2y)/(x^4 + y^2)` for (x, y) ≠ (0, 0) = 0. Show that `lim_((x,  y) -> (0,  0)) "g"(x,  y) = "k"/(1 + "k"^2)` along every parabola y = kx², k ∈ R\{0}

Show that f(x, y) = `(x^2 - y^2)/(y - 1)` s continuous at every (x, y) ∈ R² 

Let g(x, y) =  `("e"^y  sin x)/x` for x ≠ 0 and g(0, 0) = 1 shoe that g is continuous at (0, 0)

Find the partial dervatives of the following functions at indicated points.

f(x, y) = 3x² – 2xy + y² + 5x + 2, (2, – 5)

Find the partial dervatives of the following functions at indicated points.

g(x, y) = 3x² + y² + 5x + 2, (2, – 5)

Find the partial derivatives of the following functions at indicated points.

 h(x, y, z) = x sin (xy) + z²x, `(2, pi/4, 1)`

Find the partial derivatives of the following functions at the indicated points.

`"G"(x, y) = "e"^(x + 3y)  log(x^2 + y^2), (- 1, 1)`

For the following functions find the f_x, and f_y and show that f_xy = f_yx 

f(x, y) = `(3x)/(y + sinx)`

For the following functions find the f_x, and f_y and show that f_xy = f_yx 

f(x, y) = `tan^-1 (x/y)`

For the following functions find the f_x, and f_y and show that f_xy = f_yx 

f(x, y) = `cos(x^2 - 3xy)`

If U(x, y, z) = `(x^2 + y^2)/(xy) + 3z^2y`, find `(del"U")/(delx), (del"U")/(dely)` and `(del"U")/(del"z)`

If U(x, y, z) = `log(x^3 + y^3 + z^3)`,  find `(del"U")/(delx) + (del"U")/(dely) + (del"U")/(del"z)`

For the following functions find the g_xy, g_xx, g_yy and g_yx 

g(x, y) = xe^y + 3x²y

For the following functions find the g_xy, g_xx, g_yy and g_yx 

g(x, y) = log(5x + 3y)

For the following functions find the g_xy, g_xx, g_yy and g_yx 

g(x, y) = x² + 3xy – 7y + cos(5x)

Let w(x, y, z) = `1/sqrt(x^2 + y^2 + z^2)` = 1, (x, y, z) ≠ (0, 0, 0), show that `(del^2w)/(delx^2) + (del^2w)/(dely^2) + (del^2w)/(delz^2)` = 0

If V(x, y) = e^x (x cosy – y siny), then Prove that `(del^2"V")/(delx^2) + (del^2"V")/(dely^2)` = 0

If w(x, y) = xy + sin(xy), then Prove that `(del^2w)/(delydelx) = (del^2w)/(delxdely)`

If v(x, y, z) = x³ + y³ + z³ + 3xyz, Show that `(del^2"v")/(delydelz) = (del^2"v")/(delzdely)`

A from produces two types of calculates 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. Find the profit function P(x, y)

A from produces two types of calculates 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. Find `(del"P")/(delx)` (1200, 1800) and `(del"P")/(dely)` (1200, 1800) and interpret these results

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

Let z(x, y) = x²y + 3xy⁴, x, y ∈ R, Find the linear approximation for z at (2, –1)

If v(x, y) = `x^2 - xy + 1/4  y^2 + 7, x, y ∈ "R"`, find the differential dv

Let V (x, y, z) = xy + yz + zx, x, y, z ∈ R. Find the differential dV

If u(x, y) = x²y + 3xy⁴, x = e^t and y = sin t, find `"du"/"dt"` and evaluate if at t = 0

Let u(x, y, z) = xy²z³ x = sin t, y = cos t, z = 1 + e^2t, Find `"du"/"dt"`

If w(x, y, z) = x² + y² + z², x = e^t, y = e^t sin t and z = e^t cos t, find `("d"w)/"dt"`

Let U(x, y, z) = xyz, x = e^–t, y = e^–t cos t, z – sin t, t ∈ R, find `"dU"/"dt"`

Let w(x, y) = 6x³ – 3xy + 2y², x = e^s, y = cos s, s ∈ R. Find `("d"w)/"ds"` and evaluate at s = 0

Let z(x, y) = x tan^–1(xy), x = t², y = s e^t, s, t ∈ R. Find `(delz)/(del"s")` and `(delz)/(del"t")` at s = t = 1

Let U(x, y) = e^x sin y where x = st², y = s²t, s, t ∈ R. Find `(del"U")/(del"s"), (del"u")/(del"t")` and evaluate them at s = t = 1

Let z(x, y) = x³ – 3x²y³ where x = se^t, y = se^–t, s, t ∈ R. Find `(delz)/(del"s")` and `(delz)/(delt)`

W(x, y, z) = xy + yz + zx, x = u – v, y = uv, z = u + v, u, v ∈ R. Find `(del"W")/(del"u"), (del"W")/(del"v")` and evaluate them at `(1/2, 1)`

In the following, determine whether the following function is homogeneous or not. If it is so, find the degree.

f(x, y) = x²y + 6x³ + 7

In the following, determine whether the following function is homogeneous or not. If it is so, find the degree.

h(x, y) = `(6x^3y^2 - piy^5 + 9x^4y)/(2020x^2 + 2019y^2)` 

In the following, determine whether the following function is homogeneous or not. If it is so, find the degree.

g(x, y, z) = `sqrt(3x^2+ 5y^2+z^2)/(4x + 7y)`

In the following, determine whether the following function is homogeneous or not. If it is so, find the degree.

U(x, y, z) = `xy + sin((y^2 - 2z^2)/(xy))`

Prove that f(x, y) = x³ – 2x²y + 3xy² + y³ is homogeneous. What is the degree? Verify Euler’s Theorem for f

Prove that g(x, y) = `x log(y/x)` is homogeneous What is the degree? Verify Eulers Theorem for g

If `"u"(x , y) = (x^2 + y^2)/sqrt(x + y)`, prove that `x (del"v")/(delx) + y (del"u")/(dely) = 3/2 "u"`

If v(x, y) = `log((x^2 + y^2)/(x + y))`, prove that `x (del"v")/(delx) + y (del"u")/(dely) = 1`

If w(x, y, z) = `log((5x^3y^4 + 7y^2xz^4 - 75y^3zz^4)/(x^2 + y^2))`, find `x (del"w")/(delx) + y (del"w")/(dely) + z (del"w")/(delz)`

Choose the correct alternative:

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

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The percentage error of fifth root of 31 is approximately how many times the percentage error in 31?

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If u(x, y) = `"e"^(x^2 + y^2)`, then `(delu)/(delx)` is equal to

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If v(x, y) = log(e^x + e^y), then `(del"v")/(delx) + (del"u")/(dely)` is equal to

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If w(x, y) = x^y, x > 0, then `(del"w")/(delx)` is equal to

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If f(x, y) = e^xy, then `(del^2"f")/(delxdely)` is equal to

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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

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The change in the surface area S = 6x² of a cube when the edge length varies from x₀ to x₀ + dx is

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

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If g(x, y) = 3x² – 5y + 2y², x(t) = e^t and y(t) = cos t then `"dg"/"dt"` is equal to

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If f(x) = `x/(x + 1)`, then its differential is given by

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f u(x, y) = x² + 3xy + y – 2019, then `(delu)/(delx) "|"_(((4 , - 5)))` is equal to

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Linear approximation for g(x) = cos x at x = `pi/2` is

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If w(x, y, z) = x²(y – z) + y²(z – x)+ z²(x – y) then `(del"w")/(delz) + (del"w")/(dely) + (del"w")/(delz)` is 

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