Using differentials, find the approximate value of each of the following
(i) (17/81)^1/4   (ii) (33)^1/5

Solution:

We can use differentials to calculate small changes in the dependent variable of a function corresponding to small changes in the independent variable

(i) (17/81)^1/4

Let us consider y = (x)^1/4

Assume x = 16/81 and Δx = 1/81

Then,

Δy = (x + Δx)^1/4 - (x)^1/4

= (17/81)^1/4 + (16/81)^1/4

= (17/81)^1/4 - 2/3

2/3 + Δy = (17/81)^1/4

Now, dy is approximately equal to Δy and is given by,

dy = (dy/dx)Δx

= 1/4(x)^3/4 Δx

[∵ y = (x)^1/4]

= 1/4(16/81)^3/4 (1/81)

= 27/(4 x 8) x (1/81)

= 1/(32 x 3)

= 1/96

= 0.010

Hence,

(17/81)^1/4 = 2/3 + 0.010

= 0.667 + 0.010

= 0.677

Thus, the approximate value of (17/81)^1/4 is 0.677.

(ii) (33)^1/5

Consider, y = (x)^1/5

Let, x = 32 and Δx = 1

Then,

Δy = (x + Δx)^1/5 - (x)^1/5

= (33)^1/5 + (32)^1/5

= (33)^1/5 - 1/2

1/2 + Δy = (33)^1/5

Now, dy is approximately equal to Δy and is given by,

dy = (dy/dx)Δx

= - 1/5(x)^6/5 Δx

[since y = (x)^1/5]

= - 1/5(2)^6/5 (1)

= - 1/320

= - 0.003

Hence,

(33)^1/5 = 1/2 + (- 0.003)

= 0.5 - 0.003

= 0.497

Thus, the approximate value of (33)^1/5 is 0.497

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NCERT Solutions Class 12 Maths - Chapter 6 Exercise ME Question 1

Using differentials, find the approximate value of each of the following (i) (17/81)^1/4 (ii) (33)^1/5

Summary:

Using differentials, the approximate value of (i) (17/81)^1/4 is 0.677 and the approximate value of (ii) (33)^1/5 is 0.497