Question

Using differentials, find the approximate value of each of the following. (a) (b)

Answer 1

(a)

The given expression is ( 17 81 ) 1 4 .

Write the given expression in the form of x,y.

y= x 1 4 (1)

Differentiate the above equation,

dy dx = 1 4 x 3 4 dy= dx 4 ( x 1 4 ) 3 = Δx 4 ( x 1 4 ) 3 (2)

Changing x to x+Δx and y to y+Δy in equation (1)

y+Δy= ( x+Δx ) 1 4 ( x+Δx ) 1 4 = ( 17 81 ) 1 4 ( x+Δx ) 1 4 = ( 16 81 + 1 81 ) 1 4 (3)

Here, by comparison,

x 1 4 = ( 16 81 ) 1 4 = 2 3

From equation (3),

( 17 81 ) 1 4 =y+Δy ≃y+dy ≃ x 1 4 + Δx 4 ( x 1 4 ) 3

Substitute the values in the above equation,

( 17 81 ) 1 4 ≈ 2 3 + 1 81 4 ( 2 3 ) 3 ≈ 2 3 + 1 81 × 27 32 ≈ 65 96 ≈0.677

Thus, the value of the given expression is 0.677.

(b)

The given expression is ( 33 ) − 1 5 .

Write the given expression in the form of x,y.

y= x − 1 5 (1)

Differentiate the equation

dy dx = −1 5 x 6 5 dy= −dx 5 ( x 1 5 ) 6 = −Δx 5 ( x 1 5 ) 6 (2)

Changing x to x+Δx and y to y+Δy in equation (1)

y+Δy= ( x+Δx ) −1 5 ( x+Δx ) −1 5 = ( 33 ) −1 5 ( x+Δx ) −1 5 = ( 32+1 ) −1 5 (3)

Here, by comparison,

x −1 5 = ( 32 ) −1 5 = 1 2

From equation (3),

( 33 ) −1 5 =y+Δy ≃y+dy ≃ x −1 5 + Δx 5 ( x 1 5 ) 6

Substitute the values in the above equation,

( 33 ) −1 5 ≈ 1 2 + 1 5 ( 2 ) 5 ≈ 1 2 − 1 5 × 1 64 ≈ 159 320 ≈0.497

Thus, the value of the given expression is 0.497.