Question

Using differentials, find the approximate value of each of the following. (a)[17/81]^1/4 (b)(33)^-1/5

Answer 1

HELLO DEAR,

( a ) . [17/81]^¼ = 17¼ / 81¼ = 17¼/3

let f(x) = x¼

f'(x) = 1/(4x¾).

now,
{f(x + ∆x) - f(x)} = f'(x)*∆x

{f(x + ∆x) - f(x)} = 1/(4x¾)*∆x-------( 1 )

we may write 17 = (16 + 1).
putting x = 16 , ∆x = 1 ----- in ----( 1 )

we get,

f(16 + 1) - f(16) = 1/4(16)¾ * 1

f(17) - f(16) = 1/4(2³) = 1/32

f(17) = 1/32 + f(16)

f(17) = 1/32 + (16)¼)

f(17) = 0.03125 + 2

f(17) = 2.03125

(17)¼ = 2.03125

approximate value of [17/81]¼ = 2.03125/3

= 0.677

( b ). 33^{-1/5} = 1/(33)^1/5

let f(x) = x^1/5

f'(x) = 1/5x^4/5

now, we know {f(x + ∆x) - f(x)} = f'(x)*∆x

we may write, 33 = (32 - 1)
putting, x = 32 , ∆x = 1

f(32 + 1) - f(32) = 1/5(32)^4/5 * 1

f(33) - f(32) = 1/5(2⁴)

f(33) - f(32) = 1/80

f(33) = f(32) + 1/80

f(33) = 1/80 + (32)^1/5

f(33) = 0.0125 + 2

f(33) = 2.0125

(1/33)^1/5 = 1/2.0125

(33)^(-1/5) = 0.4968

I HOPE ITS HELP YOU DEAR,
THANKS