DIFFERENTIALS CALC

Question

use differentials to estimate ∜624

Answer 1

We can approximate the fourth root of 624 by using differentials if we can find an exact answer to a value close to 624.

We know that 625 = 5⁴ so the fourth root of 625 = 5

So let x = 625 and let Δx = -1

Differentials say that:

f(x + Δx) ≈ f(x) + f '(x)•Δx

For f(x) = x^1/4, f '(x) = 1/4x^-3/4

So ⁴√624 = ⁴√(625 + -1) ≈ ⁴√625 + f '(625)•(-1) ≈ 5 + 1/4(625)^-3/4•(-1) ≈ 5 + 1/4(1/125)(-1) ≈ 5 - 0.002 ≈ 4.998