Using differentials, find the approximate values of the following:
(80)1/4
Let us assume that
Also, let x = 81 so that x + Δx = 80
⇒ 81 + Δx = 80
∴ Δx = –1
On differentiating f(x) with respect to x, we get
We know
When x = 81, we have
Recall that if y = f(x) and Δx is a small increment in x, then the corresponding increment in y, Δy = f(x + Δx) – f(x), is approximately given as
Here, and Δx = –1
⇒ Δf = (0.00926)(–1)
∴ Δf = –0.00926
Now, we have f(80) = f(81) + Δf
⇒ f(80) = 3 – 0.00926
∴ f(80) = 2.99074
Thus, (80)1/4 ≈ 2.99074