Using differentials, find the approximate values of the following:
(82)1/4
Let us assume that
Also, let x = 81 so that x + Δx = 82
⇒ 81 + Δx = 82
∴ 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(82) = f(81) + Δf
⇒ f(82) = 3 + 0.00926
∴ f(82) = 3.00926
Thus, (82)1/4 ≈ 3.00926