Using differentials, find the approximate values of the following:
251/3
Let us assume that
Also, let x = 27 so that x + Δx = 25
⇒ 27 + Δx = 25
∴ Δx = –2
On differentiating f(x) with respect to x, we get
We know
When x = 27, 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 = 2
⇒ Δf = (0.03704)(–2)
∴ Δf = –0.07408
Now, we have f(25) = f(27) + Δf
⇒ f(25) = 3 – 0.07408
∴ f(25) = 2.92592
Thus, (25)1/3 ≈ 2.92592