Using differentials, find the approximate values of the following:
(3.968)3/2
Let us assume that
Also, let x = 4 so that x + Δx = 3.968
⇒ 4 + Δx = 3.968
∴ Δx = –0.032
On differentiating f(x) with respect to x, we get
We know
When x = 4, 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 = –0.032
⇒ Δf = (3)(–0.032)
∴ Δf = –0.096
Now, we have f(3.968) = f(4) + Δf
⇒ f(3.968) = 23– 0.096
⇒ f(3.968) = 8 – 0.096
∴ f(3.968) = 7.904
Thus, (3.968)3/2 ≈ 7.904