Using differentials, find the approximate values of the following:
(0.007)1/3
Let us assume that
Also, let x = 0.008 so that x + Δx = 0.007
⇒ 0.008 + Δx = 0.007
∴ Δx = –0.001
On differentiating f(x) with respect to x, we get
We know
When x = 0.008, 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 = 0.001
⇒ Δf = (8.3333)(–0.001)
∴ Δf = –0.0083333
Now, we have f(0.007) = f(0.008) + Δf
⇒ f(0.007) = 0.2 – 0.0083333
∴ f(0.007) = 0.1916667
Thus, (0.007)1/3 ≈ 0.1916667