Using differentials, find the approximate values of the following:
Let us assume that
Also, let x = 0.09 so that x + Δx = 0.082
⇒ 0.09 + Δx = 0.082
∴ Δx = –0.008
On differentiating f(x) with respect to x, we get
We know
When x = 0.09, 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 = –0.008
⇒ Δf = (1.6667)(–0.008)
∴ Δf = –0.013334
Now, we have f(0.082) = f(0.09) + Δf
⇒ f(0.082) = 0.3 – 0.013334
∴ f(0.082) = 0.286666
Thus,