Using differentials, find the approximate values of the following:
Let us assume that
Also, let x = 49 so that x + Δx = 49.5
⇒ 49 + Δx = 49.5
∴ Δx = 0.5
On differentiating f(x) with respect to x, we get
We know
When x = 49, 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 = 0.5
⇒ Δf = (0.0714286)(0.5)
∴ Δf = 0.0357143
Now, we have f(49.5) = f(49) + Δf
⇒ f(49.5) = 7 + 0.0357143
∴ f(49.5) = 7.0357143
Thus,