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,