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


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