Find the approximate value

Question:

Find the approximate value of (1.999)5.

Solution:

(1.999)5 = (2 – 0.001)5

Let x = 2 and ∆x = -0.001

Also, let y = x5

Differentiating both sides w.r.t, x, we get

dy/dx = 5x4 = 5(2)4 = 80

Now, ∆y = (dy/dx). ∆x = 80. (-0.001) = -0.080

And, (1.999)5 = y + ∆y

= x5 – 0.080 = (2)5 – 0.080 = 32 – 0.080 = 31.92

Therefore, approximate value of (1.999)5 is 31.92

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