Use a linear approximation (or differentials) to estimate the given number.

Question

(1.999)³

=_______

Answer 1

Let take function f(x) = x³ and 2 points: x₀ = 2 and x₁ = 1.999; f(x_{0) =} 2³ = 8;

f(x₁) - f(x₀) = Δf(x) ≈ dx = f'(x₀)·Δx = 3x₀²(x₁ - x₀). So, x₁³ ≈ x₀³ + 3x₀²(x₁ - x₀);

1.999³ ≈ 8 + 3·2²(1.999 - 2) = 7.988

Answer 2

Let f(x) = x³, x = 2 (2 is "close" to 1.999) and Δx = -0.001 (since 1.999 is 0.001 less than 2)

f'(x) ≈ [f(x+Δx) - f(x)] / Δx. So. f(x+Δx) ≈ f(x) + f'(x)Δx

Therefore, (1.999)³ ≈ 2³ + 3(2)²(-0.001) = 7.988