Final answer:
To approximate the value of (1.999)³, we use linear approximation with the derivative of the function f(x) = x³ at x = 2. The estimate using the tangent line approximation is approximately 7.988.
Step-by-step explanation:
To estimate the value of (1.999)³, we can use a linear approximation, which is a method of using the tangent line at a known point on a function to approximate the value of the function near that point. In this case, we can use the function f(x) = x³ and the point (2, 8), since 2 is close to 1.999 and easy to compute.
To find the linear approximation, we will first compute the derivative of f(x), which is f'(x) = 3x². At x = 2, f'(2) = 3(2)² = 12. The equation of the tangent line at x = 2 is therefore y = f'(2)(x - 2) + f(2), or y = 12(x - 2) + 8.
Substituting x = 1.999 into our tangent line equation gives us y = 12(1.999 - 2) + 8. This simplifies to y = 12(-0.001) + 8, which yields y ≈ 7.988. Thus, (1.999)³ is approximately 7.988.