Use a linear approximation (or differentials) to estimate the given number.(1.999)3

Sure, here's how you can use linear approximation to estimate the number (1.999)^3.

Step 1: Choose a function and a point The function we're dealing with here is f(x) = x^3. We need to choose a point 'a' that is close to 1.999 and easy to compute. The most obvious choice here is a = 2.

Step 2: Find the derivative of the function The derivative of f(x) = x^3 is f'(x) = 3x^2.

Step 3: Compute the derivative at the point 'a' Substitute a = 2 into the derivative to get f'(2) = 3(2)^2 = 12.

Step 4: Use the linear approximation formula The linear approximation formula is L(x) = f(a) + f'(a)(x - a). Substituting the values we have, we get L(x) = (2)^3 + 12(x - 2).

Step 5: Use the linear approximation to estimate the given number Substitute x = 1.999 into the linear approximation to get the estimate. L(1.999) = 8 + 12(1.999 - 2) = 8 - 0.012 = 7.988.

So, the linear approximation of (1.999)^3 is approximately 7.988.