Final answer:
To find the linear approximation error when using the linear approximation to estimate cube root of -0.75, we first find the linear approximation of the function using the given equation. Then, we substitute the value -0.75 into the linear approximation equation to estimate the value. Finally, we calculate the absolute difference between the actual value and the linear approximation to find the linear approximation error, which in this case is 0.
Step-by-step explanation:
To find the linear approximation error, we first need to find the linear approximation of the function. The linear approximation is given by the equation: f(x) ≈ f(a) + f'(a)(x - a), where a represents the value near which we want to find the linear approximation. In this case, a = -1. To find the linear approximation of f(x) = cube root of x near x = -1, we need to find f(-1) and f'(-1).
f(-1) = cube root of -1 = -1
To find f'(-1), we first find f'(x), which represents the derivative of the function f(x). In this case, f'(x) = (1/3)(x)^(-2/3).
f'(-1) = (1/3)(-1)^(-2/3) = (1/3)(-1) = -1/3
Using these values, we can now calculate the linear approximation: f(x) ≈ f(-1) + f'(-1)(x - (-1)) = -1 - (1/3)(x + 1).
To estimate cube root of -0.75 using the linear approximation, we substitute x = -0.75 into the linear approximation equation: f(-0.75) ≈ -1 - (1/3)(-0.75 + 1) = -1 - (1/3)(0.25) = -1 - (1/12) = -13/12
The linear approximation error can be found by calculating the absolute difference between the actual value of the function and the linear approximation: |f(-0.75) - (-13/12)| = |(-13/12) - (-13/12)| = 0.