Final answer:
To estimate the value of (8.07)^2/3 using linear approximation, we can use the formula f(x) ≈ f(a) + f'(a)(x - a), where a is a convenient point, and f'(a) is the derivative of f(x) at a. By substituting the values into the formula, we find that the estimated value is approximately 4.0233.
Step-by-step explanation:
To estimate the value of (8.07)^2/3 using linear approximation, we start by finding the linear approximation of the function f(x) = x^2/3 at a convenient point near 8.07. Let's choose x = 8.
The linear approximation is given by f(x) ≈ f(a) + f'(a)(x - a), where a is the convenient point and f'(a) is the derivative of f(x) at a.
Find f(8) ≈ (8)^2/3 = 2^2 = 4.
Next, find f'(x) = d/dx (x^2/3) = (2/3)(x^(-1/3)).
Now, evaluate f'(a) ≈ (2/3)(8^(-1/3)) ≈ (2/3) * (0.5) = 1/3.
Substitute these values into the linear approximation formula: f(x) ≈ f(a) + f'(a)(x - a) ≈ 4 + (1/3)(8.07 - 8) ≈ 4 + (1/3)(0.07) ≈ 4 + 0.0233 ≈ 4.0233.
Therefore, the estimated value of (8.07)^2/3 using linear approximation is approximately 4.0233.