Final answer:
The linear approximation of the 4th root of 17 is 2.03125, which is closest to option D) 2.5.
Step-by-step explanation:
The linear approximation of the 4th root of 17 can be found by using the concept of tangent line approximation. Let's start with a known value, such as the 4th root of 16, which is 2. Then we can calculate the derivative of the function f(x) = x^(1/4) at x = 16, which is f'(x) = 1/(4*x^(3/4)). Plugging in x = 16, we get f'(16) = 1/32. Now we can use the linear approximation formula: f(x) ≈ f(a) + f'(a) * (x - a), where a is 16 and x is 17. Plugging in the values, we get f(17) ≈ 2 + (1/32) * (17 - 16) = 2 + (1/32) = 2.03125. Therefore, the closest option is D) 2.5.