Final answer:
The linear approximation of f(x) = √x at x = 0.16 is used to estimate the value of √0.18, resulting in an approximate value of 0.425 using the formula L(x) = 0.4 + 1.25(x - 0.16).
Step-by-step explanation:
The student is asking for the linear approximation of the function f(x) = √x at x = 0.16, and subsequently using this to approximate the value of √0.18. To find the linear approximation, we can use the formula for a linear approximation or the tangent line approximation, which is L(x) = f(a) + f'(a)(x - a), where 'a' is the point of approximation, and f'(a) is the derivative of the function at point a.
To approximate √x at x = 0.16, we first find the derivative f'(x) = ½x-1/2. At x = 0.16, f'(x) becomes f'(0.16) = ½(0.16)-1/2 = ½/0.4 = 1.25. Therefore, our linear approximation L(x) near x = 0.16 is L(x) = √0.16 + 1.25(x - 0.16). Since √0.16 is 0.4, the function simplifies to L(x) = 0.4 + 1.25(x - 0.16).
To approximate √0.18, substitute 0.18 for x in our linear approximation giving us L(0.18) = 0.4 + 1.25(0.18 - 0.16) = 0.4 + 1.25(0.02) = 0.4 + 0.025 = 0.425. Thus, the approximate value of √0.18 using the linear approximation at x = 0.16 is 0.425.