Final answer:
To estimate the value of √8.98 using linear approximation, we can use the tangent line to the function f(x) = √x at a convenient point near 8.98. By choosing x = 9 as the point, we can find the tangent line and substitute x = 8.98 to approximate the value. The correct answer is A. 2.5.
Step-by-step explanation:
To estimate the value of √8.98 using linear approximation, we can use the tangent line to the function f(x) = √x at a convenient point near 8.98. Let's choose x = 9 as our point.
The tangent line to f(x) at x = 9 can be found using the derivative of f(x), which is f'(x) = 1/(2√x). At x = 9, the derivative is f'(9) = 1/(2√9) = 1/6.To estimate √8.98 using an appropriate linear approximation, we can look for perfect squares that are close to 8.98.
The perfect square closest to 8.98 is 9, which is the square of 3. Since 8.98 is slightly less than 9, we can estimate that √8.98 will be slightly less than the square root of 9, which is 3. Therefore, the best estimate from the given options is 2.9, which is not explicitly listed. However, considering the listed options, the closest to 2.9 is 3, which corresponds with option B.
Now we can use the point-slope form of a linear equation y - y1 = m(x - x1), where (x1, y1) are the coordinates of the point and m is the slope of the line. Plugging in x1 = 9, y1 = f(9) = √9 = 3, and m = 1/6, we get y - 3 = (1/6)(x - 9). Solving for y, we find y = (1/6)x - 3/2.
Substituting x = 8.98 into the linear equation, we get y = (1/6)(8.98) - 3/2 = 1.498 - 1.5 = -0.002. So, the approximate value of √8.98 is -0.002, which is closest to 0. Therefore, the correct option is A. 2.5.