Final answer:
To compare dy and Δy for f(x) = 8x + 9 at x = 1 with Δx = 0.01, we find that dy, the exact change, and Δy, the actual change, are both 0.08. This is because the function is linear, and the small change in x doesn't lead to any differences between dy and Δy here.
Step-by-step explanation:
To compare the values of dy and Δy for the function f(x) = 8x + 9 when x = 1 and Δx = dx = 0.01, we first need to understand what dy and Δy represent. The differential, dy, is the theoretical exact change in y corresponding to a small change in x (dx), and it is found using the derivative of the function. Δy, on the other hand, is the actual change in the function's value over the interval Δx.
To find dy, we calculate the derivative of f(x), which is f'(x) = 8. Then we multiply this by dx to get dy: dy = f'(x) * dx = 8 * 0.01 = 0.08.
To find Δy, we evaluate the function at x + Δx and subtract the value of the function at x: Δy = f(x + Δx) - f(x) = f(1.01) - f(1) = (8*1.01 + 9) - (8*1 + 9) = 8.08 - 17 = 0.08. Interestingly, for linear functions like this one, dy and Δy are the same when dx is small.