Final answer:
To estimate f ′(1), we can use the definition of the derivative and evaluate f(1 + h) and f(1). Simplifying the expression and taking the limit as h approaches 0, we find that the estimate for f ′(1) is -2.
Step-by-step explanation:
To estimate f ′(1), we can use the definition of the derivative:
f ′(a) = lim (h→0) [f(a + h) − f(a)] / h
Substituting a = 1:
f ′(1) = lim (h→0) [f(1 + h) − f(1)] / h
Next, let's evaluate f(1 + h) and f(1):
f(1 + h) = |2 - (1 + h)²| = |2 - (1 + 2h + h²)| = |1 - 2h - h²|
f(1) = |2 - 1²| = |1| = 1
Plugging these values back into the derivative definition:
f ′(1) = lim (h→0) [|1 - 2h - h²| - 1] / h
Now, we need to simplify the expression inside the absolute value. Since h is approaching 0, the terms involving h² and h will become negligible. Therefore, we can approximate the derivative as:
f ′(1) ≈ lim (h→0) [1 - 2h - h² - 1] / h = lim (h→0) [-2h] / h = lim (h→0) -2 = -2
So, the estimate for f ′(1) is -2.