Final answer:
To find the rate of change f(x+h) - f(x)/h, substitute the values of f(x+h) and f(x) into the equation and simplify. For f(x) = 3x², the rate of change is 6x + 3h. For f(x) = x² - 1, the rate of change is 2x + h.
Step-by-step explanation:
To find the rate of change f(x+h) - f(x)/h, we need to substitute the values of f(x+h) and f(x) into the equation and simplify. Let's calculate it for both functions:
a. f(x) = 3x²:
f(x+h) = 3(x+h)² = 3(x² + 2xh + h²) = 3x² + 6xh + 3h²
Now, substituting the values into the rate of change equation:
f(x+h) - f(x)/h = (3x² + 6xh + 3h²) - (3x²) / h = 6xh + 3h² / h = 6x + 3h
b. f(x) = x² - 1:
f(x+h) = (x + h)² - 1 = (x² + 2xh + h²) - 1 = x² + 2xh + h² - 1
Substituting the values into the rate of change equation:
f(x+h) - f(x)/h = (x² + 2xh + h² - 1) - (x² - 1) / h = 2xh + h² / h = 2x + h