Question

Given the function:

f(x)=x² - 4
A) Find the average rate of change over [1, 5].
B) Express the average rate of change as difference quotient f(x + h) – f(x)h

Answer 1

The average rate of change of the function f(x) = x² - 4 over the interval [1, 5] is 6. This rate can also be expressed as a difference quotient, resulting in 2x + h for non-zero h values.

Step-by-step explanation:

The question pertains to finding the average rate of change of the function f(x)=x² - 4 over the interval [1, 5] and also expressing this average rate of change as a difference quotient involving f(x + h) and f(x).

• A) To find the average rate of change of the function over the interval [1, 5], we calculate the difference in the function values at the endpoints of the interval divided by the change in x. This gives us:
  
  [(f(5) - f(1)] / (5 - 1) = [(25 - 4) - (1 - 4)] / 4 = (21 + 3) / 4 = 24 / 4 = 6.
• B) The difference quotient is expressed as:
  
  [f(x + h) - f(x)] / h, which simplifies to [(x + h)² - 4 - (x² - 4)] / h. By expanding and simplifying, we get:
  
  (x² + 2xh + h² - 4 - x² + 4) / h, which further simplifies to (2xh + h²) / h and ultimately 2x + h for values of h that are not equal to 0.