Question

Over which interval does the function f(x) = x² have a positive average rate of change?

Answer 1

The function f(x) = x² has a positive average rate of change over any interval where x is greater than 0. This is because the function is monotonically increasing for all positive x, and the square of any real number is always positive.

Step-by-step explanation:

The function f(x) = x² always has a positive average rate of change over any interval where x > 0. This is because the function is increasing for all positive values of x. To determine the average rate of change, you calculate the difference in the function values at two points and divide that by the difference in the points, often referred to as the slope of the secant line.

For example, consider two points x1 and x2 where x1 < x2 and both are positive real numbers. The average rate of change from x1 to x2 is computed as follows:

1. Calculate the function values: f(x1) = x1² and f(x2) = x2²
2. Find the difference in function values: f(x2) - f(x1) = x2² - x1²
3. Find the difference in x values: x2 - x1
4. Divide the difference in function values by the difference in x values: (x2² - x1²) / (x2 - x1)

This result will always be positive if x1 and x2 are positive due to how squares of real numbers work. Therefore, the function x² has a positive average rate of change over any interval where x > 0.