Average Rate of Change

Front

What is the average rate of change?

Back

The average rate of change of a function over an interval [a, b] is defined as the change in the function's value divided by the change in the input value, calculated as (f(b) - f(a)) / (b - a).

Front

How do you calculate the average rate of change for a linear function?

Back

For a linear function, the average rate of change is constant and equal to the slope of the line, which can be calculated using the formula (y2 - y1) / (x2 - x1).

Front

What is the formula for the slope of a line?

Back

The slope (m) of a line is calculated using the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.

Front

Given f(x) = 3x + 2, find the average rate of change from x = 1 to x = 4.

Back

The average rate of change is (f(4) - f(1)) / (4 - 1) = (14 - 5) / 3 = 3.

Front

What does a positive average rate of change indicate?

Back

A positive average rate of change indicates that the function is increasing over the interval.

Front

What does a negative average rate of change indicate?

Back

A negative average rate of change indicates that the function is decreasing over the interval.

Front

If f(x) = x^2, what is the average rate of change from x = 2 to x = 5?

Back

The average rate of change is (f(5) - f(2)) / (5 - 2) = (25 - 4) / 3 = 7.

Front

What is the average rate of change of f(x) = 2x^2 + 3 on the interval [1, 3]?

Back

The average rate of change is (f(3) - f(1)) / (3 - 1) = (21 - 5) / 2 = 8.

Front

How does the average rate of change relate to the derivative?

Back

The average rate of change over an interval approaches the instantaneous rate of change (derivative) as the interval shrinks to a point.

Front

What is the average rate of change of a constant function?

Back

The average rate of change of a constant function is 0, as there is no change in the function's value regardless of the input.