Average Rate of Change

Front

What is the average rate of change (ARC) of a function over an interval?

Back

The average rate of change of a function f(x) over the interval [x1, x2] is given by the formula: (f(x2) - f(x1)) / (x2 - x1).

Front

How do you calculate the average rate of change of f(x) = 2x^2 + 12x + 16 on the interval [-3, -2]?

Back

First, calculate f(-3) and f(-2). Then, use the ARC formula: (f(-2) - f(-3)) / (-2 - (-3)).

Front

What is the significance of the average rate of change in real-world applications?

Back

The average rate of change can represent the speed of an object, the growth rate of a population, or the change in cost over time.

Front

What does a positive average rate of change indicate about a function on an interval?

Back

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

Front

What does a negative average rate of change indicate about a function on an interval?

Back

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

Front

What is the average rate of change from x=0 to x=1 if f(0) = 2 and f(1) = 5?

Back

The average rate of change is (f(1) - f(0)) / (1 - 0) = (5 - 2) / (1) = 3.

Front

What is the formula for average rate of change (ARC)?

Back

The formula for average rate of change is: ARC = (y2 - y1) / (x2 - x1).

Front

How can the average rate of change be interpreted graphically?

Back

Graphically, the average rate of change represents the slope of the secant line connecting two points on the graph of the function.

Front

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

Back

For a linear function, the average rate of change is constant and equal to the slope of the line.

Front

How does the average rate of change differ from instantaneous rate of change?

Back

The average rate of change measures the change over an interval, while the instantaneous rate of change measures the change at a specific point.