What is the definition of the rate of change?

The rate of change is a measure of how a quantity changes in relation to another quantity, typically expressed as a ratio. In calculus, it is often represented as the slope of a line or the derivative of a function.

How do you calculate the average rate of change of a function f(x) over the interval [a, b]?

The average rate of change of a function f(x) over the interval [a, b] is calculated using the formula: (f(b) - f(a)) / (b - a). This gives the slope of the secant line connecting the points (a, f(a)) and (b, f(b)).

If f(x) = 2x + 3, what is the average rate of change from x = 1 to x = 4?

First, calculate f(1) = 2(1) + 3 = 5 and f(4) = 2(4) + 3 = 11. Then, use the average rate of change formula: (f(4) - f(1)) / (4 - 1) = (11 - 5) / (4 - 1) = 6 / 3 = 2.

What does a positive rate of change indicate about a function?

A positive rate of change indicates that the function is increasing. This means that as the input value increases, the output value also increases.

If the rate of change of a function f(x) is constant, what type of function is it?

If the rate of change of a function f(x) is constant, it is a linear function. This means the graph of the function is a straight line, and the slope (rate of change) does not vary.

A car travels from point A to point B, covering a distance of 150 miles in 3 hours. What is the average rate of change of the car's position?

The average rate of change is calculated as distance/time. Thus, the average rate of change = 150 miles / 3 hours = 50 miles per hour.

A function is defined as f(x) = 3x² - x + 4. What is the average rate of change from x = 2 to x = 5?

First, find f(2) = 3(2)² - 2 + 4 = 14 and f(5) = 3(5)² - 5 + 4 = 64. Then calculate the average rate of change: (f(5) - f(2)) / (5 - 2) = (64 - 14) / 3 = 50 / 3.