The average rate of change formula is used to calculate the rate of change of a variable over a given time period. It is a measure of how much a quantity changes on average over a certain time interval. The average rate of change formula is commonly used in calculus, physics, and other sciences to measure the rate of change of a variable over time or distance. For example, it can be used to calculate the average velocity of an object moving at a constant speed, or the average growth rate of a population over a given time period.
Formula for the Average Rate of Change of a Function
The formula for the average rate of change of a function is:
Average Rate of Change = (f(b) – f(a)) / (b – a)
where:
- f(b) is the value of the function at the endpoint b
- f(a) is the value of the function at the endpoint a
- b is the final value of the independent variable
- a is the initial value of the independent variable
This formula calculates the average rate at which the dependent variable changes with respect to the independent variable over a given interval [a, b]. It measures the slope of the secant line that passes through the points (a, f(a)) and (b, f(b)) on the graph of the function.
The average rate of change formula is commonly used in calculus and other fields of mathematics to study the behavior of functions over a given interval. It can be used to find the average velocity of an object over a certain time interval, or the average growth rate of a population over a certain period.
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Solved Examples
Example 1: Find the average rate of change of the function f(x) = 2x^2 – 4x + 1 over the interval [1, 4].
Solution: To find the average rate of change of the function f(x) over the interval [1, 4], we need to calculate the slope of the line that connects the points (1, f(1)) and (4, f(4)) on the graph of the function. We can use the average rate of change formula to do this:
Average Rate of Change = (f(4) – f(1)) / (4 – 1)
= ((2(4)^2 – 4(4) + 1) – (2(1)^2 – 4(1) + 1)) / 3
= (27 – (-1)) / 3
= 28 / 3
Therefore, the average rate of change of the function f(x) over the interval [1, 4] is 28/3.
Example 2: The position of a car moving along a straight line is given by the function s(t) = t^3 – 6t^2 + 9t, where s is measured in meters and t is measured in seconds. Find the average velocity of the car over the interval [0, 4].
Solution: To find the average velocity of the car over the interval [0, 4], we need to calculate the slope of the line that connects the points (0, s(0)) and (4, s(4)) on the graph of the function. We can use the average rate of change formula to do this:
Average Rate of Change = (s(4) – s(0)) / (4 – 0)
= ((4^3 – 6(4^2) + 9(4)) – (0^3 – 6(0^2) + 9(0))) / 4
= (64 – 0) / 4
= 16
Therefore, the average velocity of the car over the interval [0, 4] is 16 m/s.
