average rate of change
The average rate of change is a concept in mathematics that measures the rate at which a quantity or variable changes over a given interval
The average rate of change is a concept in mathematics that measures the rate at which a quantity or variable changes over a given interval. It is calculated by finding the difference between the values of the variable at the endpoints of the interval and dividing it by the length of the interval.
To calculate the average rate of change, you need to follow these steps:
1. Determine the two points on the graph or the values of the variable for which you want to find the average rate of change. Let’s call these points (x₁, y₁) and (x₂, y₂), or the values of the variable as a and b respectively.
2. Calculate the change in the variable. This can be done by subtracting the initial value from the final value: Δy = y₂ – y₁ or Δy = b – a.
3. Calculate the change in the independent variable or the length of the interval. This can be done by subtracting the initial value from the final value: Δx = x₂ – x₁.
4. Finally, calculate the average rate of change by dividing the change in the variable by the change in the independent variable: average rate of change = Δy/Δx or average rate of change = (b – a)/(x₂ – x₁).
The average rate of change gives you the average rate at which the variable is changing over the interval [x₁, x₂]. It measures the slope of the secant line passing through the two given points or values.
For example, let’s say we have a function f(x) = 2x + 3 and we want to find the average rate of change between x = 1 and x = 5.
Step 1: Identify the two points (x₁, y₁) = (1, 5) and (x₂, y₂) = (5, 13).
Step 2: Calculate the change in the variable: Δy = 13 – 5 = 8.
Step 3: Calculate the change in the independent variable: Δx = 5 – 1 = 4.
Step 4: Find the average rate of change: average rate of change = 8/4 = 2.
Therefore, the average rate of change of f(x) = 2x + 3 between x = 1 and x = 5 is 2, meaning that on average, the function increases by 2 units for every 1 unit increase in x over the interval [1, 5].
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