Average Rate of Change of f(x) on [a,b]
The average rate of change of a function f(x) on the interval [a, b] can be calculated by finding the difference in the values of f(x) at the endpoints (f(b) – f(a)) and dividing it by the difference in x-coordinates, which is (b – a)
The average rate of change of a function f(x) on the interval [a, b] can be calculated by finding the difference in the values of f(x) at the endpoints (f(b) – f(a)) and dividing it by the difference in x-coordinates, which is (b – a).
The formula for the average rate of change is:
Average Rate of Change = (f(b) – f(a))/(b – a)
To calculate the average rate of change, follow these steps:
1. Determine the values of f(x) at points a and b. Substituting the x-coordinate values a and b into the function f(x) will give you the corresponding y-values.
2. Subtract the value of f(a) from f(b) to find the difference in y-coordinates: (f(b) – f(a)).
3. Calculate the difference in x-coordinates by subtracting the endpoints a and b: (b – a).
4. Divide the difference in y-coordinates by the difference in x-coordinates to get the average rate of change.
Let’s look at an example:
Suppose we have the function f(x) = 2x^2 – 3x + 1 and we want to calculate the average rate of change on the interval [1, 3].
Step 1: Find the values of f(x) at points 1 and 3:
f(1) = 2(1)^2 – 3(1) + 1 = 2 – 3 + 1 = 0
f(3) = 2(3)^2 – 3(3) + 1 = 18 – 9 + 1 = 10
Step 2: Find the difference in y-coordinates:
f(b) – f(a) = 10 – 0 = 10
Step 3: Find the difference in x-coordinates:
b – a = 3 – 1 = 2
Step 4: Calculate the average rate of change:
Average Rate of Change = (f(b) – f(a))/(b – a) = 10/2 = 5
Therefore, the average rate of change of f(x) = 2x^2 – 3x + 1 on the interval [1, 3] is 5.
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