The average rate of change of a function over a given interval can be found using the formula:
average rate of change = (change in y)/(change in x)
For example if we want to find the average rate of change of the function f(x) = x^2 over the interval [1 4] we would first calculate the change in y which is f(4) – f(1) = 16 – 1 = 15. Then we would calculate the change in x which is 4 – 1 = 3. Therefore the average rate of change is 15/3 = 5.
We can also use the formula to find the average rate of change of a function over a specific interval. For example if we want to find the average rate of change of the function f(x) = x^2 over the interval [2 4] we would first calculate the change in y which is f(4) – f(2) = 16 – 4 = 12. Then we would calculate the change in x which is 4 – 2 = 2. Therefore the average rate of change is 12/2 = 6.
If we want to find the average rate of change of a function over a more general interval we can take the limit as the interval approaches a specific point. For example if we want to find the average rate of change of the function f(x) = x^2 at x = 2 we would take the limit as the interval [2 4] approaches 2. This gives us:
lim_(x→2)((f(4)-f(2))/(4-2))
= lim_(x→2)((16-4)/(4-2))
= lim_(x→2)(12/2)
= 6
Therefore the average rate of change of the function f(x) = x^2 at x = 2 is 6.
How do you find the average rate of change of a function?
The average rate of change of a function is the rate of change of the function over a certain interval divided by the length of the interval.