The average rate of change of a function over a specific interval can be calculated using the formula:
Average Rate of Change = (f(b) – f(a)) / (b – a)
In this case, we want to find the average rate of change of the function y = cos(2x) over the interval from x = 0 to x = π/2.
First, we need to evaluate the function at the endpoints of the interval:
1. Calculate f(0):
– We substitute x = 0 into the function:
f(0) = cos(2 * 0) = cos(0) = 1
2. Calculate f(π/2):
– We substitute x = π/2 into the function:
f(π/2) = cos(2 * (π/2)) = cos(π) = -1
Now we have:
– f(0) = 1
– f(π/2) = -1
Next, we can plug these values into the average rate of change formula:
1. Set a = 0 and b = π/2
Average Rate of Change = (f(π/2) – f(0)) / (π/2 – 0)
Average Rate of Change = (-1 – 1) / (π/2 – 0)
Average Rate of Change = (-2) / (π/2)
To simplify:
Average Rate of Change = -2 * (2/π) = -4/π
Therefore, the average rate of change of the function y = cos(2x) over the interval from 0 to π/2 is:
-4/π
This negative value indicates that the function is decreasing over this interval.