The rate of change between the interval of x = 0 and x = π/2 can be understood by evaluating the change in a given function over this interval. Typically, this is analyzed in the context of calculus, where the derivative of a function provides the rate of change at a specific point.
For example, if we consider the function f(x) = sin(x), the derivative f'(x) = cos(x) gives us the rate of change of the function. At the interval endpoints:
- At x = 0, f'(0) = cos(0) = 1.
- At x = π/2, f'(π/2) = cos(π/2) = 0.
This indicates that the function is increasing at x = 0 and the rate of increase slows as x approaches π/2.
To calculate the overall change in function value over this interval, you can compute:
f(π/2) – f(0) = sin(π/2) – sin(0) = 1 – 0 = 1.
Therefore, the average rate of change over the interval from 0 to π/2 can be calculated as:
Average rate of change = (f(π/2) – f(0)) / (π/2 – 0) = 1 / (π/2) = 2/π.
This shows that the average rate of change of the function sin(x) between 0 and π/2 is 2/π. Depending on the function you analyze, the process of finding the rate of change can yield different results, but the fundamental approach remains the same.