To find the average rate of change of a function over a given interval, you can use the formula:
Average Rate of Change = (f(b) – f(a)) / (b – a)
In this case, we are looking at the function y = 2e^x over the interval from x = 0 to x = 2. Here, a = 0 and b = 2.
1. **Calculate f(a)**:
Substituting a = 0 into the function:
f(0) = 2e^0 = 2(1) = 2
2. **Calculate f(b)**:
Substituting b = 2 into the function:
f(2) = 2e^2
Now, since e^2 is approximately 7.3891, we find:
f(2) = 2 * 7.3891 ≈ 14.7782
3. **Plugging in the values into the average rate of change formula:**
Average Rate of Change = (f(2) – f(0)) / (2 – 0)
Average Rate of Change = (14.7782 – 2) / 2
Average Rate of Change = 12.7782 / 2
Average Rate of Change ≈ 6.3891
Thus, the average rate of change of the function y = 2e^x over the interval from x = 0 to x = 2 is approximately 6.3891.