Find the average rate of change of the function y = 2e^x over the interval from x = 0 to x = 2.

Solution:

Given, y = 2e^x

We have to find the average rate of change in f(x) over the interval (1, 5)

The average rate of change of the function f(x) over the interval (a,b) is equal to

\(\frac{[f(b)-f(a)]}{(b-a)}\)

Here, a = 0 and b = 2

f(0) = 2e⁰

f(0) = 2(1)

f(0) = 2

f(2) = 2e²

Now, to find average rate of change in f(x)

\(\frac{[f(b)-f(a)]}{(b-a)}\) = \(\frac{[2e^2-2]}{(2-0)}\)

= \(\frac{2(e^2-1)}{2}\)

= e² - 1

Therefore, the average rate of change of the given function is e² - 1.

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Find the average rate of change of the function y = 2e^x over the interval from x = 0 to x = 2.

Summary:

The average rate of change of the function y = 2ev over the interval from x = 0 to x = 2 is e² - 1.