a) Compute the average rate of change of $f(x)$ on the following intervals:
i) $[0,2]$:
ii) $[0.75,1.25]$:
iii) $[0.99, 1.01]$:

Compare these values to $e^{1}$ What do you notice?

A. As the interval gets smaller, the average rate of change gets closer to $e^1$.
B. As the interval gets larger, the average rate of change gets closer to $e^1$.
C. As the interval gets smaller, the average rate of change gets farther from $e^1$.
D. As the interval gets smaller, the average rate of change approaches 0.

b) Compute the average rate of change of $f(x)$ on the following intervals:
i) $[1,3]$:
ii) $[1.75,2.25]$:
iii) $[1.99, 2.01]$:

Compare these values to $e^{2}$ What do you notice?

A. As the interval gets smaller, the average rate of change gets closer to $e^2$.
B. As the interval gets larger, the average rate of change gets closer to $e^2$.
C. As the interval gets smaller, the average rate of change gets farther from $e^2$.
D. As the interval gets smaller, the average rate of change approaches 0.

c) Make a prediction about the average rate of change of $f(x)$ at any particular $x$ value.

A. We predict that as the distance between the right and left endpoints of an interval gets smaller, the average rate of change of $f(x)$ on that interval will approach $e^x$ for any $x$ value.
B. We predict that as the distance between the right and left endpoints of an interval gets larger, the average rate of change of $f(x)$ on that interval will approach $e^x$ for any $x$ value.
C. We predict that as the distance between $e^1$ and $f(x)$ gets larger, the average rate of change of $f(x)$ will approach 0 for any $x$ value.
D. We predict that as $f(x)$ approaches 0, the average rate of change of $f(x)$ on any interval will approach 0 for any $x$ value near 0.

Hint: