For a function f(x), its average rate of change on an interval [a, b] is given by what's called the "difference quotient",
[tex]\dfrac{f(b)-f(a)}{b-a}[/tex]
and is basically the ratio of the difference in values of f(x) at the endpoints to the difference in the endpoints themselves.
I assume the 6 intervals below each table correspond to each table immediately above them.
From the first table, we get the following average rates of change (after arranging the intervals from left to right):
• [-3, -2] : (f(-2) - f(-3)) / (-2 - (-3)) = (7 - 12)/1 = -5
• [-2, -1] : (f(-1) - f(-2)) / (-1 - (-2)) = (4 - 7)/1 = -3
• [-1, 0] : (f(0) - f(-1)) / (0 - (-1)) = (3 - 4)/1 = -1
• [0, 1] : (f(1) - f(0)) / (1 - 0) = (4 - 3)/1 = 1
• [1, 2] : (f(2) - f(1)) / (2 - 1) = (7 - 4)/1 = 3
• [2, 3] : (f(3) - f(2)) / (3 - 2) = (12 - 7)/1 = 5
Just do the same for the other two tables. Note that each listed interval has length 1, so practically, each average rate of change over [a, b] is exactly f(b) - f(a).
For parts (c) and (d):
• (c) You'll notice that each of the average rates of change are odd numbers …
• (d) … but they're not necessarily all the same size, and they follow slightly different patterns.
