Question

these tables represent an exponential function. find the average rate of change for the interval from x = 8 to x = 9.

a. 6561
b. 19,683
c. 3
d. 13,122

Explanation:

Step1: Find the pattern of average rate of change

From the table of average rate of change, we can see that each subsequent average rate of change is 3 times the previous one.
The average rate of change from \(0\) to \(1\) is \(2\), from \(1\) to \(2\) is \(2\times3 = 6\), from \(2\) to \(3\) is \(6\times3=18\), and so on.

Step2: Calculate the number of multiplications by 3

The interval from \(x = 8\) to \(x=9\) is \(8\) intervals away from the interval \(0\) to \(1\).
We know that the formula for the \(n\) - th term of a geometric sequence \(a_n=a_1r^{n - 1}\), where \(a_1 = 2\) (the average rate of change from \(0\) to \(1\)), \(r = 3\) (the common ratio), and \(n=9\) (since we want the 9th term corresponding to the interval \(8\) to \(9\)).

Step3: Calculate the average rate of change

Using the formula \(a_n=a_1r^{n - 1}\), substitute \(a_1 = 2\), \(r = 3\), and \(n = 9\)

$$a_9=2\times3^{9 - 1}=2\times3^8=2\times6561 = 13122$$

Answer:

D. 13,122