Answer:
b = 9
Explanation:
To find the number "b" such that the average rate of change of the function
on the interval [2, b] is -1/18, we need to calculate the average rate of change and then set it equal to -1/18.
The average rate of change of a function on an interval [a,b] is defined as the change in the function's value divided by the change in the input variable (in this case, x) over the interval. Mathematically, it is given by:

Where:
- f(b) represents the value of the function at x = b
- f(a) represents the value of the function at x = a

In our case,
, and the interval is [2, b]. So, a = 2 and b = b.
Now, we need to calculate f(a) and f(b):
For f(a); a = 2:

For f(b); a = b:

Next, we can plug all this information into the average rate of change formula:

Now, we can solve for "b." To do this, we can first cross-multiply to eliminate the fractions:

Expand the right side of the equation and simplify:

Now, let's get rid of the fractions by multiplying everything by b (the common denominator):

Move all the terms to one side to set the equation to zero:

Combine like terms:

Now we need to solve this quadratic equation for "b." We can either factor or use the quadratic formula. I will factor:

From this, we get two potential solutions: b = 2 or b = 9.
However, we need to check if both of these solutions satisfy the condition for the average rate of change to be -1/18.
For b = 2:

For b = 9:

Thus, the correct value for "b" that satisfies the condition is b = 9.

Additional Terminology:
Average Rate of Change: The average rate of change of a function over an interval measures the average rate at which the function's output values (y-values) change concerning the input values (x-values) within that interval. It represents the slope of the secant line between two points on the function's graph corresponding to the interval endpoints.
Interval Notation: In mathematics, interval notation is used to describe a range of values. The interval [2, b] represents all the values of "x" that lie between 2 and "b," including both 2 and "b." The square brackets denote that the endpoints 2 and "b" are included in the interval.
Factoring: A method used to find the roots (solutions) of a quadratic equation by rewriting it as a product of binomials.
Roots/Solutions: The values of the variable that satisfy the equation and make it true.