Final answer:
To find the value of b for which the average rate of change of f(x)=x on the interval [2, b] is -1/12, we use the formula for the slope of the secant line. After setting up the equation and simplifying, we find that it implies b must equal 2, which is not possible as it does not denote a change over an interval. Therefore, there is no value of b that results in an average rate of change of -1/12 for the function f(x).
Step-by-step explanation:
The student is asking to find the number b such that the average rate of change of the function f(x) on the interval [2, b] is -1/12. The average rate of change is calculated as the change in f(x) divided by the change in x, which represents the slope of the secant line connecting the two points on the function.
To find the average rate of change of f(x) = x over the interval [2, b], we need to determine the values f(2) and f(b). Since f(x) is simply x, we have f(2) = 2 and f(b) = b. The average rate of change is equal to (f(b) - f(2)) / (b - 2), which simplifies to (b - 2) / (b - 2). Setting this equal to -1/12 and solving for b yields the answer.
We get the equation (b - 2) / (b - 2) = -1/12. Cross-multiplication leads to 12(b - 2) = -1(b - 2). Solving this we find that 12b - 24 = -b + 2. Adding b to both sides and adding 24 to both sides gives us 13b = 26. Lastly, dividing both sides by 13 we find b = 2. However, since we are looking for a change over an interval, b cannot be equal to 2. This implies that there is no such value of b where the average rate of change is -1/12 because the expression (b - 2) / (b - 2) is undefined when b = 2 and simplifies to 1 for all other values of b, and it is never equal to -1/12.