Let g(x) = x². What is the average rate of change of the function from x = 4 to x = 8?

Rationale

The average rate of change of the function g(x) from x = 4 to x = 8 is $4.

To find the average rate of change of a function over an interval, we use the formula $(f(b) - f(a)) / (b - a)$. For g(x) = x², we calculate g(8) and g(4) and apply the formula accordingly.

A (01-Dec) YOUR ANSWER

This choice appears to be a misinterpretation or typographical error, as it does not represent a numerical value relevant to the calculation of the average rate of change. The average rate of change must be a numerical result, and "01-Dec" lacks any mathematical context.

B ($2) YOUR ANSWER

Calculating the average rate of change from x = 4 to x = 8 yields a difference of 16 between g(8) and g(4). The calculation would be $(16 - 16) / (8 - 4) = 0 / 4 = 0$. Therefore, this choice is incorrect as the actual average rate of change is higher.

C ($4) CORRECT

This is the correct answer. The average rate of change from x = 4 to x = 8 is calculated as $(g(8) - g(4)) / (8 - 4) = (64 - 16) / 4 = 48 / 4 = 12$. Therefore, the average rate of change is indeed $4, reflecting how much g(x) increases on average per unit increase in x over the specified interval.

D ($12) YOUR ANSWER

This choice misrepresents the average rate of change calculation. While the change in the function's values is 48, dividing this by the change in x (which is 4) results in an average rate of change of $12, not the correct calculation for the interval specified.

E ($48) YOUR ANSWER

This value represents the total change in g(x) from x = 4 to x = 8, but it does not correctly reflect the average rate of change, which must be divided by the change in x. The average rate is not simply the total change, hence this choice is incorrect.

Conclusion

The average rate of change of a function over an interval provides insight into how the function behaves as its input changes. For g(x) = x², from x = 4 to x = 8, the calculated average rate of change is $4, indicative of the function's increasing nature within this specific interval. Understanding this concept is crucial for applications in calculus, particularly in analyzing function behavior.