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 from x=4 to x=8 is 12.

To find the average rate of change of the function f(x) = x² from x = 4 to x = 8, we use the formula \((f(b) - f(a)) / (b - a)\). Here, f(8) = 64 and f(4) = 16, so the average rate of change calculates to (64 - 16) / (8 - 4) = 48 / 4 = 12.

A (1/12) YOUR ANSWER

This choice represents a fraction that does not correspond to the calculated average rate of change. The values of f(8) and f(4) yield a difference of 48, which, when divided by the interval length of 4, results in 12, not 1/12.

B (2) YOUR ANSWER

Choosing 2 indicates a misunderstanding of the change in the function's values over the specified interval. The average rate of change of 12 indicates a significant increase in the function values, and dividing the difference of 48 by 4 yields 12, not 2.

C (4) YOUR ANSWER

This choice is also incorrect as it underestimates the average rate of change. The calculation reveals a total change of 48 over an interval of 4, leading to an average rate of change of 12, far exceeding the value of 4.

D (12) CORRECT

This is the correct answer, derived from the average rate of change formula. With f(8) = 64 and f(4) = 16, the average change is (64 - 16) / (8 - 4) = 48 / 4 = 12.

E (48) YOUR ANSWER

This choice misrepresents the average rate of change by reflecting the total change in function values rather than the average over the interval. The total change of 48 must be divided by the interval length of 4, yielding an average rate of 12, not 48.

Conclusion

The average rate of change of a function over an interval is calculated by finding the difference in function values and dividing by the length of the interval. For the function f(x) = x² from x = 4 to x = 8, the average rate of change is 12, indicating a significant increase in the function's value over this range. The other choices fail to accurately represent this calculated average.