Question 13

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), where a = 4 and b = 8. Calculating this gives us (f(8) - f(4)) / (8 - 4) = (64 - 16) / 4 = 48 / 4 = 12.

A) 1/12

Option A is incorrect as it underestimates the average rate of change. The calculation clearly shows that the average rate of change from x=4 to x=8 results in 12, not a fraction such as 1/12.

B) 2

Option B is also incorrect. An average rate of change of 2 would imply a much smaller difference in the output values of the function over the interval, which does not align with the function's values computed at x=4 and x=8.

C) 4

Option C is incorrect as well. A rate of 4 does not accurately represent the change observed in the function values between x=4 and x=8, as shown by the calculated average rate of change being 12.

D) 12

Option D is correct. The calculations demonstrate that the average rate of change from x=4 to x=8 for the function f(x) = x² is indeed 12, confirming that this option accurately reflects the computed value.

E) 48

Option E is incorrect. While 48 is the difference in function values (f(8) - f(4)), it does not represent the average rate of change over the interval since the average is calculated by dividing this difference by the interval length.

Conclusion

The correct answer is 12, which directly results from applying the formula for the average rate of change. All other options fail to reflect the accurate computation based on the function's behavior over the specified interval, either by miscalculating the values or misrepresenting the average.