Question

Find the average rate of change of the function on the interval specified for real number, h, in simplest form:

a(t) = 1 / (t + 3) on [9, 9 + h]

Option 1: (1 + h) / (12 + h)
Option 2: (1 + h) / (9 + h)
Option 3: (1 + h) / (3 + h)
Option 4: (1 + h) / (6 + h)

Answer 1

The average rate of change of the function a(t) = 1 / (t + 3) on the interval [9, 9 + h] is -1 / (12 + h), but the options given are all positive and none match the correct expression. The provided options may be incorrect or based on a misunderstanding of the question.

Step-by-step explanation:

The student is asking to find the average rate of change of the function a(t) = 1 / (t + 3) on the interval [9, 9 + h]. The average rate of change is calculated as the difference in function values over the difference in t values, which translates to (a(9 + h) - a(9))/(9 + h - 9). Plugging the values into the function gives us:

a(9+h) = 1 / (9 + h + 3) = 1 / (12 + h)

a(9) = 1 / (9 + 3) = 1 / 12

Therefore, the average rate of change is:

((1 / (12 + h)) - (1 / 12)) / h = (1 / (12 + h) - 1 / 12) / h = (12 - (12 + h)) / (12(12 + h)h) = -h / (12(12 + h)h) = -1 / (12 + h)

Since the change is from a larger value to a smaller value, the average rate of change is negative. However, the options provided do not include a negative sign, implying there may be a misunderstanding in the options presented or the question asked. Assuming the absolute value is sought, the positive form would be 1 / (12 + h), which is not listed in the options provided.