To find the average rate of change, we need to calculate the change in the function value over the interval and divide it by the change in the input values.
First, find the value of the function at x = 1:
f(1) = 3(1)^2 + 2(1) + 25
f(1) = 3(1) + 2 + 25
f(1) = 3 + 2 + 25
f(1) = 30
Now, find the value of the function at x = 12:
f(12) = 3(12)^2 + 2(12) + 25
f(12) = 3(144) + 24 + 25
f(12) = 432 + 24 + 25
f(12) = 481
The change in the function values over the interval is:
481 - 30 = 451
The change in the input values is:
12 - 1 = 11
Now, calculate the average rate of change:
Average Rate of Change = Change in Function Values / Change in Input Values
Average Rate of Change = 451 / 11
Average Rate of Change = 41
Therefore, the average rate of change for the function over the interval 1 ≤ x ≤ 12 is 41.
The function f(x) = 3x2 + 2x + 25 models the sales of a new product over time.
Find the average rate of change for the function over the interval 1 ≤ x ≤ 12.
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