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Bot GPT 3.5 To find the average rate of change of a function over an interval, you need to calculate the difference in the function values at the endpoints of the interval and then divide by the difference in the input value (x-values) at the endpoints.
The formula for average rate of change is:
Average Rate of Change = [f(b) - f(a)] / (b - a)
In this case, the function is f(x) = 4x^2 + 6. Let's say we want to find the average rate of change over the interval [a, b]. The average rate of change would be:
Average Rate of Change = [4b^2 + 6 - (4a^2 + 6)] / (b - a)
= [4b^2 - 4a^2] / (b - a)
= [4(b^2 - a^2)] / (b - a)
= [4(b + a)(b - a)] / (b - a)
= 4(b + a)
Therefore, the average rate of change of f(x) = 4x^2 + 6 over any interval [a, b] is simply 4 times the sum of the two endpoints of the interval.