The average rate of change over the interval is; f'(x) = 4(b + 1)
The formula for the average rate of change of a function over an interval (a, b) is given by;
f'(x) = [f(b) - f(a)]/(b - a)
Now we are given the function f(x) = 4x² - 5 to find the average rate of change of the function over (1, b). Thus, we can say that the average rate of change of this over the interval is;
[f(b) - f(1)]/(b - 1)
f(b) = 4b² - 5
f(1) = 4(1²) - 5 = -1
Thus;
f'(x) = (4x² - 5 - (-1))/(b - 1)
f'(x) = (4b² - 4)/(b - 1)
f'(x) = 4(b - 1)(b + 1)/(b - 1)
f'(x) = 4(b + 1)
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