The average rate of change of the function f(x) = 8x² - 6 is given as 8h + 16x
It quantifies how much the function changed per unit on average throughout that time span. The slope of the straight line connecting the interval's ends on the function's graph is used to calculate it.
f(x) = 8x^2 - 6
Avge rate : [ f(x+h) - f(x)]/(x +h - x)
f(x+h) = 8(x+h)^2 - 6
f(x) = 8x^2 - 6
f(x+h) - f(x) = 8x^2 + 8h^2 + 16xh + 6 - 8x^2 -6 = 8h^2 + 16xh
Avg rate = ( 8h^2 + 16xh)/h
= 8h + 16x
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The average rate of change of the function over the interval is 8(b + 1)
The given parameters are
Function: f(x) = 8x^2 - 6
Interval: [1, b]
Calculate f(1) and f(b) using the function definition
So, we have
f(1) = 8(1)^2 - 6 = 2
f(b) = 8(b)^2 - 6 = 8b^2 - 6
The average rate of change of the function over the interval is then calculated as
Rate = [f(b) - f(1)]/[b - 1]
This gives
Rate = [8b^2 - 6 - 2]/[b - 1]
Evaluate
Rate = [8b^2 - 8]/[b - 1]
Factorize the numerator
Rate = [8(b + 1)(b - 1)]/[b - 1]
Divide through by b - 1
Rate = 8(b + 1)
Hence, the average rate of change of the function over the interval is 8(b + 1)
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