what is the average rate of change for the function g(x)=3x^2-4 over the interval (-1,3)?
-6 <_ x <_ 3
To find the average rate of change of a function over a given interval, you need to calculate the difference in the function's values at the endpoints of the interval and divide it by the difference in the inputs (x-values) of the endpoints.
In this case, the given function is g(x) = 3x^2 - 4, and the interval is (-1, 3).
Step 1: Calculate the difference in the function values:
Evaluate g(x) at the endpoints of the interval:
g(-1) = 3(-1)^2 - 4 = 3 - 4 = -1
g(3) = 3(3)^2 - 4 = 3(9) - 4 = 27 - 4 = 23
Therefore, the difference in the function values is: g(3) - g(-1) = 23 - (-1) = 24
Step 2: Calculate the difference in the inputs:
The difference in the inputs (x-values) is: 3 - (-1) = 4
Step 3: Calculate the average rate of change:
Divide the difference in function values by the difference in inputs:
Average Rate of Change = (g(3) - g(-1)) / (3 - (-1))
= 24 / 4
= 6
Therefore, the average rate of change for the function g(x) = 3x^2 - 4 over the interval (-1, 3) is 6.
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