Given the function g, of, x, equals, minus, x, squared, minus, 2, x, plus, 6g(x)=−x 2 −2x+6, determine the average rate of change of the function over the interval minus, 7, is less than or equal to, x, is less than or equal to, 3−7≤x≤3.
Question
Given the function g, of, x, equals, minus, x, squared, minus, 2, x, plus, 6g(x)=−x 2 −2x+6, determine the average rate of change of the function over the interval minus, 7, is less than or equal to, x, is less than or equal to, 3−7≤x≤3.
Solution
The average rate of change of a function over an interval [a, b] is given by the formula:
(f(b) - f(a)) / (b - a)
Here, our function is g(x) = -x^2 - 2x + 6 and the interval is [-7, 3].
First, we need to find the values of the function at the endpoints of the interval.
g(-7) = -(-7)^2 - 2*(-7) + 6 = -49 + 14 + 6 = -29 g(3) = -(3)^2 - 2*3 + 6 = -9 - 6 + 6 = -9
Then, we substitute these values into the formula:
(g(3) - g(-7)) / (3 - (-7)) = (-9 - (-29)) / (3 - (-7)) = 20 / 10 = 2
So, the average rate of change of the function g(x) = -x^2 - 2x + 6 over the interval -7 ≤ x ≤ 3 is 2.
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