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Given the function g(x)=-x^(2)+6x+12, determine the average rate of change of the function over the interval -1 <= x <= 6. Answer:

Given the function g(x)=−x2+6x+12 g(x)=-x^{2}+6 x+12 , determine the average rate of change of the function over the interval −1≤x≤6 -1 \leq x \leq 6 .\newlineAnswer:

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Q. Given the function g(x)=−x2+6x+12 g(x)=-x^{2}+6 x+12 , determine the average rate of change of the function over the interval −1≤x≤6 -1 \leq x \leq 6 .\newlineAnswer:
  1. Define Function: We have the function g(x)=−x2+6x+12g(x) = -x^2 + 6x + 12. To find the average rate of change over the interval [−1,6][-1, 6], we will use the formula for the average rate of change, which is (g(b)−g(a))/(b−a)(g(b) - g(a)) / (b - a), where aa and bb are the endpoints of the interval.
  2. Find g(−1)g(-1): First, we need to find the value of g(−1)g(-1). We substitute x=−1x = -1 into the function g(x)g(x).
    g(−1)=−(−1)2+6(−1)+12g(-1) = -(-1)^2 + 6(-1) + 12
    g(−1)=−1−6+12g(-1) = -1 - 6 + 12
    g(−1)=5g(-1) = 5
  3. Find g(6)g(6): Next, we need to find the value of g(6)g(6). We substitute x=6x = 6 into the function g(x)g(x).g(6)=−(6)2+6(6)+12g(6) = -(6)^2 + 6(6) + 12g(6)=−36+36+12g(6) = -36 + 36 + 12g(6)=12g(6) = 12
  4. Calculate Average Rate of Change: Now that we have g(−1)=5g(-1) = 5 and g(6)=12g(6) = 12, we can calculate the average rate of change using the values of g(a)g(a) and g(b)g(b) we found in the previous steps.\newlineAverage rate of change = (g(6)−g(−1))/(6−(−1))(g(6) - g(-1)) / (6 - (-1))\newlineAverage rate of change = (12−5)/(6−(−1))(12 - 5) / (6 - (-1))\newlineAverage rate of change = 7/77 / 7\newlineAverage rate of change = 11

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