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Given the function g(x)=-x^(2)-8x+24, determine the average rate of change of the function over the interval -6 <= x <= 1. Answer:

Given the function g(x)=−x2−8x+24 g(x)=-x^{2}-8 x+24 , determine the average rate of change of the function over the interval −6≤x≤1 -6 \leq x \leq 1 .\newlineAnswer:

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Q. Given the function g(x)=−x2−8x+24 g(x)=-x^{2}-8 x+24 , determine the average rate of change of the function over the interval −6≤x≤1 -6 \leq x \leq 1 .\newlineAnswer:
  1. Calculate g(−6)g(-6): We have the function g(x)=−x2−8x+24g(x) = -x^2 - 8x + 24. To find the average rate of change over the interval [−6,1][-6, 1], we will use the formula for 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. Calculate g(1)g(1): First, we need to find the value of g(−6)g(-6). We substitute x=−6x = -6 into the function g(x)g(x).\newlineg(−6)=−(−6)2−8(−6)+24g(-6) = -(-6)^2 - 8(-6) + 24\newlineg(−6)=−36+48+24g(-6) = -36 + 48 + 24\newlineg(−6)=36g(-6) = 36
  3. Calculate Average Rate of Change: Next, 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)2−8(1)+24g(1) = -(1)^2 - 8(1) + 24g(1)=−1−8+24g(1) = -1 - 8 + 24g(1)=15g(1) = 15
  4. Calculate Average Rate of Change: Next, 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)2−8(1)+24g(1) = -(1)^2 - 8(1) + 24g(1)=−1−8+24g(1) = -1 - 8 + 24g(1)=15g(1) = 15Now that we have g(−6)=36g(-6) = 36 and g(1)=15g(1) = 15, we can calculate the average rate of change using the formula (g(b)−g(a))/(b−a)(g(b) - g(a)) / (b - a) with a=−6a = -6 and x=1x = 100.Average rate of change = x=1x = 111Average rate of change = x=1x = 122Average rate of change = x=1x = 133Average rate of change = x=1x = 144

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