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Given the function g(x)=x^(2)+8x+12, determine the average rate of change of the function over the interval -10 <= x <= -1. Answer:

Given the function g(x)=x2+8x+12 g(x)=x^{2}+8 x+12 , determine the average rate of change of the function over the interval −10≤x≤−1 -10 \leq x \leq-1 .\newlineAnswer:

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Q. Given the function g(x)=x2+8x+12 g(x)=x^{2}+8 x+12 , determine the average rate of change of the function over the interval −10≤x≤−1 -10 \leq x \leq-1 .\newlineAnswer:
  1. Given function: We are given the function g(x)=x2+8x+12g(x) = x^2 + 8x + 12. To find the average rate of change over the interval [−10,−1][-10, -1], 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(−10)g(-10): First, we need to find the value of g(−10)g(-10). We substitute x=−10x = -10 into the function g(x)g(x).\newlineg(−10)=(−10)2+8∗(−10)+12g(-10) = (-10)^2 + 8*(-10) + 12\newlineg(−10)=100−80+12g(-10) = 100 - 80 + 12\newlineg(−10)=32g(-10) = 32
  3. Find g(−1)g(-1): Next, we need to find the value of g(−1)g(-1). We substitute x=−1x = -1 into the function g(x)g(x).\newlineg(−1)=(−1)2+8∗(−1)+12g(-1) = (-1)^2 + 8*(-1) + 12\newlineg(−1)=1−8+12g(-1) = 1 - 8 + 12\newlineg(−1)=5g(-1) = 5
  4. Calculate average rate of change: Now we have the values g(−10)=32g(-10) = 32 and g(−1)=5g(-1) = 5. 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=−10a = -10 and b=−1b = -1.\newlineAverage rate of change = (g(−1)−g(−10))/(−1−(−10))(g(-1) - g(-10)) / (-1 - (-10))\newlineAverage rate of change = (5−32)/(9)(5 - 32) / (9)\newlineAverage rate of change = −27/9-27 / 9\newlineAverage rate of change = −3-3

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