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Given the function g(x)=-x^(2)-7x+15, determine the average rate of change of the function over the interval -5 <= x <= -1. Answer:

Given the function g(x)=−x2−7x+15 g(x)=-x^{2}-7 x+15 , determine the average rate of change of the function over the interval −5≤x≤−1 -5 \leq x \leq-1 .\newlineAnswer:

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Q. Given the function g(x)=−x2−7x+15 g(x)=-x^{2}-7 x+15 , determine the average rate of change of the function over the interval −5≤x≤−1 -5 \leq x \leq-1 .\newlineAnswer:
  1. Given Function and Interval: We are given the function g(x)=−x2−7x+15g(x) = -x^2 - 7x + 15 and asked to find the average rate of change over the interval [−5,−1][-5, -1]. The average rate of change is calculated using the formula:\newlineAverage rate of change = (g(b)−g(a))/(b−a)(g(b) - g(a)) / (b - a)\newlinewhere aa and bb are the endpoints of the interval. In this case, a=−5a = -5 and b=−1b = -1.
  2. Calculate g(a)g(a) for a=−5a = -5: First, we need to find the value of g(a)g(a) where a=−5a = -5. We substitute x=−5x = -5 into the function g(x)g(x):
    g(−5)=−(−5)2−7(−5)+15g(-5) = -(-5)^2 - 7(-5) + 15
    g(−5)=−(25)+35+15g(-5) = -(25) + 35 + 15
    g(−5)=−25+35+15g(-5) = -25 + 35 + 15
    g(−5)=25g(-5) = 25
  3. Calculate g(b)g(b) for b=−1b = -1: Next, we need to find the value of g(b)g(b) where b=−1b = -1. We substitute x=−1x = -1 into the function g(x)g(x):
    g(−1)=−(−1)2−7(−1)+15g(-1) = -(-1)^2 - 7(-1) + 15
    g(−1)=−(1)+7+15g(-1) = -(1) + 7 + 15
    g(−1)=−1+7+15g(-1) = -1 + 7 + 15
    g(−1)=21g(-1) = 21
  4. Calculate Average Rate of Change: Now that we have g(a)g(a) and g(b)g(b), we can calculate the average rate of change:\newlineAverage rate of change = (g(b)−g(a))/(b−a)(g(b) - g(a)) / (b - a)\newlineAverage rate of change = (g(−1)−g(−5))/(−1−(−5))(g(-1) - g(-5)) / (-1 - (-5))\newlineAverage rate of change = (21−25)/(−1+5)(21 - 25) / (-1 + 5)\newlineAverage rate of change = (−4)/4(-4) / 4\newlineAverage rate of change = −1-1

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