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Given the function g(x)=x^(2)+x-5, determine the average rate of change of the function over the interval -1 <= x <= 3. Answer:

Given the function g(x)=x2+x−5 g(x)=x^{2}+x-5 , determine the average rate of change of the function over the interval −1≤x≤3 -1 \leq x \leq 3 .\newlineAnswer:

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Q. Given the function g(x)=x2+x−5 g(x)=x^{2}+x-5 , determine the average rate of change of the function over the interval −1≤x≤3 -1 \leq x \leq 3 .\newlineAnswer:
  1. Substitute xx into g(x)g(x): We have the function g(x)=x2+x−5g(x) = x^2 + x - 5. To find the average rate of change over the interval [−1,3][-1, 3], 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. Calculate 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).\newlineg(−1)=(−1)2+(−1)−5g(-1) = (-1)^2 + (-1) - 5\newlineg(−1)=1−1−5g(-1) = 1 - 1 - 5\newlineg(−1)=−5g(-1) = -5
  3. Calculate g(3)g(3): Next, we need to find the value of g(3)g(3). We substitute x=3x = 3 into the function g(x)g(x).\newlineg(3)=(3)2+(3)−5g(3) = (3)^2 + (3) - 5\newlineg(3)=9+3−5g(3) = 9 + 3 - 5\newlineg(3)=7g(3) = 7
  4. Calculate average rate of change: Now we have the values g(−1)=−5g(-1) = -5 and g(3)=7g(3) = 7. 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=−1a = -1 and b=3b = 3.\newlineAverage rate of change = (g(3)−g(−1))/(3−(−1))(g(3) - g(-1)) / (3 - (-1))\newlineAverage rate of change = (7−(−5))/(3−(−1))(7 - (-5)) / (3 - (-1))\newlineAverage rate of change = (7+5)/(3+1)(7 + 5) / (3 + 1)\newlineAverage rate of change = 12/412 / 4\newlineAverage rate of change = 33

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