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Given the function g(x)=x^(2)-9x+12, determine the average rate of change of the function over the interval 1 <= x <= 7. Answer:

Given the function g(x)=x2−9x+12 g(x)=x^{2}-9 x+12 , determine the average rate of change of the function over the interval 1≤x≤7 1 \leq x \leq 7 .\newlineAnswer:

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Q. Given the function g(x)=x2−9x+12 g(x)=x^{2}-9 x+12 , determine the average rate of change of the function over the interval 1≤x≤7 1 \leq x \leq 7 .\newlineAnswer:
  1. Define function g(x)g(x): We have the function g(x)=x2−9x+12g(x) = x^2 - 9x + 12. We need to find the average rate of change over the interval [1,7][1, 7]. 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=1a = 1 and b=7b = 7.
  2. Calculate g(1)g(1): Calculate the value of g(1)g(1) by substituting x=1x = 1 into the function g(x)g(x).
    g(1)=(1)2−9(1)+12g(1) = (1)^2 - 9(1) + 12
    g(1)=1−9+12g(1) = 1 - 9 + 12
    g(1)=4g(1) = 4
  3. Calculate g(7)g(7): Calculate the value of g(7)g(7) by substituting x=7x = 7 into the function g(x)g(x).
    g(7)=(7)2−9(7)+12g(7) = (7)^2 - 9(7) + 12
    g(7)=49−63+12g(7) = 49 - 63 + 12
    g(7)=−14+12g(7) = -14 + 12
    g(7)=−2g(7) = -2
  4. Calculate average rate of change: Now that we have g(1)g(1) and g(7)g(7), we can calculate the average rate of change using the values from Step 22 and Step 33.\newlineAverage rate of change = (g(7)−g(1))/(7−1)(g(7) - g(1)) / (7 - 1)\newlineAverage rate of change = (−2−4)/(7−1)(-2 - 4) / (7 - 1)\newlineAverage rate of change = (−6)/6(-6) / 6\newlineAverage rate of change = −1-1

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