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Given the function g(x)=-x^(2)-6x+14, determine the average rate of change of the function over the interval -5 <= x <= 2. Answer:

Given the function g(x)=−x2−6x+14 g(x)=-x^{2}-6 x+14 , determine the average rate of change of the function over the interval −5≤x≤2 -5 \leq x \leq 2 .\newlineAnswer:

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Q. Given the function g(x)=−x2−6x+14 g(x)=-x^{2}-6 x+14 , determine the average rate of change of the function over the interval −5≤x≤2 -5 \leq x \leq 2 .\newlineAnswer:
  1. Define Function: We have the function g(x)=−x2−6x+14g(x) = -x^2 - 6x + 14. To find the average rate of change over the interval [−5,2][-5, 2], 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(−5)g(-5): First, we need to find the value of g(−5)g(-5). We substitute x=−5x = -5 into the function g(x)g(x).
    g(−5)=−(−5)2−6(−5)+14g(-5) = -(-5)^2 - 6(-5) + 14
    g(−5)=−(25)+30+14g(-5) = -(25) + 30 + 14
    g(−5)=−25+30+14g(-5) = -25 + 30 + 14
    g(−5)=5+14g(-5) = 5 + 14
    g(−5)=19g(-5) = 19
  3. Find g(2)g(2): Next, we need to find the value of g(2)g(2). We substitute x=2x = 2 into the function g(x)g(x).
    g(2)=−(2)2−6(2)+14g(2) = -(2)^2 - 6(2) + 14
    g(2)=−4−12+14g(2) = -4 - 12 + 14
    g(2)=−16+14g(2) = -16 + 14
    g(2)=−2g(2) = -2
  4. Calculate Average Rate of Change: Now we have the values g(−5)=19g(-5) = 19 and g(2)=−2g(2) = -2. 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=−5a = -5 and b=2b = 2.\newlineAverage rate of change = (g(2)−g(−5))/(2−(−5))(g(2) - g(-5)) / (2 - (-5))\newlineAverage rate of change = (−2−19)/(2−(−5))(-2 - 19) / (2 - (-5))\newlineAverage rate of change = (−21)/(7)(-21) / (7)\newlineAverage rate of change = −3-3

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