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Given the function g(x)=-x^(2)+4x+10, determine the average rate of change of the function over the interval -1 <= x <= 3. Answer:

Given the function g(x)=−x2+4x+10 g(x)=-x^{2}+4 x+10 , 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+4x+10 g(x)=-x^{2}+4 x+10 , determine the average rate of change of the function over the interval −1≤x≤3 -1 \leq x \leq 3 .\newlineAnswer:
  1. Define function g(x)g(x): We have the function g(x)=−x2+4x+10g(x) = -x^2 + 4x + 10. 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. Find 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):g(−1)=−(−1)2+4(−1)+10=−1−4+10=5.g(-1) = -(-1)^2 + 4(-1) + 10 = -1 - 4 + 10 = 5.
  3. Find 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):g(3)=−(3)2+4(3)+10=−9+12+10=13.g(3) = -(3)^2 + 4(3) + 10 = -9 + 12 + 10 = 13.
  4. Calculate average rate of change: Now we have g(−1)=5g(-1) = 5 and g(3)=13g(3) = 13. We can calculate the average rate of change using the values of g(−1)g(-1) and g(3)g(3) and the endpoints of the interval, −1-1 and 33:\newlineAverage rate of change = (g(3)−g(−1))/(3−(−1))=(13−5)/(3−(−1))=8/4=2(g(3) - g(-1)) / (3 - (-1)) = (13 - 5) / (3 - (-1)) = 8 / 4 = 2.

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