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Given the function g(x)=-x^(2)+10 x+28, determine the average rate of change of the function over the interval 4 <= x <= 10. Answer:

Given the function g(x)=−x2+10x+28 g(x)=-x^{2}+10 x+28 , determine the average rate of change of the function over the interval 4≤x≤10 4 \leq x \leq 10 .\newlineAnswer:

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Q. Given the function g(x)=−x2+10x+28 g(x)=-x^{2}+10 x+28 , determine the average rate of change of the function over the interval 4≤x≤10 4 \leq x \leq 10 .\newlineAnswer:
  1. Given function and interval: We are given the function g(x)=−x2+10x+28g(x) = -x^2 + 10x + 28. We need to find the average rate of change over the interval [4,10][4, 10]. 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, and g(x)g(x) is the function. In this case, a=4a = 4 and b=10b = 10.
  2. Calculate g(a)g(a): First, we need to find the value of g(a)g(a) where a=4a = 4.\newlineg(4)=−42+10(4)+28g(4) = -4^2 + 10(4) + 28\newlineg(4)=−16+40+28g(4) = -16 + 40 + 28\newlineg(4)=52g(4) = 52
  3. Calculate g(b)g(b): Next, we need to find the value of g(b)g(b) where b=10b = 10.
    g(10)=−102+10(10)+28g(10) = -10^2 + 10(10) + 28
    g(10)=−100+100+28g(10) = -100 + 100 + 28
    g(10)=28g(10) = 28
  4. Calculate average rate of change: Now we have both g(a)g(a) and g(b)g(b), we can calculate the average rate of change.\newlineAverage rate of change = g(10)−g(4)10−4\frac{g(10) - g(4)}{10 - 4}\newlineAverage rate of change = 28−5210−4\frac{28 - 52}{10 - 4}\newlineAverage rate of change = −246\frac{-24}{6}\newlineAverage rate of change = −4-4

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