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Given the function f(x)=-x^(2)-7x+20, determine the average rate of change of the function over the interval -8 <= x <= 0. Answer:

Given the function f(x)=−x2−7x+20 f(x)=-x^{2}-7 x+20 , determine the average rate of change of the function over the interval −8≤x≤0 -8 \leq x \leq 0 .\newlineAnswer:

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Q. Given the function f(x)=−x2−7x+20 f(x)=-x^{2}-7 x+20 , determine the average rate of change of the function over the interval −8≤x≤0 -8 \leq x \leq 0 .\newlineAnswer:
  1. Given function: We have the function f(x)=−x2−7x+20f(x) = -x^2 - 7x + 20. We need to find the average rate of change over the interval [−8,0][-8, 0].\newlineTo find the average rate of change, we use the formula:\newlineAverage rate of change = (f(b)−f(a))/(b−a)(f(b) - f(a)) / (b - a)\newlinewhere aa and bb are the endpoints of the interval. In this case, a=−8a = -8 and b=0b = 0.
  2. Find f(a)f(a): First, we need to find the value of f(a)f(a) where a=−8a = -8. Substitute −8-8 into the function f(x)f(x): f(−8)=−(−8)2−7(−8)+20f(-8) = -(-8)^2 - 7(-8) + 20 f(−8)=−(64)+56+20f(-8) = -(64) + 56 + 20 f(−8)=−64+56+20f(-8) = -64 + 56 + 20 f(−8)=−8+20f(-8) = -8 + 20 f(−8)=12f(-8) = 12
  3. Find f(b)f(b): Next, we need to find the value of f(b)f(b) where b=0b = 0. Substitute 00 into the function f(x)f(x): f(0)=−(0)2−7(0)+20f(0) = -(0)^2 - 7(0) + 20 f(0)=0−0+20f(0) = 0 - 0 + 20 f(0)=20f(0) = 20
  4. Calculate average rate of change: Now that we have f(a)f(a) and f(b)f(b), we can calculate the average rate of change.\newlineAverage rate of change = (f(b)−f(a))/(b−a)(f(b) - f(a)) / (b - a)\newlineAverage rate of change = (f(0)−f(−8))/(0−(−8))(f(0) - f(-8)) / (0 - (-8))\newlineAverage rate of change = (20−12)/(0+8)(20 - 12) / (0 + 8)\newlineAverage rate of change = 8/88 / 8\newlineAverage rate of change = 11

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