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Given the function f(x)=-x^(2)-x+7, determine the average rate of change of the function over the interval -4 <= x <= 2. Answer:

Given the function f(x)=−x2−x+7 f(x)=-x^{2}-x+7 , determine the average rate of change of the function over the interval −4≤x≤2 -4 \leq x \leq 2 .\newlineAnswer:

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Q. Given the function f(x)=−x2−x+7 f(x)=-x^{2}-x+7 , determine the average rate of change of the function over the interval −4≤x≤2 -4 \leq x \leq 2 .\newlineAnswer:
  1. Identify formula and interval: Identify the average rate of change formula and the given interval.\newlineThe average rate of change of a function f(x)f(x) over an interval [a,b][a, b] is given by the formula:\newlineAverage rate of change = f(b)−f(a)b−a\frac{f(b) - f(a)}{b - a}\newlineFor the given function f(x)=−x2−x+7f(x) = -x^2 - x + 7, we need to find the average rate of change over the interval [−4,2][-4, 2].
  2. Calculate f(−4)f(-4): Calculate the value of f(−4)f(-4).\newlineSubstitute x=−4x = -4 into the function f(x)f(x) to find f(−4)f(-4).\newlinef(−4)=−(−4)2−(−4)+7f(-4) = -(-4)^2 - (-4) + 7\newlinef(−4)=−(16)+4+7f(-4) = -(16) + 4 + 7\newlinef(−4)=−16+4+7f(-4) = -16 + 4 + 7\newlinef(−4)=−12+7f(-4) = -12 + 7\newlinef(−4)=−5f(-4) = -5
  3. Calculate f(2)f(2): Calculate the value of f(2)f(2).\newlineSubstitute x=2x = 2 into the function f(x)f(x) to find f(2)f(2).\newlinef(2)=−(2)2−(2)+7f(2) = -(2)^2 - (2) + 7\newlinef(2)=−4−2+7f(2) = -4 - 2 + 7\newlinef(2)=−6+7f(2) = -6 + 7\newlinef(2)=1f(2) = 1
  4. Use average rate of change formula: Use the average rate of change formula with the values found for f(−4)f(-4) and f(2)f(2).\newlineAverage rate of change = f(2)−f(−4)2−(−4)\frac{f(2) - f(-4)}{2 - (-4)}\newlineAverage rate of change = 1−(−5)2+4\frac{1 - (-5)}{2 + 4}\newlineAverage rate of change = 1+56\frac{1 + 5}{6}\newlineAverage rate of change = 66\frac{6}{6}\newlineAverage rate of change = 11

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