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Given the function f(x)=-x^(2)+3x+6, determine the average rate of change of the function over the interval -1 <= x <= 5. Answer:

Given the function f(x)=−x2+3x+6 f(x)=-x^{2}+3 x+6 , determine the average rate of change of the function over the interval −1≤x≤5 -1 \leq x \leq 5 .\newlineAnswer:

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Full solution

Q. Given the function f(x)=−x2+3x+6 f(x)=-x^{2}+3 x+6 , determine the average rate of change of the function over the interval −1≤x≤5 -1 \leq x \leq 5 .\newlineAnswer:
  1. Identify Rate of Change Formula: Identify the average rate of change formula, which is (f(b)−f(a))/(b−a)(f(b) - f(a)) / (b - a), where aa and bb are the endpoints of the interval.
  2. Calculate f(−1)f(-1): Calculate the value of f(−1)f(-1) by substituting x=−1x = -1 into the function f(x)=−x2+3x+6f(x) = -x^2 + 3x + 6.
    f(−1)=−(−1)2+3(−1)+6f(-1) = -(-1)^2 + 3(-1) + 6
    f(−1)=−1−3+6f(-1) = -1 - 3 + 6
    f(−1)=2f(-1) = 2
  3. Calculate f(5)f(5): Calculate the value of f(5)f(5) by substituting x=5x = 5 into the function f(x)=−x2+3x+6f(x) = -x^2 + 3x + 6.
    f(5)=−(5)2+3(5)+6f(5) = -(5)^2 + 3(5) + 6
    f(5)=−25+15+6f(5) = -25 + 15 + 6
    f(5)=−4f(5) = -4
  4. Calculate Average Rate of Change: Use the values of f(−1)f(-1) and f(5)f(5) to calculate the average rate of change over the interval [−1,5][-1, 5].\newlineAverage rate of change = f(5)−f(−1)5−(−1)\frac{f(5) - f(-1)}{5 - (-1)}\newlineAverage rate of change = (−4−2)5−(−1)\frac{(-4 - 2)}{5 - (-1)}\newlineAverage rate of change = −66\frac{-6}{6}\newlineAverage rate of change = −1-1

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