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Given the function f(x)=x^(2)+9x+12, determine the average rate of change of the function over the interval -5 <= x <= -1. Answer:

Given the function f(x)=x2+9x+12 f(x)=x^{2}+9 x+12 , determine the average rate of change of the function over the interval −5≤x≤−1 -5 \leq x \leq-1 .\newlineAnswer:

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Full solution

Q. Given the function f(x)=x2+9x+12 f(x)=x^{2}+9 x+12 , determine the average rate of change of the function over the interval −5≤x≤−1 -5 \leq x \leq-1 .\newlineAnswer:
  1. Identify average 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(−5)f(-5): Calculate the value of f(−5)f(-5) by substituting x=−5x = -5 into the function f(x)=x2+9x+12f(x) = x^2 + 9x + 12.\newlinef(−5)=(−5)2+9∗(−5)+12f(-5) = (-5)^2 + 9*(-5) + 12\newlinef(−5)=25−45+12f(-5) = 25 - 45 + 12\newlinef(−5)=−8f(-5) = -8
  3. Calculate f(−1)f(-1): Calculate the value of f(−1)f(-1) by substituting x=−1x = -1 into the function f(x)=x2+9x+12f(x) = x^2 + 9x + 12.
    f(−1)=(−1)2+9∗(−1)+12f(-1) = (-1)^2 + 9*(-1) + 12
    f(−1)=1−9+12f(-1) = 1 - 9 + 12
    f(−1)=4f(-1) = 4
  4. Use average rate of change formula: Use the average rate of change formula with f(−5)=−8f(-5) = -8 and f(−1)=4f(-1) = 4.\newlineAverage rate of change = (f(−1)−f(−5))/(−1−(−5))(f(-1) - f(-5)) / (-1 - (-5))\newlineAverage rate of change = (4−(−8))/(−1+5)(4 - (-8)) / (-1 + 5)\newlineAverage rate of change = (4+8)/4(4 + 8) / 4\newlineAverage rate of change = 12/412 / 4\newlineAverage rate of change = 33

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