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Given the function h(x)=x^(2)+6x+3, determine the average rate of change of the function over the interval -8 <= x <= -2. Answer:

Given the function h(x)=x2+6x+3 h(x)=x^{2}+6 x+3 , determine the average rate of change of the function over the interval −8≤x≤−2 -8 \leq x \leq-2 .\newlineAnswer:

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Q. Given the function h(x)=x2+6x+3 h(x)=x^{2}+6 x+3 , determine the average rate of change of the function over the interval −8≤x≤−2 -8 \leq x \leq-2 .\newlineAnswer:
  1. Define Function: We have the function h(x)=x2+6x+3h(x) = x^2 + 6x + 3. To find the average rate of change over the interval [−8,−2][-8, -2], we will use the formula for the average rate of change, which is (h(b)−h(a))/(b−a)(h(b) - h(a)) / (b - a), where aa and bb are the endpoints of the interval.
  2. Find h(−8)h(-8): First, we need to find the value of h(−8)h(-8). We substitute xx with −8-8 in the function h(x)h(x).\newlineh(−8)=(−8)2+6∗(−8)+3h(-8) = (-8)^2 + 6*(-8) + 3\newlineh(−8)=64−48+3h(-8) = 64 - 48 + 3\newlineh(−8)=19h(-8) = 19
  3. Find h(−2)h(-2): Next, we need to find the value of h(−2)h(-2). We substitute xx with −2-2 in the function h(x)h(x).
    h(−2)=(−2)2+6∗(−2)+3h(-2) = (-2)^2 + 6*(-2) + 3
    h(−2)=4−12+3h(-2) = 4 - 12 + 3
    h(−2)=−5h(-2) = -5
  4. Calculate Average Rate of Change: Now we have both h(−8)h(-8) and h(−2)h(-2). We can calculate the average rate of change using the values we found.\newlineAverage rate of change = h(−2)−h(−8)−2−(−8)\frac{h(-2) - h(-8)}{-2 - (-8)}\newlineAverage rate of change = −5−19−2−(−8)\frac{-5 - 19}{-2 - (-8)}\newlineAverage rate of change = −5−196\frac{-5 - 19}{6}\newlineAverage rate of change = −246\frac{-24}{6}\newlineAverage rate of change = −4-4

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