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Given the function h(x)=x^(2)+8x+10, determine the average rate of change of the function over the interval -9 <= x <= -1. Answer:

Given the function h(x)=x2+8x+10 h(x)=x^{2}+8 x+10 , determine the average rate of change of the function over the interval −9≤x≤−1 -9 \leq x \leq-1 .\newlineAnswer:

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Q. Given the function h(x)=x2+8x+10 h(x)=x^{2}+8 x+10 , determine the average rate of change of the function over the interval −9≤x≤−1 -9 \leq x \leq-1 .\newlineAnswer:
  1. Given Function: We are given the function h(x)=x2+8x+10h(x) = x^2 + 8x + 10. We need to find the average rate of change over the interval [−9,−1][-9, -1]. The average rate of change is calculated using the formula:\newlineAverage rate of change = (h(b)−h(a))/(b−a)(h(b) - h(a)) / (b - a)\newlinewhere aa and bb are the endpoints of the interval. In this case, a=−9a = -9 and b=−1b = -1.
  2. Calculate h(a)h(a): First, we need to find the value of h(a)h(a) where a=−9a = -9.
    h(−9)=(−9)2+8∗(−9)+10h(-9) = (-9)^2 + 8*(-9) + 10
    h(−9)=81−72+10h(-9) = 81 - 72 + 10
    h(−9)=19h(-9) = 19
  3. Calculate h(b)h(b): Next, we need to find the value of h(b)h(b) where b=−1b = -1.\newlineh(−1)=(−1)2+8∗(−1)+10h(-1) = (-1)^2 + 8*(-1) + 10\newlineh(−1)=1−8+10h(-1) = 1 - 8 + 10\newlineh(−1)=3h(-1) = 3
  4. Calculate Average Rate of Change: Now we have both h(a)h(a) and h(b)h(b). We can calculate the average rate of change.\newlineAverage rate of change = (h(−1)−h(−9))/(−1−(−9))(h(-1) - h(-9)) / (-1 - (-9))\newlineAverage rate of change = (3−19)/(−1+9)(3 - 19) / (-1 + 9)\newlineAverage rate of change = (−16)/8(-16) / 8\newlineAverage rate of change = −2-2

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