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Given the function h(x)=-x^(2)+7x+20, determine the average rate of change of the function over the interval -2 <= x <= 9. Answer:

Given the function h(x)=−x2+7x+20 h(x)=-x^{2}+7 x+20 , determine the average rate of change of the function over the interval −2≤x≤9 -2 \leq x \leq 9 .\newlineAnswer:

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Q. Given the function h(x)=−x2+7x+20 h(x)=-x^{2}+7 x+20 , determine the average rate of change of the function over the interval −2≤x≤9 -2 \leq x \leq 9 .\newlineAnswer:
  1. Given Function: We have the function h(x)=−x2+7x+20h(x) = -x^2 + 7x + 20. We need to find the average rate of change over the interval [−2,9][-2, 9].\newlineThe average rate of change formula is (h(b)−h(a))/(b−a)(h(b) - h(a)) / (b - a), where aa and bb are the endpoints of the interval.\newlineIn this case, a=−2a = -2 and b=9b = 9.
  2. Calculate h(−2)h(-2): First, we find the value of h(−2)h(-2).\newlineSubstitute −2-2 for xx in the function h(x)h(x).\newlineh(−2)=−(−2)2+7(−2)+20h(-2) = -(-2)^2 + 7(-2) + 20\newlineh(−2)=−4−14+20h(-2) = -4 - 14 + 20\newlineh(−2)=2h(-2) = 2
  3. Calculate h(9)h(9): Next, we find the value of h(9)h(9). Substitute 99 for xx in the function h(x)h(x). h(9)=−(9)2+7(9)+20h(9) = -(9)^2 + 7(9) + 20 h(9)=−81+63+20h(9) = -81 + 63 + 20 h(9)=−81+83h(9) = -81 + 83 h(9)=2h(9) = 2
  4. Calculate Average Rate of Change: Now we have both h(−2)h(-2) and h(9)h(9). We can calculate the average rate of change.\newlineAverage rate of change = h(9)−h(−2)9−(−2)\frac{h(9) - h(-2)}{9 - (-2)}\newlineAverage rate of change = 2−29+2\frac{2 - 2}{9 + 2}\newlineAverage rate of change = 011\frac{0}{11}\newlineAverage rate of change = 00

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