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Given the function h(x)=x^(2)+7x+5, determine the average rate of change of the function over the interval -6 <= x <= 1. Answer:

Given the function h(x)=x2+7x+5 h(x)=x^{2}+7 x+5 , determine the average rate of change of the function over the interval −6≤x≤1 -6 \leq x \leq 1 .\newlineAnswer:

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Q. Given the function h(x)=x2+7x+5 h(x)=x^{2}+7 x+5 , determine the average rate of change of the function over the interval −6≤x≤1 -6 \leq x \leq 1 .\newlineAnswer:
  1. Calculate Function Value at −6-6: To find the average rate of change of the function h(x)=x2+7x+5h(x) = x^2 + 7x + 5 over the interval from x=−6x = -6 to x=1x = 1, we need to calculate the difference in the function values at these points and divide by the difference in the xx-values.
  2. Calculate Function Value at 11: First, we calculate the function value at x=−6x = -6: h(−6)=(−6)2+7∗(−6)+5h(-6) = (-6)^2 + 7*(-6) + 5.
  3. Find Difference in Function Values: Performing the calculation: h(−6)=36−42+5=−1h(-6) = 36 - 42 + 5 = -1.
  4. Find Difference in X-Values: Next, we calculate the function value at x=1x = 1: h(1)=(1)2+7∗(1)+5h(1) = (1)^2 + 7*(1) + 5.
  5. Calculate Average Rate of Change: Performing the calculation: h(1)=1+7+5=13h(1) = 1 + 7 + 5 = 13.
  6. Calculate Average Rate of Change: Performing the calculation: h(1)=1+7+5=13h(1) = 1 + 7 + 5 = 13.Now, we find the difference in the function values: h(1)−h(−6)=13−(−1)=14h(1) - h(-6) = 13 - (-1) = 14.
  7. Calculate Average Rate of Change: Performing the calculation: h(1)=1+7+5=13h(1) = 1 + 7 + 5 = 13.Now, we find the difference in the function values: h(1)−h(−6)=13−(−1)=14h(1) - h(-6) = 13 - (-1) = 14.We also find the difference in the xx-values: 1−(−6)=1+6=71 - (-6) = 1 + 6 = 7.
  8. Calculate Average Rate of Change: Performing the calculation: h(1)=1+7+5=13h(1) = 1 + 7 + 5 = 13.Now, we find the difference in the function values: h(1)−h(−6)=13−(−1)=14h(1) - h(-6) = 13 - (-1) = 14.We also find the difference in the x-values: 1−(−6)=1+6=71 - (-6) = 1 + 6 = 7.Finally, we calculate the average rate of change by dividing the difference in function values by the difference in x-values: Average rate of change = (h(1)−h(−6))/(1−(−6))(h(1) - h(-6)) / (1 - (-6)).
  9. Calculate Average Rate of Change: Performing the calculation: h(1)=1+7+5=13h(1) = 1 + 7 + 5 = 13.Now, we find the difference in the function values: h(1)−h(−6)=13−(−1)=14h(1) - h(-6) = 13 - (-1) = 14.We also find the difference in the x-values: 1−(−6)=1+6=71 - (-6) = 1 + 6 = 7.Finally, we calculate the average rate of change by dividing the difference in function values by the difference in x-values: Average rate of change = (h(1)−h(−6))/(1−(−6))(h(1) - h(-6)) / (1 - (-6)).Performing the calculation: Average rate of change = 14/7=214 / 7 = 2.

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