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Given the function h(x)=x^(2)-7x+8, determine the average rate of change of the function over the interval 1 <= x <= 8. Answer:

Given the function h(x)=x2−7x+8 h(x)=x^{2}-7 x+8 , determine the average rate of change of the function over the interval 1≤x≤8 1 \leq x \leq 8 .\newlineAnswer:

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Full solution

Q. Given the function h(x)=x2−7x+8 h(x)=x^{2}-7 x+8 , determine the average rate of change of the function over the interval 1≤x≤8 1 \leq x \leq 8 .\newlineAnswer:
  1. Use Average Rate of Change Formula: To find the average rate of change of the function h(x)=x2−7x+8h(x) = x^2 - 7x + 8 over the interval [1,8][1, 8], 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. Calculate h(1)h(1): First, we need to calculate the value of h(x)h(x) at the lower endpoint of the interval, which is x=1x = 1.\newlineh(1)=(1)2−7(1)+8=1−7+8=2h(1) = (1)^2 - 7(1) + 8 = 1 - 7 + 8 = 2.
  3. Calculate h(8)h(8): Next, we need to calculate the value of h(x)h(x) at the upper endpoint of the interval, which is x=8x = 8.\newlineh(8)=(8)2−7(8)+8=64−56+8=16h(8) = (8)^2 - 7(8) + 8 = 64 - 56 + 8 = 16.
  4. Calculate Average Rate of Change: Now we have the values of h(x)h(x) at both endpoints of the interval. We can calculate the average rate of change using the formula:\newlineAverage rate of change = h(8)−h(1)8−1\frac{h(8) - h(1)}{8 - 1}.
  5. Substitute Values and Calculate: Substitute the values we found into the formula:\newlineAverage rate of change = (16−2)/(8−1)=14/7=2(16 - 2) / (8 - 1) = 14 / 7 = 2.

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