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Given the function h(x)=x^(2)-6x+1, determine the average rate of change of the function over the interval -1 <= x <= 4. Answer:

Given the function h(x)=x2−6x+1 h(x)=x^{2}-6 x+1 , determine the average rate of change of the function over the interval −1≤x≤4 -1 \leq x \leq 4 .\newlineAnswer:

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Q. Given the function h(x)=x2−6x+1 h(x)=x^{2}-6 x+1 , determine the average rate of change of the function over the interval −1≤x≤4 -1 \leq x \leq 4 .\newlineAnswer:
  1. Given Function and Interval: We are given the function h(x)=x2−6x+1h(x) = x^2 - 6x + 1 and asked to find the average rate of change over the interval [−1,4][-1, 4]. 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=−1a = -1 and b=4b = 4.
  2. Calculate h(a)h(a): First, we need to find the value of h(a)h(a) where a=−1a = -1.
    h(−1)=(−1)2−6(−1)+1h(-1) = (-1)^2 - 6(-1) + 1
    h(−1)=1+6+1h(-1) = 1 + 6 + 1
    h(−1)=8h(-1) = 8
  3. Calculate h(b)h(b): Next, we need to find the value of h(b)h(b) where b=4b = 4.\newlineh(4)=(4)2−6(4)+1h(4) = (4)^2 - 6(4) + 1\newlineh(4)=16−24+1h(4) = 16 - 24 + 1\newlineh(4)=−7h(4) = -7
  4. Calculate Average Rate of Change: Now we have both h(a)h(a) and h(b)h(b), so we can calculate the average rate of change.\newlineAverage rate of change = h(b)−h(a)b−a\frac{h(b) - h(a)}{b - a}\newlineAverage rate of change = (−7)−(8)(4)−(−1)\frac{(-7) - (8)}{(4) - (-1)}\newlineAverage rate of change = −155\frac{-15}{5}\newlineAverage rate of change = −3-3

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