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Given the function f(x)=x^(2)-10 x+18, determine the average rate of change of the function over the interval -1 <= x <= 6. Answer:

Given the function f(x)=x2−10x+18 f(x)=x^{2}-10 x+18 , determine the average rate of change of the function over the interval −1≤x≤6 -1 \leq x \leq 6 .\newlineAnswer:

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Q. Given the function f(x)=x2−10x+18 f(x)=x^{2}-10 x+18 , determine the average rate of change of the function over the interval −1≤x≤6 -1 \leq x \leq 6 .\newlineAnswer:
  1. Given Function and Interval: We are given the function f(x)=x2−10x+18f(x) = x^2 - 10x + 18 and asked to find the average rate of change over the interval [−1,6][-1, 6]. The average rate of change is given by the formula:\newlineAverage rate of change = (f(b)−f(a))/(b−a)(f(b) - f(a)) / (b - a)\newlinewhere aa and bb are the endpoints of the interval. In this case, a=−1a = -1 and b=6b = 6.
  2. Find f(a)f(a): First, we need to find the value of f(a)f(a) where a=−1a = -1.\newlinef(−1)=(−1)2−10(−1)+18f(-1) = (-1)^2 - 10(-1) + 18\newlinef(−1)=1+10+18f(-1) = 1 + 10 + 18\newlinef(−1)=29f(-1) = 29
  3. Find f(b)f(b): Next, we need to find the value of f(b)f(b) where b=6b = 6.\newlinef(6)=(6)2−10(6)+18f(6) = (6)^2 - 10(6) + 18\newlinef(6)=36−60+18f(6) = 36 - 60 + 18\newlinef(6)=−6f(6) = -6
  4. Calculate Average Rate of Change: Now we can calculate the average rate of change using the values of f(a)f(a) and f(b)f(b) we found.\newlineAverage rate of change = (f(b)−f(a))/(b−a)(f(b) - f(a)) / (b - a)\newlineAverage rate of change = (−6−29)/(6−(−1))(-6 - 29) / (6 - (-1))\newlineAverage rate of change = (−35)/(7)(-35) / (7)\newlineAverage rate of change = −5-5

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