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Given the function g(x)=x^(2)-8x+14, determine the average rate of change of the function over the interval -1 <= x <= 10. Answer:

Given the function g(x)=x2−8x+14 g(x)=x^{2}-8 x+14 , determine the average rate of change of the function over the interval −1≤x≤10 -1 \leq x \leq 10 .\newlineAnswer:

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Q. Given the function g(x)=x2−8x+14 g(x)=x^{2}-8 x+14 , determine the average rate of change of the function over the interval −1≤x≤10 -1 \leq x \leq 10 .\newlineAnswer:
  1. Identify formula and interval: Identify the average rate of change formula and the interval for the function g(x)g(x). The average rate of change of a function over an interval [a,b][a, b] is given by the formula: Average rate of change =g(b)−g(a)b−a= \frac{g(b) - g(a)}{b - a} For the function g(x)=x2−8x+14g(x) = x^2 - 8x + 14, we need to find the average rate of change over the interval [−1,10][-1, 10].
  2. Calculate value at −1-1: Calculate the value of g(x)g(x) at the beginning of the interval, which is x=−1x = -1.\newlineSubstitute x=−1x = -1 into the function g(x)g(x):\newlineg(−1)=(−1)2−8(−1)+14g(-1) = (-1)^2 - 8(-1) + 14\newlineg(−1)=1+8+14g(-1) = 1 + 8 + 14\newlineg(−1)=23g(-1) = 23
  3. Calculate value at 1010: Calculate the value of g(x)g(x) at the end of the interval, which is x=10x = 10.\newlineSubstitute x=10x = 10 into the function g(x)g(x):\newlineg(10)=(10)2−8(10)+14g(10) = (10)^2 - 8(10) + 14\newlineg(10)=100−80+14g(10) = 100 - 80 + 14\newlineg(10)=34g(10) = 34
  4. Calculate average rate of change: Use the values from Step 22 and Step 33 to calculate the average rate of change over the interval [−1,10][-1, 10].\newlineAverage rate of change = (g(10)−g(−1))/(10−(−1))(g(10) - g(-1)) / (10 - (-1))\newlineAverage rate of change = (34−23)/(10+1)(34 - 23) / (10 + 1)\newlineAverage rate of change = 11/1111 / 11\newlineAverage rate of change = 11

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