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Given the function h(x)=x^(2)-6x+7, determine the average rate of change of the function over the interval 0 <= x <= 4. Answer:

Given the function h(x)=x2−6x+7 h(x)=x^{2}-6 x+7 , determine the average rate of change of the function over the interval 0≤x≤4 0 \leq x \leq 4 .\newlineAnswer:

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Q. Given the function h(x)=x2−6x+7 h(x)=x^{2}-6 x+7 , determine the average rate of change of the function over the interval 0≤x≤4 0 \leq x \leq 4 .\newlineAnswer:
  1. Identify formula and interval: Identify the average rate of change formula and the interval over which it needs to be calculated.\newlineThe average rate of change of a function h(x)h(x) over an interval [a,b][a, b] is given by the formula:\newlineAverage rate of change = h(b)−h(a)b−a\frac{h(b) - h(a)}{b - a}\newlineFor the given function h(x)=x2−6x+7h(x) = x^2 - 6x + 7, we need to find the average rate of change over the interval [0,4][0, 4].
  2. Calculate h(x)h(x) at x=0x=0: Calculate the value of h(x)h(x) at the beginning of the interval, which is x=0x = 0.\newlineSubstitute x=0x = 0 into the function h(x)h(x):\newlineh(0)=(0)2−6(0)+7h(0) = (0)^2 - 6(0) + 7\newlineh(0)=0−0+7h(0) = 0 - 0 + 7\newlineh(0)=7h(0) = 7
  3. Calculate h(x)h(x) at x=4x=4: Calculate the value of h(x)h(x) at the end of the interval, which is x=4x = 4. Substitute x=4x = 4 into the function h(x)h(x): h(4)=(4)2−6(4)+7h(4) = (4)^2 - 6(4) + 7 h(4)=16−24+7h(4) = 16 - 24 + 7 h(4)=−8+7h(4) = -8 + 7 h(4)=−1h(4) = -1
  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 [0,4][0, 4].\newlineAverage rate of change = (h(4)−h(0))/(4−0)(h(4) - h(0)) / (4 - 0)\newlineAverage rate of change = (−1−7)/(4−0)(-1 - 7) / (4 - 0)\newlineAverage rate of change = (−8)/4(-8) / 4\newlineAverage rate of change = −2-2

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