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g(x)=-(x^(2))/(4)+7 What is the average rate of change of g over the interval [-2,4] ?

g(x)=−x24+7g(x)=-\frac{x^{2}}{4}+7\newlineWhat is the average rate of change of g g over the interval [−2,4] [-2,4] ?

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Q. g(x)=−x24+7g(x)=-\frac{x^{2}}{4}+7\newlineWhat is the average rate of change of g g over the interval [−2,4] [-2,4] ?
  1. Calculate Function Values: To find the average rate of change of the function g(x)g(x) over the interval [−2,4][-2,4], we need to calculate the difference in the function values at the endpoints of the interval and divide by the length of the interval.
  2. Find Difference: First, we calculate the function value at the left endpoint of the interval, which is x=−2x = -2.g(−2)=−((−2)2)/4+7=−4/4+7=−1+7=6g(-2) = -((-2)^2)/4 + 7 = -4/4 + 7 = -1 + 7 = 6.
  3. Calculate Length: Next, we calculate the function value at the right endpoint of the interval, which is x=4x = 4.g(4)=−((4)24)+7=−164+7=−4+7=3g(4) = -\left(\frac{(4)^2}{4}\right) + 7 = -\frac{16}{4} + 7 = -4 + 7 = 3.
  4. Calculate Average Rate: Now, we find the difference in the function values at x=4x = 4 and x=−2x = -2.Δg=g(4)−g(−2)=3−6=−3\Delta g = g(4) - g(-2) = 3 - 6 = -3.
  5. Calculate Average Rate: Now, we find the difference in the function values at x=4x = 4 and x=−2x = -2.Δg=g(4)−g(−2)=3−6=−3\Delta g = g(4) - g(-2) = 3 - 6 = -3.We also need to find the length of the interval from x=−2x = -2 to x=4x = 4.Δx=4−(−2)=4+2=6\Delta x = 4 - (-2) = 4 + 2 = 6.
  6. Calculate Average Rate: Now, we find the difference in the function values at x=4x = 4 and x=−2x = -2.Δg=g(4)−g(−2)=3−6=−3\Delta g = g(4) - g(-2) = 3 - 6 = -3.We also need to find the length of the interval from x=−2x = -2 to x=4x = 4.Δx=4−(−2)=4+2=6\Delta x = 4 - (-2) = 4 + 2 = 6.Finally, we calculate the average rate of change of g(x)g(x) over the interval [−2,4][-2,4] by dividing the difference in function values by the length of the interval.Average rate of change=ΔgΔx=−36=−12\text{Average rate of change} = \frac{\Delta g}{\Delta x} = \frac{-3}{6} = -\frac{1}{2}.

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