QuestionJune 24, 2025

Find the average rate of change of g(x)=4x^2+(3)/(x^3) on the interval [-2,1] square

Find the average rate of change of g(x)=4x^2+(3)/(x^3) on the interval [-2,1] square
Find the average rate of change of g(x)=4x^2+(3)/(x^3) on the interval [-2,1] square

Solution
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Answer

-\frac{75}{24} Explanation 1. Calculate g(-2) Substitute x = -2 into g(x). g(-2) = 4(-2)^2 + \frac{3}{(-2)^3} = 16 - \frac{3}{-8} = 16 + \frac{3}{8} = \frac{128}{8} + \frac{3}{8} = \frac{131}{8}. 2. Calculate g(1) Substitute x = 1 into g(x). g(1) = 4(1)^2 + \frac{3}{1^3} = 4 + 3 = 7. 3. Apply Average Rate of Change Formula Use the formula **\frac{g(b) - g(a)}{b - a}** where a = -2 and b = 1. \frac{g(1) - g(-2)}{1 - (-2)} = \frac{7 - \frac{131}{8}}{3} = \frac{\frac{56}{8} - \frac{131}{8}}{3} = \frac{-75/8}{3} = -\frac{75}{24}.

Explanation

1. Calculate $g(-2)$<br /> Substitute $x = -2$ into $g(x)$. $g(-2) = 4(-2)^2 + \frac{3}{(-2)^3} = 16 - \frac{3}{-8} = 16 + \frac{3}{8} = \frac{128}{8} + \frac{3}{8} = \frac{131}{8}$.<br />2. Calculate $g(1)$<br /> Substitute $x = 1$ into $g(x)$. $g(1) = 4(1)^2 + \frac{3}{1^3} = 4 + 3 = 7$.<br />3. Apply Average Rate of Change Formula<br /> Use the formula **$\frac{g(b) - g(a)}{b - a}$** where $a = -2$ and $b = 1$. $\frac{g(1) - g(-2)}{1 - (-2)} = \frac{7 - \frac{131}{8}}{3} = \frac{\frac{56}{8} - \frac{131}{8}}{3} = \frac{-75/8}{3} = -\frac{75}{24}$.
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