Problem

Find the average rate of change of $g(x)=x^{3}-3 x^{2}+2 x$ from $x=-2$ to $x=1$. Simplify your answer as much as possible.

Answer

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Answer

The average rate of change of \(g(x)=x^{3}-3 x^{2}+2 x\) from \(x=-2\) to \(x=1\) is \(\boxed{8}\).

Steps

Step 1 :We are given the function \(g(x)=x^{3}-3 x^{2}+2 x\) and we need to find the average rate of change from \(x=-2\) to \(x=1\).

Step 2 :The formula for the average rate of change of a function \(f(x)\) from \(x=a\) to \(x=b\) is \(\frac{f(b)-f(a)}{b-a}\).

Step 3 :Substitute \(x=-2\) into the function \(g(x)\) to get \(g(-2)\).

Step 4 :Substitute \(x=1\) into the function \(g(x)\) to get \(g(1)\).

Step 5 :Substitute \(g(-2)\) and \(g(1)\) into the formula to get the average rate of change.

Step 6 :Simplify the expression to get the final answer.

Step 7 :The average rate of change of \(g(x)=x^{3}-3 x^{2}+2 x\) from \(x=-2\) to \(x=1\) is \(\boxed{8}\).

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