Average Rate Change C059D3

1. **State the problem:** We need to find the average rate of change of the function $$g(x)=x^3-2x+1$$ over three intervals: from $$-3$$ to $$-2$$, from $$1$$ to $$3$$, and from $$-1$$ to $$1$$. 2. **Formula for average rate of change:** The average rate of change of a function $$g(x)$$ over an interval $$[a,b]$$ is given by: $$\text{Average rate of change} = \frac{g(b)-g(a)}{b-a}$$ 3. **Calculate for each interval:** **Interval 1: from $$-3$$ to $$-2$$** - Calculate $$g(-3) = (-3)^3 - 2(-3) + 1 = -27 + 6 + 1 = -20$$ - Calculate $$g(-2) = (-2)^3 - 2(-2) + 1 = -8 + 4 + 1 = -3$$ - Average rate of change: $$\frac{g(-2)-g(-3)}{-2 - (-3)} = \frac{-3 - (-20)}{-2 + 3} = \frac{-3 + 20}{1} = 17$$ **Interval 2: from $$1$$ to $$3$$** - Calculate $$g(1) = 1^3 - 2(1) + 1 = 1 - 2 + 1 = 0$$ - Calculate $$g(3) = 3^3 - 2(3) + 1 = 27 - 6 + 1 = 22$$ - Average rate of change: $$\frac{g(3)-g(1)}{3 - 1} = \frac{22 - 0}{2} = \frac{22}{2} = 11$$ **Interval 3: from $$-1$$ to $$1$$** - Calculate $$g(-1) = (-1)^3 - 2(-1) + 1 = -1 + 2 + 1 = 2$$ - Calculate $$g(1) = 0$$ (from above) - Average rate of change: $$\frac{g(1)-g(-1)}{1 - (-1)} = \frac{0 - 2}{1 + 1} = \frac{-2}{2} = -1$$ 4. **Final answers:** - Average rate of change from $$-3$$ to $$-2$$ is $$17$$. - Average rate of change from $$1$$ to $$3$$ is $$11$$. - Average rate of change from $$-1$$ to $$1$$ is $$-1$$.