Find the average rate of change of $$ g(x)=-5 x^{3}+2 $$ from $$ x=-2 $$ to $$ x=1 $$. Question Help: $$ \square $$ Video Submit Question

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1. The average rate of change of a function $$ g(x) $$ from $$ x = a $$ to $$ x = b $$ is given by
   $$
   \frac{g(b)-g(a)}{b-a}.
   $$

2. Here, $$ g(x) = -5x^3 + 2 $$, $$ a = -2 $$, and $$ b = 1 $$.

3. Compute $$ g(1) $$:
   $$
   g(1) = -5(1)^3 + 2 = -5 + 2 = -3.
   $$

4. Compute $$ g(-2) $$:
   $$
   g(-2) = -5(-2)^3 + 2.
   $$
   Since $$(-2)^3 = -8$$, this becomes
   $$
   g(-2) = -5(-8) + 2 = 40 + 2 = 42.
   $$

5. Substitute these values into the average rate of change formula:
   $$
   \frac{g(1)-g(-2)}{1-(-2)} = \frac{-3 - 42}{1 + 2} = \frac{-45}{3} = -15.
   $$

Thus, the average rate of change of $$ g(x) $$ from $$ x = -2 $$ to $$ x = 1 $$ is $$ -15 $$.
The average rate of change of $$ g(x) $$ from $$ x = -2 $$ to $$ x = 1 $$ is $$ -15 $$.

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