Find the average rate of change of $$ f(x)=-2 x^{3}+4 x^{2} $$ from $$ x=1 $$ to $$ x=3 $$, Simplify your answer as much as possible. UII ,

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UpStudy AI · Accepted Answer

**Step 1: Write down the formula for the average rate of change.**  
The average rate of change of $$ f(x) $$ from $$ x=a $$ to $$ x=b $$ is given by:  
$$
\frac{f(b)-f(a)}{b-a}
$$

**Step 2: Identify the values.**  
Here, $$ a=1 $$ and $$ b=3 $$. The function is  
$$
f(x) = -2x^3 + 4x^2
$$

**Step 3: Compute $$ f(3) $$.**  
$$
f(3) = -2(3)^3 + 4(3)^2 = -2(27) + 4(9) = -54 + 36 = -18
$$

**Step 4: Compute $$ f(1) $$.**  
$$
f(1) = -2(1)^3 + 4(1)^2 = -2(1) + 4(1) = -2 + 4 = 2
$$

**Step 5: Substitute into the formula and simplify.**  
$$
\text{Average Rate of Change} = \frac{f(3)-f(1)}{3-1} = \frac{-18 - 2}{2} = \frac{-20}{2} = -10
$$

**Final Answer:**  
The average rate of change of $$ f(x) $$ from $$ x=1 $$ to $$ x=3 $$ is $$ -10 $$.
The average rate of change is -10.

Find the average rate of change of from to ,
Simplify your answer as much as possible.
UII ,

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Find the average rate of change of from to , Simplify your answer as much as possible. UII ,