Find the average rate of change of $$ f(x)=-3 x^{2}-3 x $$ from $$ x=3 $$ to $$ x=5 $$. Simplify. your answer as much as possible.

Question

UpStudy AI · Accepted Answer

Compute the average rate of change using the formula

$$
\frac{f(b)-f(a)}{b-a}.
$$

Here, $$ a=3 $$ and $$ b=5 $$.

1. Evaluate $$ f(3) $$:

$$
f(3) = -3(3)^2 - 3(3) = -3(9) - 9 = -27 - 9 = -36.
$$

2. Evaluate $$ f(5) $$:

$$
f(5) = -3(5)^2 - 3(5) = -3(25) - 15 = -75 - 15 = -90.
$$

3. Compute the average rate of change:

$$
\frac{f(5)-f(3)}{5-3} = \frac{-90 - (-36)}{2} = \frac{-90 + 36}{2} = \frac{-54}{2} = -27.
$$

Thus, the average rate of change is $$-27$$.
The average rate of change is $$-27$$.

Find the average rate of change of \( f(x)=-3 x^{2}-3 x \) from \( x=3 \) to \( x=5 \). Simplify. your answer as much as possible.