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

Solution Steps

The function given is

\[ f(x) = -2x^2 - 3x \]

We need to evaluate the function at the endpoints \(x = 2\) and \(x = 5\).

Calculating \(f(2)\):

\[ f(2) = -2(2^2) - 3(2) = -2(4) - 6 = -8 - 6 = -14 \]

Calculating \(f(5)\):

\[ f(5) = -2(5^2) - 3(5) = -2(25) - 15 = -50 - 15 = -65 \]

The average rate of change from \(x = 2\) to \(x = 5\) is given by the formula:

\[ \text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \]

Substituting the values we found:

\[ \text{Average Rate of Change} = \frac{-65 - (-14)}{5 - 2} = \frac{-65 + 14}{3} = \frac{-51}{3} = -17 \]

Final Answer