Question
O Graphs and Functions Finding the average rate of change of a function Find the average rate of change of \( f(x)=-2 x^{2}-3 x \) from \( x=2 \) to \( x=5 \). Simplify your answer as much as possible.
Ask by French Parsons. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The average rate of change of the function from \( x = 2 \) to \( x = 5 \) is \( -17 \).
Solution
**Step 1: Calculate \( f(2) \)**
\[
f(2) = -2(2)^2 - 3(2) = -2(4) - 6 = -8 - 6 = -14
\]
**Step 2: Calculate \( f(5) \)**
\[
f(5) = -2(5)^2 - 3(5) = -2(25) - 15 = -50 - 15 = -65
\]
**Step 3: Use the average rate of change formula**
The average rate of change from \( x = 2 \) to \( x = 5 \) is given by
\[
\frac{f(5) - f(2)}{5 - 2}.
\]
Substitute the values:
\[
\frac{-65 - (-14)}{5 - 2} = \frac{-65 + 14}{3} = \frac{-51}{3}.
\]
**Step 4: Simplify the expression**
\[
\frac{-51}{3} = -17.
\]
The average rate of change of the function from \( x = 2 \) to \( x = 5 \) is \( -17 \).
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Simplify this solution Extra Insights
To find the average rate of change of the function \( f(x) = -2x^2 - 3x \) from \( x = 2 \) to \( x = 5 \), we can use the formula: \[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \] where \( a = 2 \) and \( b = 5 \). First, we calculate \( f(2) \): \[ f(2) = -2(2^2) - 3(2) = -2(4) - 6 = -8 - 6 = -14 \] Next, we calculate \( f(5) \): \[ f(5) = -2(5^2) - 3(5) = -2(25) - 15 = -50 - 15 = -65 \] Now, we can find the average rate of change: \[ \text{Average Rate of Change} = \frac{-65 - (-14)}{5 - 2} = \frac{-65 + 14}{3} = \frac{-51}{3} = -17 \] So the average rate of change from \( x = 2 \) to \( x = 5 \) is \(-17\).
