AP Calculus AB 2.1 The Average Rate of Change of a Polynomial

To find the average rate of change of a polynomial function between two given \( x \)-values, follow these steps:

### Step 1: Understand the Formula for Average Rate of Change

The average rate of change of a function \( f(x) \) over an interval \([x_1, x_2]\) is given by:
\[
\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]
This formula is essentially the slope of the secant line that passes through the points \( (x_1, f(x_1)) \) and \( (x_2, f(x_2)) \).

### Step 2: Evaluate the Function at the Given \( x \)-Values

1. **Identify the two \( x \)-values**, say \( x_1 \) and \( x_2 \).
2. **Substitute** each \( x \)-value into the polynomial to find the corresponding function values \( f(x_1) \) and \( f(x_2) \).

### Step 3: Apply the Formula

1. **Find the difference in function values**:
\[
f(x_2) - f(x_1)
\]
2. **Find the difference in \( x \)-values**:
\[
x_2 - x_1
\]
3. **Divide** the difference in function values by the difference in \( x \)-values:
\[
\frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

This result represents the average rate of change of the polynomial between the two points \( x_1 \) and \( x_2 \).

### Example

Consider the polynomial \( f(x) = x^2 + 3x - 4 \), and let the two \( x \)-values be \( x_1 = 1 \) and \( x_2 = 3 \).

1. **Evaluate the function** at the given \( x \)-values:
\[
f(1) = (1)^2 + 3(1) - 4 = 1 + 3 - 4 = 0
\]
\[
f(3) = (3)^2 + 3(3) - 4 = 9 + 9 - 4 = 14
\]

2. **Apply the average rate of change formula**:
\[
\text{Average Rate of Change} = \frac{f(3) - f(1)}{3 - 1} = \frac{14 - 0}{2} = \frac{14}{2} = 7
\]

Thus, the average rate of change of the function \( f(x) = x^2 + 3x - 4 \) between \( x_1 = 1 \) and \( x_2 = 3 \) is 7. This means that, on average, the function increases by 7 units for each unit increase in \( x \) over this interval.

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Nick Perich
Norristown Area High School
Norristown Area School District
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