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To solve this problem, we need to calculate the **average rate of change** of the function \( f(x) = x^3 - 2x + 1 \) over the interval from \( x = 1 \) to \( x = 3 \).

The formula for the average rate of change of a function \( f(x) \) over an interval \([a, b]\) is given by:

\[
\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}
\]

In this case:
- \( f(x) = x^3 - 2x + 1 \)
- \( a = 1 \)
- \( b = 3 \)

### Step 1: Calculate \( f(1) \)
\[
f(1) = 1^3 - 2(1) + 1 = 1 - 2 + 1 = 0
\]

### Step 2: Calculate \( f(3) \)
\[
f(3) = 3^3 - 2(3) + 1 = 27 - 6 + 1 = 22
\]

### Step 3: Apply the formula for the average rate of change
\[
\text{Average rate of change} = \frac{f(3) - f(1)}{3 - 1} = \frac{22 - 0}{2} = \frac{22}{2} = 11
\]

Thus, the average rate of change is **11**, which matches the first option.

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Would you like further details or have any other questions?

Here are 5 related questions:
1. What is the significance of the average rate of change in a function?
2. How would the result change if we calculated the instantaneous rate of change at \( x = 2 \)?
3. Can the average rate of change be negative? Under what conditions?
4. How does the average rate of change compare to the slope of a secant line?
5. What is the relationship between the average rate of change and derivatives?

**Tip:** The average rate of change over an interval can be seen as the slope of the secant line connecting two points on the curve of the function.

Sea f una función definida por f(x) = x^3 - 2x + 1. Calcule la razón de cambio promedio de f respecto a la variable x, desde x=1 hasta x=3.

Learn how to calculate the average rate of change of the function f(x) = x^3 - 2x + 1 over the interval from x=1 to x=3. This algebraic problem explores polynomial functions and rate of change concepts. Suitable for students in Grades 9-12.

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