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The task asks for the calculation of the "mittlere Änderungsrate" (average rate of change) of the function \( f(x) = x^2 + 2 \) over the interval \([1; 3]\).

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

\[
\text{Mittlere Änderungsrate} = \frac{f(b) - f(a)}{b - a}
\]

Here, \( a = 1 \) and \( b = 3 \), and the function is \( f(x) = x^2 + 2 \).

1. First, calculate \( f(1) \):
\[
f(1) = 1^2 + 2 = 1 + 2 = 3
\]

2. Then, calculate \( f(3) \):
\[
f(3) = 3^2 + 2 = 9 + 2 = 11
\]

3. Now, calculate the average rate of change:
\[
\text{Mittlere Änderungsrate} = \frac{f(3) - f(1)}{3 - 1} = \frac{11 - 3}{3 - 1} = \frac{8}{2} = 4
\]

The average rate of change is 4.

Would you like further explanation or details on any part of the calculation?

Here are five related questions to expand this topic:
1. How does the average rate of change relate to the derivative of the function?
2. What is the geometric interpretation of the average rate of change on a graph?
3. How would you find the instantaneous rate of change at a specific point?
4. How does the average rate of change differ for a linear function compared to a quadratic one?
5. How does the interval size affect the average rate of change?

**Tip:** The average rate of change over an interval is equivalent to the slope of the secant line connecting the two points on the graph.

Berechne die mittlere Änderungsrate von f(x) = x^2 + 2 im Intervall [1; 3].

Learn how to calculate the average rate of change of the quadratic function f(x) = x^2 + 2 over the interval [1; 3]. This problem covers key algebraic concepts, such as functions, quadratic equations, and the formula for the rate of change. Suitable for students in Grades 9-10.

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