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.