What is the average rate of change of a constant function f on an interval [a, b], a 6= b? (i) 1 (ii) it depends on the interval [a, b] (iii) it depends on the function f (iv) 0

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

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

For a constant function \( f(x) = c \) (where \( c \) is a constant), the values of \( f(a) \) and \( f(b) \) are the same, i.e., \( f(a) = f(b) = c \).

Substituting into the formula:

\[
\frac{f(b) - f(a)}{b - a} = \frac{c - c}{b - a} = \frac{0}{b - a} = 0
\]

Therefore, the correct answer is:

**(iv) 0**

Would you like any further details or have any questions?

Here are 5 related questions:

1. How do you calculate the average rate of change for a linear function?
2. What does the average rate of change represent geometrically?
3. Can the average rate of change be negative? If so, under what conditions?
4. How does the average rate of change differ for non-linear functions?
5. What is the significance of the interval \([a, b]\) when calculating the average rate of change?

**Tip:** The average rate of change is essentially the slope of the secant line connecting two points on the function's graph over the interval \([a, b]\).

Explore how to calculate the average rate of change for a constant function on an interval [a, b]. Understand the mathematical formula (f(b) - f(a)) / (b - a) and its application. This explanation covers key calculus concepts and is suitable for high school students.

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