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Let's walk through the questions step by step from the image provided:

### (a) Find the average rate of change of \( f \) on the interval \([10, 60]\).
The average rate of change of a function \( f \) on the interval \([a, b]\) is calculated by:
\[
\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}
\]
From the graph:
- \( f(60) \) appears to be 800.
- \( f(10) \) is approximately 500.

Now we calculate:
\[
\frac{f(60) - f(10)}{60 - 10} = \frac{800 - 500}{60 - 10} = \frac{300}{50} = 6
\]
The correct answer is \( 6 \), and this was marked as correct.

### (b) Identify an interval on which the average rate of change of \( f \) is 0.
The average rate of change is 0 if the function does not change over the interval (i.e., it's constant). From the graph, the function is flat between \( x = 10 \) and \( x = 50 \), meaning the slope is 0 in that region.

The correct interval is \([10, 50]\), which was selected and marked correct.

### (c) Compute the following:
\[
\frac{f(40) - f(0)}{40 - 0}
\]
From the graph:
- \( f(40) \) appears to be approximately 500.
- \( f(0) \) is approximately 500.

Thus:
\[
\frac{f(40) - f(0)}{40 - 0} = \frac{500 - 500}{40 - 0} = \frac{0}{40} = 0
\]
This is marked as incorrect in the problem, but the calculation is actually correct. The value is \( 0 \).

#### What does this value represent geometrically?
Since the rate of change is 0, it means that the slope of the line segment from \( (0, f(0)) \) to \( (40, f(40)) \) is 0. The correct interpretation geometrically is that it represents "the slope of the line segment from \( (0, f(0)) \) to \( (40, f(40)) \)." This was marked as correct.

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Do you have any questions or need further details on any of these solutions?

### 5 related questions:
1. How is the average rate of change different from the instantaneous rate of change?
2. What does a zero average rate of change imply about the function over the interval?
3. How can we visually identify where the average rate of change is zero on a graph?
4. What role do the endpoints of an interval play in determining the average rate of change?
5. How would you compute the instantaneous rate of change from a graph?

### Tip:
When calculating the average rate of change, always ensure you have accurately read the values of \( f(a) \) and \( f(b) \) from the graph or function before performing the computation.

The graph of a function f is shown. Find the average rate of change of f on the interval [10, 60]. Identify an interval on which the average rate of change of f is 0. Compute the following: (f(40) - f(0)) / (40 - 0), and interpret its geometric meaning.

Understand how to find the average rate of change of a function over a given interval using the formula (f(b) - f(a)) / (b - a). Learn to identify intervals with a zero average rate of change and interpret geometric meanings of slopes from graphical data. Suitable for students in Grades 9-12.

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