https://file.aiquickdraw.com/mathbot/file/1727405327865vq4yzetz.jpg

To analyze the problem, let’s break down the given information and steps:

1. **Given Graph**: We have a graph of \( f(x) \) with three key points: \( A = (a, f(a)) \), \( B = (b, f(b)) \), and \( C = (c, f(c)) \).

2. **Values to Order**: We need to arrange the following items in ascending order:
   - \( f'(a) \): The slope of \( f(x) \) at point \( A \).
   - \( f'(b) \): The slope of \( f(x) \) at point \( B \).
   - \( f'(c) \): The slope of \( f(x) \) at point \( C \).
   - Slope of \( AB \): The average rate of change between points \( A \) and \( B \).
   - The number \( 0 \).
   - The number \( 1 \).

3. **Estimating the Slopes**:
   - \( f'(a) \): The tangent line at \( A \) is steeper and negative, so \( f'(a) < 0 \).
   - \( f'(b) \): The tangent line at \( B \) is positive but less than 1.
   - \( f'(c) \): The tangent line at \( C \) appears to be larger than 1.
   - Slope of \( AB \): The slope between \( A \) and \( B \) is positive and less than 1.

4. **Arranging in Order**:
   Based on the visual estimations:
   - \( f'(a) \) is the smallest since it's negative.
   - The number \( 0 \).
   - Slope of \( AB \).
   - \( f'(b) \).
   - The number \( 1 \).
   - \( f'(c) \) is the largest.

### Final Answer
The values in ascending order are:

1. **Smallest**: \( f'(a) \)
2. \( 0 \)
3. Slope of \( AB \)
4. \( f'(b) \)
5. \( 1 \)
6. **Largest**: \( f'(c) \)

Would you like me to go into further details about how these slopes were determined?

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**Relative Questions**:
1. How do you find the slope of a secant line between two points?
2. What is the geometric interpretation of the derivative at a point?
3. How can you visually estimate whether a derivative is greater than or less than 1?
4. What are other ways to compare the derivatives visually without exact values?
5. How would the ordering change if \( f(x) \) was decreasing throughout?

**Tip**: When interpreting graphs, pay close attention to how the slope changes at key points. A steeper slope corresponds to a higher value of the derivative, while a flatter slope means a lower derivative value.

Using the graph of f(x), arrange the values f'(a), f'(b), f'(c), slope AB, the number 0, and the number 1 in ascending order.

Learn how to arrange derivatives and slopes from a graph of f(x) in ascending order. This problem covers key calculus concepts, including the interpretation of derivatives and slope analysis. Suitable for students in Grades 11-12.

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