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?

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.