https://file.aiquickdraw.com/mathbot/file/1728259272142fw50pl4v.jpg

Let's analyze the graph and the questions step by step.

### Graph Interpretation:
The graph of the function \( f(x) \) consists of several line segments, some of which contain open and closed points. The points of interest here include:
- \( f(1) = 1 \)
- \( f(2) = 2 \) (open circle at \( (2, 1) \), indicating \( f(2) \) is not defined)
- \( f(3) = 1 \)
- \( f(4) = -1 \)

### First Question: \( f'(2) \)
To describe \( f'(2) \), the derivative represents the slope of the tangent line to the curve at \( x = 2 \). However, since there is an open point at \( x = 2 \), \( f(2) \) is not defined, and we can't calculate a tangent line slope. Therefore:
- **\( f'(2) \) does not exist**.

### Second Question: \( f'( \frac{271}{100}) \)
The point \( \frac{271}{100} = 2.71 \) is located between \( x = 2 \) and \( x = 3 \), where the function is decreasing linearly. We can calculate the slope of this segment.

The slope between points \( (2, 1) \) and \( (3, 1) \) is:
\[
\text{slope} = \frac{1 - 1}{3 - 2} = 0
\]
So:
- **\( f'( \frac{271}{100}) = 0 \)**.

### Third Question: Derivative Definition Limit
We are given the limit expression for the derivative:
\[
\lim_{h \to 0} \frac{f\left( \frac{271}{100} + h \right) - f\left( \frac{271}{100} \right)}{h}
\]
Since we have already calculated that \( f'\left( \frac{271}{100} \right) = 0 \), this limit evaluates to:
- **\( 0 \)**.

Let me know if you'd like further details or have any questions!

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Here are 5 related questions you can explore:

1. What is the value of \( f'(1) \)?
2. What happens to the derivative at \( x = 3 \)?
3. Can the derivative exist at a point where \( f(x) \) is not continuous?
4. What is the geometric meaning of the derivative on a linear section of a graph?
5. How does the derivative behave at a corner or cusp?

**Tip**: A function can have a discontinuity, but this means its derivative may not exist at that point (like at \( x = 2 \) here).

Learn how to analyze the derivative at specific points, especially where the function includes open points or discontinuities. This problem involves interpreting a piecewise linear graph to determine the existence and value of derivatives, ideal for Grades 11-12.

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