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!

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).