Let's analyze the piecewise linear graph you provided to determine the derivative values at the specific points.

General Approach for Derivatives of Linear Segments:

• Since the graph consists of straight line segments, the derivative (slope) at each point where the function is smooth is constant.
• At points where the graph has a sharp turn (like at $x = 0$ and $x = 1$), the derivative is undefined, as the graph is not smooth at those points.

Let's answer the derivative rules based on the graph:

a. $f'(-2.5)$:

• From $x = -8$ to $x = 0$, the graph is a straight line increasing. We can calculate the slope between two points. The slope of this line is calculated by $\Delta y / \Delta x$. For example, from $(-8, -1)$ to $(0, 1)$, the slope is: $\text{slope} = \frac{1 - (-1)}{0 - (-8)} = \frac{2}{8} = 0.25$ Therefore, $f'(-2.5) = 0.25$.

b. $f'(0)$:

• At $x = 0$, there is a sharp corner (cusp), indicating a non-differentiable point. Therefore, $f'(0)$ is undefined.

c. $f'(1)$:

• At $x = 1$, there is another sharp turn in the graph, indicating non-differentiability. Therefore, $f'(1)$ is undefined.

d. $f'(4)$:

• From $x = 1$ to $x = 4$, the graph is decreasing. The slope can be calculated between $(1, 0)$ and $(4, -2)$: $\text{slope} = \frac{-2 - 0}{4 - 1} = \frac{-2}{3} = -\frac{2}{3}$ Therefore, $f'(4) = -\frac{2}{3}$.

e. $f'(6)$:

• From $x = 4$ to $x = 8$, the graph is sharply increasing. The slope can be calculated between $(4, -2)$ and $(8, 8)$: $\text{slope} = \frac{8 - (-2)}{8 - 4} = \frac{10}{4} = 2.5$ Therefore, $f'(6) = 2.5$.

Summary of Derivatives:

• $f'(-2.5) = 0.25$
• $f'(0)$ is undefined
• $f'(1)$ is undefined
• $f'(4) = -\frac{2}{3}$
• $f'(6) = 2.5$

Would you like further explanation on any of these calculations or details on slopes?

Related Questions:

1. What is the derivative at $x = -5$?
2. How does differentiability change at points where the graph has sharp turns?
3. What is the physical meaning of a positive versus negative derivative?
4. How would you interpret a zero derivative from a graph like this?
5. Can you describe the general shape of the graph based on its slopes?

Tip: The derivative at a point provides information about the slope or rate of change at that point. If the graph has a sharp corner, the derivative is not defined there.