https://file.aiquickdraw.com/mathbot/file/1726711610304cj04hh8v.jpg

use this graph to find the following derivative rules.
a. f'(-2.5)=
b. f'(0)=
c. f'(1)=
d. f'(4)=
e. f'(6)=

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.

Use this graph to find the following derivative rules.
a. f'(-2.5)=
b. f'(0)=
c. f'(1)=
d. f'(4)=
e. f'(6)=

Analyze the derivative rules using a piecewise linear graph. The problem includes calculating slopes and identifying points where the derivative is undefined, involving key calculus concepts suitable for Grades 11-12.

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