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To estimate the values of the derivatives based on the graph, we need to examine the slope of the function \( f(x) \) at various points. The derivative at a point gives us the slope of the tangent line to the graph at that point. Let me walk through each point and explain the slope visually.

### Estimation Process:
1. **\( f'(0) \)**: Look at \( x = 0 \). The graph appears to have a positive slope, but not too steep. The slope seems to be around **1**.
   
2. **\( f'(1) \)**: At \( x = 1 \), the graph reaches a local maximum. The tangent at a maximum is horizontal, meaning the slope is **0**.

3. **\( f'(1.5) \)**: This is a point of steep decline, indicating a negative slope. It appears quite steep, likely around **-2**.

4. **\( f'(2) \)**: At \( x = 2 \), the graph appears to have a minimum. Similar to a maximum, the slope here should be horizontal, so the derivative is **0**.

5. **\( f'(3) \)**: This point is on a gentle upward slope. The slope is positive but not very steep, so an estimate might be around **0.5**.

6. **\( f'(4) \)**: The slope at this point is zero again because it looks like a peak (local maximum).

7. **\( f'(5) \)**: The graph is declining steeply here, suggesting a negative slope. It seems to be around **-1.5**.

### Estimated Derivatives:
- \( f'(0) \approx 1 \)
- \( f'(1) = 0 \)
- \( f'(1.5) \approx -2 \)
- \( f'(2) = 0 \)
- \( f'(3) \approx 0.5 \)
- \( f'(4) = 0 \)
- \( f'(5) \approx -1.5 \)

Would you like me to further clarify any specific part of this process?

Here are 5 related questions to expand on this topic:
1. What is the significance of a zero derivative at a point on the graph?
2. How would you estimate the derivative at a point that is not clearly a maximum or minimum?
3. How does the steepness of the graph relate to the magnitude of the derivative?
4. What does it mean when the derivative is negative at a point?
5. How can we use the second derivative to further understand the behavior of the graph?

**Tip**: The derivative of a function at a maximum or minimum is always zero, as the tangent line is horizontal at these points.

The graph of y = f(x) is given. Use the graph to estimate the values of the following derivatives: f'(0), f'(1), f'(1.5), f'(2), f'(3), f'(4), f'(5).

Learn how to estimate derivatives from a graph of f(x) at various points, including f'(0), f'(1), and f'(2). This problem covers key calculus concepts such as derivatives, slopes, and critical points. Suitable for high school calculus students.

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