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Let's work through the given graph to estimate the derivatives of \( f(x) \) at the specified points. Derivatives represent the slope of the tangent line to the curve at a given point.

### (a) \( f'(-3) \)
At \( x = -3 \), the slope of the tangent line is negative and appears to be steep. It looks close to \( -1 \), which is given correctly.

### (b) \( f'(-2) \)
At \( x = -2 \), the curve is flat, indicating that the tangent line is horizontal. Therefore, \( f'(-2) = 0 \), which is correct.

### (c) \( f'(-1) \)
At \( x = -1 \), the slope of the tangent line is positive but not as steep as 2. It looks closer to \( 1 \) instead. So, \( f'(-1) \approx 1 \).

### (d) \( f'(0) \)
At \( x = 0 \), the tangent line is steep and positive. The slope is about \( 1 \), which is correctly estimated.

### (e) \( f'(1) \)
At \( x = 1 \), the tangent line has a steep negative slope. The correct slope is approximately \( -2 \), not \( -3 \). 

### (f) \( f'(2) \)
At \( x = 2 \), the curve is flat again, so \( f'(2) = 0 \), which is correct.

### (g) \( f'(3) \)
At \( x = 3 \), the slope is slightly negative but close to zero, so \( f'(3) \) should be approximately \( -1 \).

### Summary of Correct Estimates:
- (a) \( f'(-3) = -1 \)
- (b) \( f'(-2) = 0 \)
- (c) \( f'(-1) \approx 1 \)
- (d) \( f'(0) = 1 \)
- (e) \( f'(1) \approx -2 \)
- (f) \( f'(2) = 0 \)
- (g) \( f'(3) \approx -1 \)

Would you like further explanation on any part of this problem?

Here are some related questions you might find useful:
1. How is the derivative geometrically interpreted as the slope of the tangent line?
2. Why do horizontal tangent lines indicate a zero derivative?
3. How do positive and negative slopes correspond to increasing or decreasing functions?
4. What does it mean if a derivative is larger or smaller in magnitude at different points?
5. How can you estimate slopes more accurately using a grid?

**Tip:** Always check for horizontal tangent lines (flat areas) as they correspond to zero derivatives.

Use the given graph to estimate the value of each derivative: f'(-3), f'(-2), f'(-1), f'(0), f'(1), f'(2), f'(3).

Learn how to estimate the value of derivatives at given points on a graph of a function. This problem involves understanding the concept of the derivative as the slope of the tangent line and visualizing slope changes. Suitable for high school or introductory college calculus.

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