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.