To estimate the value of each derivative $f'(x)$ at the given points from the graph of $f(x)$, we need to evaluate the slope of the tangent line at each specified point. The derivative at a given point represents the slope of the tangent line to the curve at that point.

Let’s analyze the graph:

• At $x = -3$: The slope is negative, as the curve is descending.
• At $x = -2$: The curve seems to be flat, meaning the slope (derivative) is zero, indicating a local minimum.
• At $x = -1$: The slope is positive, as the curve is increasing after the minimum.
• At $x = 0$: The slope continues to be positive.
• At $x = 1$: The slope is zero again, indicating a local maximum.
• At $x = 2$: The slope is negative, as the curve starts to descend.
• At $x = 3$: The slope is negative but less steep than at $x = 2$.

Estimated values of the derivatives:

(a) $f'(-3) \approx -1$ (negative slope) (b) $f'(-2) = 0$ (flat, local minimum) (c) $f'(-1) \approx 1$ (positive slope) (d) $f'(0) \approx 1$ (positive slope) (e) $f'(1) = 0$ (flat, local maximum) (f) $f'(2) \approx -1$ (negative slope) (g) $f'(3) \approx -0.5$ (negative but less steep)

Would you like a more detailed explanation of how each value was estimated, or do you have any other questions?

Here are 5 relative questions to deepen understanding:

1. What does the derivative tell us about the behavior of a function at a point?
2. Why is the derivative zero at local maxima and minima?
3. How can the slope of the tangent line be approximated visually from a graph?
4. What is the significance of positive versus negative slopes in a graph of $f(x)$?
5. How could this graph be used to identify concavity or inflection points?

Tip: When estimating derivatives from a graph, look at how steep the curve is and whether it’s increasing or decreasing—this helps gauge the sign and magnitude of the slope.