The task is to determine graphically the value of $f'(-2)$ from the given curve, which appears to represent a function $f(x)$, along with a tangent line at the point $A$ on the curve where $x = -2$.

To solve this:

1. Tangent line at $x = -2$: $f'(-2)$ represents the derivative of the function $f(x)$ at $x = -2$. The derivative at this point is equal to the slope of the tangent line drawn at $x = -2$ on the curve.

2. Slope of the tangent line: The blue line shown passing through point $A$ is the tangent line at $x = -2$. To calculate the slope of this line, we need to determine the rise over the run, or the change in $y$-coordinates divided by the change in $x$-coordinates between two points on this tangent line.

   From the graph:

   • One point on the tangent line is $A(-2, 1)$ (since $A$ lies on the curve at $x = -2$).
   • Another visible point on the tangent line is approximately $(-5, -8)$.

   The slope $m$ of the tangent line can be calculated using: $m = \frac{\Delta y}{\Delta x} = \frac{1 - (-8)}{-2 - (-5)} = \frac{9}{3} = 3$

Conclusion:

The value of $f'(-2)$, which is the slope of the tangent line at $x = -2$, is $3$.

Would you like a more detailed explanation or have further questions?

Here are 5 related questions for further exploration:

1. What is the relationship between the slope of the tangent line and the derivative?
2. How can the derivative at different points on the curve be determined graphically?
3. How does the concavity of the function relate to the sign of its second derivative?
4. What are some examples of functions where the derivative can be calculated easily from the graph?
5. How can we estimate the derivative at non-integer points using the graph?

Tip: When analyzing a graph, always look for clear intersections or well-marked points to simplify calculations like slopes or coordinates!