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In this figure we see the graph of a function f(x) and it tangent in the point (3, -1). Use the figure to determine the value of f'(3).

To determine the value of \( f'(3) \) from the graph, we need to understand that the derivative \( f'(3) \) represents the slope of the tangent line at the point where \( x = 3 \).

### Steps to Solve:
1. **Identify the Tangent Line:**
   The dotted line shown in the graph is the tangent line at the point \( (3, -1) \).

2. **Determine the Slope:**
   The slope \( m \) of the tangent line can be found by using the slope formula:
   \[
   m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
   \]
   where \( (x_1, y_1) \) and \( (x_2, y_2) \) are two points on the tangent line.

3. **Find Two Points on the Tangent Line:**
   By inspecting the graph, we can choose two clear points on the tangent line. For instance:
   - The point where the tangent line intersects \( x = 3 \) (this is the point of tangency) is \( (3, -1) \).
   - Another point on the tangent line could be \( (2, -3) \), which seems to lie on the dotted line.

4. **Calculate the Slope:**
   Substituting these points into the slope formula:
   \[
   m = \frac{-1 - (-3)}{3 - 2} = \frac{-1 + 3}{1} = \frac{2}{1} = 2
   \]

So, the value of \( f'(3) \) is 2.

Would you like more details or have any questions on this? Here are some additional questions to explore:

1. What does the sign of the derivative tell us about the function at \( x = 3 \)?
2. How would you interpret the derivative if the tangent line were horizontal?
3. What does the second derivative at \( x = 3 \) tell you about the concavity of \( f(x) \)?
4. How would the graph look if \( f'(3) \) were negative?
5. Can you find the equation of the tangent line at \( x = 3 \)?

**Tip:** When determining the slope from a graph, always ensure you choose points that clearly lie on the line to avoid calculation errors.

Learn how to calculate the derivative f'(3) using the tangent line at the point (3, -1) from the graph of the function f(x). Detailed step-by-step solution including identification of the tangent line, slope calculation using the slope formula, and interpretation of the derivative's significance.

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