https://file.aiquickdraw.com/mathbot/file/172609556860796cwt063.jpg

To determine \( f'(a) \) for the given function \( f(x) \) at specific points, we need to estimate the slope of the tangent line to the curve at those points. The slope of the tangent line represents the derivative, \( f'(x) \), at that point.

Let's analyze the graph provided for each value of \( a = 1, 2, 4, 7 \):

### Step-by-step Analysis:

1. **At \( a = 1 \):**
   - The graph is relatively flat at \( x = 1 \). 
   - The slope of the tangent line at this point appears to be close to zero.
   - Therefore, \( f'(1) \approx 0 \).

2. **At \( a = 2 \):**
   - The graph is increasing linearly at \( x = 2 \).
   - The slope of the tangent line appears to be constant and positive. 
   - If we estimate the slope by using two points on this line, we get a positive slope. 
   - So, \( f'(2) \approx 1 \).

3. **At \( a = 4 \):**
   - At \( x = 4 \), the graph reaches its maximum point.
   - The slope of the tangent line at the maximum point is horizontal.
   - Therefore, \( f'(4) = 0 \).

4. **At \( a = 7 \):**
   - The graph is steeply decreasing at \( x = 7 \).
   - The slope of the tangent line appears to be negative and quite steep.
   - An approximate value for the negative slope could be around \( -2 \).
   - So, \( f'(7) \approx -2 \).

### Conclusion:

\[
\begin{align*}
f'(1) &\approx 0 \\
f'(2) &\approx 1 \\
f'(4) &= 0 \\
f'(7) &\approx -2 \\
\end{align*}
\]

Would you like further details on how to estimate the slope or more practice problems? Here are five related questions for deeper understanding:

1. How do you find the derivative of a function at a point using the limit definition?
2. What does it mean for a derivative to be zero at a point on a graph?
3. How can you determine if a point is a local maximum or minimum using the first derivative?
4. What are some methods for finding the derivative of a non-linear function analytically?
5. How do you estimate the slope of a tangent line if given only a graph and not the function?

**Tip:** When analyzing a graph, remember that the steeper the slope of the tangent line, the larger the magnitude of the derivative (positive or negative).

Let f(x) be the function whose graph is shown. Determine f'(a) for a = 1, 2, 4, 7.

Learn how to find the derivative of a function from a graph at specific points (a = 1, 2, 4, 7). Analyze the slope of tangent lines at various points to estimate f'(a). Suitable for students in Grades 11-12 studying calculus and derivative concepts.

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