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The question in the image asks: **"Given a function \( f \) and a point \( a \) in its domain, what does \( f'(a) \) represent?"**

Here are the provided answer choices:
- **A.** The instantaneous rate of change at any point in the domain.
- **B.** The instantaneous rate of change at \( a \).
- **C.** The value of the function at point \( a \).

### Explanation:
\( f'(a) \) represents the derivative of the function \( f \) at the point \( a \). The derivative at a specific point \( a \) gives us the **instantaneous rate of change** of the function at that point. In other words, it tells us how the function \( f \) is changing at \( a \), which corresponds to the slope of the tangent line at that point.

Therefore, the correct answer is:

- **B. The instantaneous rate of change at \( a \).**

Would you like more details on how derivatives relate to the rate of change, or do you have any other questions?

Here are five related questions to explore:
1. What is the difference between the instantaneous rate of change and the average rate of change?
2. How can you calculate the derivative of a function using limits?
3. What does it mean if the derivative \( f'(a) \) is zero?
4. How do you interpret the second derivative \( f''(a) \) at a point?
5. What role do derivatives play in understanding the concavity of a function?

**Tip:** The derivative \( f'(a) \) gives the slope of the tangent line to the graph of the function at point \( a \). This is crucial in understanding the behavior of functions at specific points.

Given a function f and a point a in its domain, what does f'(a) represent?

Explore the concept of the derivative f'(a) and learn how it represents the instantaneous rate of change of a function at a given point. This calculus problem is essential for understanding the slope of tangent lines and the behavior of functions. Suitable for high school students in Grades 11-12 or college level.

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