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.