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The image presents problem 71 related to the average rate of change and average value of a function. Here's a summary of the given problem:

- **Given**: A function \( f \) is differentiable on the interval \([a, b]\).
  
- The average rate of change of \( f \) over \([a, b]\) is:
  \[
  \frac{f(b) - f(a)}{b - a}
  \]
  
- The instantaneous rate of change at any point \( x \) is \( f'(x) \).
  
- The chapter defines the *average value* of a function \( f' \) over the interval \([a, b]\), and for consistency, it is required that:
  \[
  \frac{f(b) - f(a)}{b - a} = \text{average value of } f' \text{ on } [a, b].
  \]

This leads to the idea that the average value of \( f'(x) \) over the interval \([a, b]\) is the same as the average rate of change of \( f \) over \([a, b]\).

This can be mathematically expressed using the mean value theorem for integrals:
\[
\frac{1}{b-a} \int_a^b f'(x) \, dx = \frac{f(b) - f(a)}{b - a}.
\]

This shows that the average value of the derivative \( f'(x) \) over \([a, b]\) is equivalent to the difference quotient, which is the average rate of change of the function \( f \) itself.

Would you like further elaboration on how this connects to the mean value theorem, or a detailed example? Here are five related questions to explore this further:

1. What is the geometric interpretation of the average rate of change of a function?
2. How does the Mean Value Theorem relate to the concept of the average rate of change?
3. Can the equality between the average value of \( f'(x) \) and the average rate of change fail for any differentiable function?
4. How can you apply this concept to real-world problems involving average and instantaneous rates of change?
5. How do you calculate the average value of a general continuous function over an interval?

**Tip**: Remember that the average value of a function over an interval is found by integrating the function and dividing by the length of the interval.

Let f be a function that is differentiable on [a, b]. The problem involves comparing the average rate of change of f over [a, b] to the average value of f' over the same interval. Specifically, we are given that the average rate of change of f over [a, b] is \( \frac{f(b) - f(a)}{b - a} \), and the average value of f' over [a, b] must be consistent with this. The problem requires us to verify the relationship between the average rate of change and the average value of f' using the mean value theorem for integrals.

Understand the relationship between the average rate of change of a function f over an interval [a, b] and the average value of its derivative f'. This problem applies the Mean Value Theorem for integrals and helps in visualizing the connection between average and instantaneous rates of change.

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