https://file.aiquickdraw.com/mathbot/file/1725740376507h6nesyfy.jpg

The image presents a function \( f(x) \) along with four integrals (A, B, C, D), and asks to rank them from smallest to largest in terms of their values.

### Let's break down each integral based on the graph provided:

1. **\( A = \int_0^3 f'(x) \, dx \)**:
   - This integral represents the total change in \( f(x) \) from \( x = 0 \) to \( x = 3 \), as it integrates the derivative of \( f(x) \). From the graph, we can observe that \( f(x) \) increases, reaches a maximum, and then decreases up to \( x = 3 \). So, this value is not very large, and could even be slightly negative due to the oscillations.

2. **\( B = \int_0^3 f(x) \, dx \)**:
   - This is the area under the curve of \( f(x) \) between \( x = 0 \) and \( x = 3 \). By visually inspecting the graph, the area consists of both positive and negative sections. The negative area seems to outweigh the positive area slightly, suggesting \( B \) is likely negative.

3. **\( C = \int_0^3 |f(x)| \, dx \)**:
   - This integral calculates the total area under the curve but considers the absolute value of \( f(x) \). Since both the positive and negative portions are included as positive, this will result in a larger value than \( B \).

4. **\( D = \int_0^3 f(x) \, dx \)** (not absolute value, same as \( B \)):
   - This seems to be a repetition of \( B \), which confirms that the correct ranking was \( D = B \).

### Ranking the Integrals:
From the explanations above:
- \( B \) (or \( D \)) is negative (since the negative area dominates).
- \( A \), being based on the derivative, seems slightly larger than \( B \) but not large.
- \( C \) is the largest since it sums the absolute areas.

Thus, the correct order is:
\[ D, A, B, C \]

This matches the answer provided in the image: **D, A, B, C**.

Would you like more details on integrals or graph behavior? Here are five related questions to explore further:

1. How is the integral of a derivative related to the original function?
2. What happens when you integrate a function with both positive and negative values?
3. How does the absolute value in an integral affect the result?
4. Why would \( A \) (integral of the derivative) be different from \( B \) or \( D \)?
5. How can you visually estimate the area under a curve?

**Tip**: When estimating integrals from a graph, always check whether you're dealing with the actual values of the function or their absolute values, as this significantly affects the result.

For the function f whose graph is given below, list the following quantities in increasing order, from smallest to largest.

Explore how to rank definite integrals of a function f(x), including integrals of the function, its derivative, and the absolute value. This calculus problem is perfect for advanced high school students learning about integration and graph analysis.

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