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To solve this problem, we need to use the Maclaurin series for \( \cos(x) \) and apply it to the function \( f(x) = x^3 \cos(5x) \).

### Step-by-Step Solution

1. **Maclaurin Series for \( \cos(x) \):**
   The Maclaurin series expansion for \( \cos(x) \) is:
   \[
   \cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dots
   \]

2. **Maclaurin Series for \( \cos(5x) \):**
   To find the series for \( \cos(5x) \), substitute \( 5x \) for \( x \) in the series:
   \[
   \cos(5x) = 1 - \frac{(5x)^2}{2!} + \frac{(5x)^4}{4!} - \frac{(5x)^6}{6!} + \dots
   \]
   Simplifying each term:
   \[
   \cos(5x) = 1 - \frac{25x^2}{2} + \frac{625x^4}{24} - \frac{15625x^6}{720} + \dots
   \]

3. **Multiply by \( x^3 \):**
   Now, multiply each term in the series for \( \cos(5x) \) by \( x^3 \) to get the series for \( f(x) = x^3 \cos(5x) \):
   \[
   f(x) = x^3 \left(1 - \frac{25x^2}{2} + \frac{625x^4}{24} - \dots \right)
   \]
   Expanding this, we get:
   \[
   f(x) = x^3 - \frac{25x^5}{2} + \frac{625x^7}{24} - \dots
   \]

4. **Identify the First Three Non-Zero Terms:**
   The first three non-zero terms in this series are:
   \[
   x^3 - \frac{25x^5}{2} + \frac{625x^7}{24}
   \]

### Conclusion

The correct answer is:
\[
x^3 - \frac{25x^5}{2} + \frac{625x^7}{24}
\]
This matches the second option in the provided answer choices.

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Would you like a detailed explanation of each term or any other questions? Here are some related questions for practice:

1. What is the general formula for the Maclaurin series expansion?
2. How would you find the Maclaurin series for \( \sin(x) \) and use it in a similar problem?
3. How does substituting \( 5x \) in \( \cos(x) \) change the Maclaurin series?
4. Can we approximate \( f(x) \) using just the first two non-zero terms?
5. What are the limitations of using Maclaurin series expansions?

**Tip:** Remember, multiplying the series term-by-term is an efficient way to adjust a known series expansion for functions like \( x^3 \cos(5x) \).

Using the Maclaurin series for cos(x), find the first three non-zero terms of the Maclaurin series for f(x) = x^3 cos(5x).

Learn how to find the first three non-zero terms of the Maclaurin series for f(x) = x^3 cos(5x) using the Maclaurin series for cos(x). This calculus problem involves power series expansions and trigonometric functions, suitable for students in undergraduate calculus.

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