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To find the first three non-zero terms of the Maclaurin series for \( f(x) = x^2 \cos(3x) \), we’ll start by finding the Maclaurin series for \( \cos(x) \) and then substitute \( 3x \) in place of \( x \) before multiplying by \( x^2 \).

### Step 1: Maclaurin Series for \( \cos(x) \)

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

### Step 2: Substitute \( 3x \) for \( x \)

Now, substitute \( 3x \) in place of \( x \) to get the Maclaurin series for \( \cos(3x) \):
\[
\cos(3x) = 1 - \frac{(3x)^2}{2!} + \frac{(3x)^4}{4!} - \frac{(3x)^6}{6!} + \cdots
\]
Simplify each term:
\[
\cos(3x) = 1 - \frac{9x^2}{2} + \frac{81x^4}{24} - \frac{729x^6}{720} + \cdots
\]
\[
\cos(3x) = 1 - \frac{9x^2}{2} + \frac{27x^4}{8} - \frac{27x^6}{80} + \cdots
\]

### Step 3: Multiply by \( x^2 \) to Find \( f(x) = x^2 \cos(3x) \)

Now, multiply each term in \( \cos(3x) \) by \( x^2 \) to get the series for \( f(x) = x^2 \cos(3x) \):
\[
f(x) = x^2 \cdot \left(1 - \frac{9x^2}{2} + \frac{27x^4}{8} - \frac{27x^6}{80} + \cdots \right)
\]
\[
f(x) = x^2 - \frac{9x^4}{2} + \frac{27x^6}{8} - \frac{27x^8}{80} + \cdots
\]

### Step 4: Identify the First Three Non-Zero Terms

The first three non-zero terms in the Maclaurin series for \( f(x) \) are:
\[
x^2, \quad -\frac{9x^4}{2}, \quad \text{and} \quad \frac{27x^6}{8}
\]

### Final Answer:

So, the first three non-zero terms of the Maclaurin series for \( f(x) = x^2 \cos(3x) \) are:
\[
\boxed{x^2 - \frac{9x^4}{2} + \frac{27x^6}{8}}
\]

Would you like more clarification on any step?

Here are 5 related questions for further practice:
1. How does the Maclaurin series for \( \sin(x) \) differ from \( \cos(x) \)?
2. What would be the first three non-zero terms of the Maclaurin series for \( x^3 \sin(2x) \)?
3. How do you find the radius of convergence for a Maclaurin series?
4. How would the series change if \( f(x) = x^3 \cos(3x) \)?
5. Can you determine the Maclaurin series for \( e^{3x} \)?

**Tip:** When working with Maclaurin series, make sure to simplify terms at each step to avoid complex calculations later on.

Using the Maclaurin series for cos(x), find the first three non-zero terms of the Maclaurin series for f(x) = x^2 cos(3x).

Explore how to find the first three non-zero terms of the Maclaurin series for f(x) = x^2 cos(3x) using the Maclaurin series for cos(x). This solution includes step-by-step calculations and substitutions. Ideal for college-level calculus students.

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