https://file.aiquickdraw.com/mathbot/file/1729928979467nbfu320r.jpg

To solve this problem, let's start by finding the first three non-zero terms in the Maclaurin series for \( f(x) = x^2 \cos(3x) \).

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

### Step 2: Substitute \( x = 3x \)
To find \( \cos(3x) \), we substitute \( 3x \) into the series:
\[
\cos(3x) = 1 - \frac{(3x)^2}{2!} + \frac{(3x)^4}{4!} - \frac{(3x)^6}{6!} + \cdots
\]

Now, calculate each term:
1. The constant term is \( 1 \).
2. The \( x^2 \) term: 
   \[
   -\frac{(3x)^2}{2} = -\frac{9x^2}{2}
   \]
3. The \( x^4 \) term:
   \[
   \frac{(3x)^4}{24} = \frac{81x^4}{24} = \frac{27x^4}{8}
   \]

Thus, we have:
\[
\cos(3x) = 1 - \frac{9x^2}{2} + \frac{27x^4}{8} + \cdots
\]

### Step 3: Multiply by \( x^2 \) for \( f(x) = x^2 \cos(3x) \)
Now, multiply each term by \( x^2 \):
\[
f(x) = x^2 \cdot \cos(3x) = x^2 - \frac{9x^4}{2} + \frac{27x^6}{8} + \cdots
\]

### Step 4: Identify the First Three Non-Zero Terms
The first three non-zero terms of \( f(x) \) are:
\[
x^2 - \frac{9x^4}{2} + \frac{27x^6}{8}
\]

### Step 5: Match with the Answer Choices
Looking at the choices in the image:
- The correct answer is the one with \( x^2 - \frac{9x^4}{2} + \frac{27x^6}{8} \).

It appears that **none of the answer choices exactly match this series** based on our calculation. If you could recheck or clarify the choices, that might help, or you may need to choose the closest approximation.

Would you like further details or explanations?

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### Related Questions
1. How do you derive the Maclaurin series for functions involving other trigonometric functions?
2. What is the general formula for the Maclaurin series expansion?
3. How can the error be calculated for a truncated Maclaurin series?
4. How does the substitution \( x \to kx \) affect the series terms?
5. What applications do Maclaurin series have in engineering and physics?

### Tip
Always double-check your expansions and substitutions to ensure terms align correctly with each power of \( x \) in a series expansion problem.

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

Learn how to use the Maclaurin series expansion to find the first three non-zero terms for \( f(x) = x^2 \cos(3x) \). This problem covers calculus concepts, including trigonometric series expansion and Maclaurin series, with detailed steps for substituting \( 3x \) and multiplying by \( x^2 \).

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