To find the Maclaurin series of $\cos(3x)$, we need to expand $\cos(3x)$ around $x = 0$. The Maclaurin series for a function $f(x)$ is:

$f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots$

Step-by-Step Solution

1. Identify $f(x) = \cos(3x)$ and compute derivatives at $x = 0$:

   We know: $f(x) = \cos(3x)$

2. Compute the derivatives and evaluate them at $x = 0$:

   • $f(x) = \cos(3x)$, so $f(0) = \cos(0) = 1$.

   • First derivative: $f'(x) = -3 \sin(3x)$.
     At $x = 0$: $f'(0) = -3 \sin(0) = 0$.

   • Second derivative: $f''(x) = -9 \cos(3x)$. At $x = 0$: $f''(0) = -9 \cos(0) = -9$.

   • Third derivative: $f'''(x) = 27 \sin(3x)$. At $x = 0$: $f'''(0) = 27 \sin(0) = 0$.

   • Fourth derivative: $f^{(4)}(x) = -81 \cos(3x)$. At $x = 0$: $f^{(4)}(0) = -81 \cos(0) = -81$.

3. Construct the series:

   Now substitute these values into the Maclaurin series formula:

   $f(x) = f(0) + f'(0) x + \frac{f''(0)}{2!} x^2 + \frac{f'''(0)}{3!} x^3 + \frac{f^{(4)}(0)}{4!} x^4 + \cdots$

   Substitute the values we calculated:

   $\cos(3x) = 1 + 0 \cdot x + \frac{-9}{2} x^2 + 0 \cdot x^3 + \frac{-81}{24} x^4 + \cdots$

4. Simplify terms:

   This simplifies to:

   $\cos(3x) = 1 - \frac{9}{2} x^2 - \frac{81}{24} x^4 + \cdots$

   Further simplifying $\frac{81}{24} = \frac{27}{8}$, we get:

   $\cos(3x) = 1 - \frac{9}{2} x^2 + \frac{27}{8} x^4 + \cdots$

Final Answer

The Maclaurin series for $\cos(3x)$ is:

$\cos(3x) = 1 - \frac{9}{2} x^2 + \frac{27}{8} x^4 + \cdots$

Would you like more details on each step or have any questions?

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Related Questions

1. What is the Maclaurin series of $\sin(3x)$?
2. How does the Maclaurin series for $\cos(x)$ compare to $\cos(3x)$?
3. What are the convergence properties of the Maclaurin series for cosine functions?
4. How would you find the Maclaurin series of $e^{3x}$?
5. How can the Taylor series expansion of $\cos(x)$ help in calculating higher-order derivatives?

Tip:

The Maclaurin series for trigonometric functions like $\sin(x)$ and $\cos(x)$ always have alternating signs and factorial denominators, which is a helpful pattern to remember.