Find the first 4 terms for the Maclaurin series of f(x) = tan x.

To find the first four terms of the Maclaurin series for \( f(x) = \tan(x) \), we need to compute the derivatives of \( f(x) = \tan(x) \) at \( x = 0 \) and use the formula for the Maclaurin series:

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

### Step-by-Step Process:

1. **Calculate the function and its derivatives at \( x = 0 \):**

   Let's find the first few derivatives of \( f(x) = \tan(x) \).

   - \( f(x) = \tan(x) \)

   - First derivative: \( f'(x) = \sec^2(x) \)

   - Second derivative: \( f''(x) = 2 \sec^2(x) \tan(x) \)

   - Third derivative: \( f'''(x) = 2 \sec^2(x) (1 + 2 \tan^2(x)) \)

   - Fourth derivative: \( f^{(4)}(x) = 8 \sec^2(x) \tan(x) (1 + \tan^2(x)) \)

2. **Evaluate these derivatives at \( x = 0 \):**

   At \( x = 0 \), we know:

   - \( \tan(0) = 0 \)
   - \( \sec(0) = 1 \)

   Therefore:

   - \( f(0) = \tan(0) = 0 \)

   - \( f'(0) = \sec^2(0) = 1 \)

   - \( f''(0) = 2 \sec^2(0) \tan(0) = 2 \cdot 1 \cdot 0 = 0 \)

   - \( f'''(0) = 2 \sec^2(0) (1 + 2 \tan^2(0)) = 2 \cdot 1 \cdot (1 + 0) = 2 \)

   - \( f^{(4)}(0) = 8 \sec^2(0) \tan(0) (1 + \tan^2(0)) = 8 \cdot 1 \cdot 0 = 0 \)

3. **Construct the Maclaurin series using these values:**

   Substitute the values of the derivatives into the Maclaurin series formula:

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

Substituting the values we found:

\[
\tan(x) \approx 0 + 1 \cdot x + \frac{0}{2!}x^2 + \frac{2}{3!}x^3 + \frac{0}{4!}x^4 + \cdots
\]

Simplify:

\[
\tan(x) \approx x + \frac{x^3}{3} + O(x^5)
\]

### Conclusion:

The first four terms of the Maclaurin series for \( \tan(x) \) are:

\[
\tan(x) \approx x + \frac{x^3}{3}
\]

### Additional Questions:
1. How do we find the Maclaurin series for other trigonometric functions like \( \sin(x) \) or \( \cos(x) \)?
2. What is the radius of convergence for the Maclaurin series of \( \tan(x) \)?
3. How would the Maclaurin series change if we were looking for a different function, such as \( \arctan(x) \)?
4. What are some common applications of Maclaurin series in physics or engineering?
5. How do you compute higher-order derivatives for more complicated functions?

**Tip:** Understanding how to derive and apply series expansions is useful for approximating functions, especially when dealing with small values of \( x \) in practical applications.

Explore the first four terms of the Maclaurin series for the function tan(x). This calculus problem involves the computation of derivatives and constructing the series expansion. Key concepts include series expansions and trigonometric functions, suitable for undergraduate calculus students.

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