The Taylor series expansion of a function \( f(x) \) about \( x = 0 \) (Maclaurin series) is given by:

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

For \( f(x) = e^{-x^2} \), let's find the first four terms of the Taylor series expansion.

1. \( f(x) = e^{-x^2} \)
   
   At \( x = 0 \), \( f(0) = e^{0} = 1 \).

2. First derivative:
   \[
   f'(x) = \frac{d}{dx}(e^{-x^2}) = -2x e^{-x^2}
   \]
   At \( x = 0 \), \( f'(0) = 0 \).

3. Second derivative:
   \[
   f''(x) = \frac{d}{dx}(-2x e^{-x^2}) = -2 e^{-x^2} + 4x^2 e^{-x^2}
   \]
   At \( x = 0 \), \( f''(0) = -2 \).

4. Third derivative:
   \[
   f^{(3)}(x) = \frac{d}{dx}(-2 e^{-x^2} + 4x^2 e^{-x^2}) = 12x e^{-x^2}
   \]
   At \( x = 0 \), \( f^{(3)}(0) = 0 \).

5. Fourth derivative:
   \[
   f^{(4)}(x) = \frac{d}{dx}(12x e^{-x^2}) = 12 e^{-x^2} - 24x^2 e^{-x^2}
   \]
   At \( x = 0 \), \( f^{(4)}(0) = 12 \).

Now, substituting these values into the Taylor series formula, we get the first four terms:

\[
e^{-x^2} \approx 1 - \frac{2}{2!}x^2 + \frac{12}{4!}x^4
\]
\[
e^{-x^2} \approx 1 - x^2 + \frac{x^4}{2}
\]

This is the Taylor series expansion of \( e^{-x^2} \) up to four terms.

Would you like more details or have any questions?

Here are five related questions to explore further:
1. How is the radius of convergence of a Taylor series determined?
2. What is the general formula for the Taylor series expansion of any function?
3. How does the Taylor series for \( e^{-x^2} \) compare to the series for \( e^{x} \)?
4. Can the Taylor series for \( e^{-x^2} \) be used to approximate values of this function for small \( x \)?
5. How can we generalize this approach to find higher-order terms?

**Tip**: The more terms you use in a Taylor series, the better the approximation near the center point (here \( x = 0 \)).

Explore the Taylor series expansion of e^-x^2 up to four terms. Learn how to derive the series, compute derivatives, and understand the role of Taylor and Maclaurin series. Suitable for students in Grades 11-12 and college-level calculus.

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