find the mclauren polinomial of the function e^(-2x) to the 4 degree

The Maclaurin polynomial is a type of Taylor polynomial centered at \( x = 0 \). For the function \( f(x) = e^{-2x} \), we need to compute the Maclaurin series up to the 4th degree.

The general form of a Maclaurin series is:

\[
f(x) = f(0) + f'(0) \frac{x}{1!} + f''(0) \frac{x^2}{2!} + f^{(3)}(0) \frac{x^3}{3!} + f^{(4)}(0) \frac{x^4}{4!} + \cdots
\]

### Step 1: Compute the derivatives of \( e^{-2x} \)
1. \( f(x) = e^{-2x} \)
2. \( f'(x) = -2e^{-2x} \)
3. \( f''(x) = 4e^{-2x} \)
4. \( f^{(3)}(x) = -8e^{-2x} \)
5. \( f^{(4)}(x) = 16e^{-2x} \)

### Step 2: Evaluate these derivatives at \( x = 0 \)
1. \( f(0) = e^{0} = 1 \)
2. \( f'(0) = -2 \)
3. \( f''(0) = 4 \)
4. \( f^{(3)}(0) = -8 \)
5. \( f^{(4)}(0) = 16 \)

### Step 3: Construct the Maclaurin polynomial
Using the formula for the Maclaurin series, we plug in the values for the function and its derivatives at \( x = 0 \):

\[
P_4(x) = 1 - 2 \frac{x}{1!} + 4 \frac{x^2}{2!} - 8 \frac{x^3}{3!} + 16 \frac{x^4}{4!}
\]

Now simplify each term:

\[
P_4(x) = 1 - 2x + 2x^2 - \frac{4x^3}{3} + \frac{x^4}{3}
\]

Thus, the 4th-degree Maclaurin polynomial for \( e^{-2x} \) is:

\[
P_4(x) = 1 - 2x + 2x^2 - \frac{4x^3}{3} + \frac{x^4}{3}
\]

Would you like a detailed explanation of how each derivative was calculated or clarification on any of the steps?

### Related Questions:
1. How is the general Taylor series derived from a function?
2. What are the convergence criteria for a Maclaurin series?
3. Can we derive the Maclaurin series for \( e^{2x} \) and compare?
4. What is the relationship between the Taylor series and the function's radius of convergence?
5. How does the error bound in a truncated Taylor series work?

### Tip:
When working with exponential functions in a Maclaurin or Taylor series, it’s useful to recognize patterns in their derivatives to speed up computations.

Learn how to find the 4th-degree Maclaurin polynomial for the function e^(-2x). This problem involves calculating derivatives, applying the Maclaurin series formula, and simplifying each term. Ideal for college-level calculus students.

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