Question

Find the Maclaurin polynomial of degree \(n\) for the given function. $$f(x)=e^{2 x}, \quad n=4$$

Short answer

The Maclaurin polynomial of degree 4 for \( e^{2x} \) is \( 1 + 2x + 2x^2 + \frac{4}{3}x^3 + \frac{2}{3}x^4 \).

Step-by-step solution

Recall the Maclaurin Series Format

The Maclaurin series of a function \( f(x) \) is given by:\[ f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots \]To find the Maclaurin polynomial of degree \( n \), we calculate the terms up to \( x^n \).

Calculate the Derivatives

Compute the first four derivatives of \( f(x) = e^{2x} \):- \( f(x) = e^{2x} \)- \( f'(x) = 2e^{2x} \)- \( f''(x) = 4e^{2x} \)- \( f'''(x) = 8e^{2x} \)- \( f^{(4)}(x) = 16e^{2x} \)

Evaluate the Derivatives at x = 0

Substitute \( x = 0 \) into each derivative:- \( f(0) = e^0 = 1 \)- \( f'(0) = 2e^0 = 2 \)- \( f''(0) = 4e^0 = 4 \)- \( f'''(0) = 8e^0 = 8 \)- \( f^{(4)}(0) = 16e^0 = 16 \)

Assemble the Maclaurin Polynomial

Substitute the calculated terms into the Maclaurin series formula:\[ P_4(x) = 1 + 2x + \frac{4}{2!}x^2 + \frac{8}{3!}x^3 + \frac{16}{4!}x^4 \]Simplify the coefficients:\[ P_4(x) = 1 + 2x + 2x^2 + \frac{4}{3}x^3 + \frac{2}{3}x^4 \]

Key concepts

Understanding Derivatives

Derivatives are a fundamental concept in calculus, representing the rate at which a function changes at any given point. When dealing with the function \( f(x) = e^{2x} \), derivatives help capture the behavior of the function as we calculate the Maclaurin series.
In essence, finding derivatives involves taking the function and applying a differentiation process:

• The first derivative \( f'(x) = 2e^{2x} \) tells us how the function grows at a particular point.
• Each subsequent derivative represents the rate change of the previous derivative. For instance, \( f''(x) = 4e^{2x} \) represents how the growth rate of \( f'(x) \) itself changes.
• Higher-order derivatives, like the third \( f'''(x) = 8e^{2x} \) and the fourth \( f^{(4)}(x) = 16e^{2x} \), continue this pattern, stripping away layers to reveal intricate details about the function's behavior.

At each derivative level, setting \( x = 0 \) allows us to assess these rates directly at the center point of the Maclaurin series, creating core components to build the polynomial. This evaluation forms the scaffold for constructing a polynomial that approximates the original function around this center.

Diving into Series Expansion

A series expansion represents a function as an infinite sum of terms. The Maclaurin series is a specific type of series expansion that expresses a function \( f(x) \) around the point \( x = 0 \).In each step of the Maclaurin series, we calculate derivatives and evaluate them at \( x = 0 \):

• The \( f(0) \) term gives the function's base point from which expansions start.
• Each derivative, such as \( f'(0) \), \( f''(0) \), contributes differently scaled terms like \( x^1, x^2 \).
• We divide these contributions by factorial numbers (e.g., \( 2! \), \( 3! \)) to maintain accuracy when summing terms.

This series is infinitely long; however, for practical purposes, we truncate the series. Creating a truncated series up to a certain degree \( n \), like in this exercise, gives a polynomial approximation of the function.

Polishing with Polynomial Approximation

Polynomial approximation involves creating a polynomial that closely mimics a function's behavior around a specific point. In our exercise, reaching the final Maclaurin polynomial means effectively approximating \( f(x) = e^{2x} \) using a fourth-degree polynomial \( P_4(x) \).The purpose of this approximation is to simplify calculations:

• Exact representation: The derived polynomial \( P_4(x) = 1 + 2x + 2x^2 + \frac{4}{3}x^3 + \frac{2}{3}x^4 \) closely follows the original function near \( x = 0 \).
• Solving: Polynomial expressions are easier to evaluate and analyze than functions with exponential terms.
• Flexibility: This form enables practical computation in scenarios where exact function values are unwieldy or impossible to derive quickly.

Using polynomial approximations like this is crucial in numerical methods where concise, approximate solutions facilitate problem-solving without diving into more complex original functions.