Question

For some function \(f(x)\), the Maclaurin polynomial of degree 4 is \(p_{4}(x)=6+3 x-4 x^{2}+5 x^{3}-7 x^{4}\). What is \(p_{2}(x) ?\)

Short answer

The polynomial \( p_2(x) \) is \( 6 + 3x - 4x^2 \).

Step-by-step solution

Understanding Maclaurin Polynomial

A Maclaurin polynomial is a Taylor polynomial centered at zero. For a function \( f(x) \), its Maclaurin series is given by: \[ p_n(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + ... + \frac{f^{(n)}(0)}{n!}x^n \] Here, we need the polynomial of degree 2, \( p_2(x) \), which means we are looking for the terms up to \( x^2 \).

Extract Terms for \( p_2(x) \)

To find \( p_2(x) \), we take the constant term, the linear \( x \) term, and the quadratic \( x^2 \) term from \( p_4(x) \). Thus, we extract: - Constant term: 6 - Linear term: 3x - Quadratic term: -4x^2

Form the Polynomial \( p_2(x) \)

Combine the extracted terms to form the polynomial of degree 2:\[ p_2(x) = 6 + 3x - 4x^2 \] This is the desired Maclaurin polynomial when considering terms up to \( x^2 \).

Key concepts

Taylor Series

When we talk about Maclaurin or Taylor series, we're discussing ways to approximate functions with polynomials. A **Taylor series** is a type of polynomial series that helps us understand how functions behave near a particular point. The heart of the process involves creating a series based on the function's derivatives at a specific point.- **Maclaurin Series**: A special case of Taylor series centered at zero (0). The formula is: \[ p_n(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \ldots + \frac{f^{(n)}(0)}{n!}x^n \] - **Purpose**: Taylor series allow complicated functions to be expressed as infinite polynomials, making them easier to analyze or compute. However, in practice, we often use a finite version.In our problem, the Maclaurin polynomial for the function is given, and we're extracting terms to form a simpler polynomial of a lower degree.

Polynomial Degree

The **degree** of a polynomial refers to the highest power of the variable (like \( x \)) that appears in the polynomial with a non-zero coefficient. The degree tells us about the behavior and shape of the polynomial.- **Why It Matters**: The degree gives essential information about the polynomial's characteristics: - A degree of 0 (constant polynomial) means it's just a plain number. - A degree of 1 is a straight line. - Higher degrees (2, 3, 4...) indicate more complex curves.In the problem, we start with a polynomial of degree 4: \[ p_4(x) = 6 + 3x - 4x^2 + 5x^3 - 7x^4 \] We aim to reduce it to a degree 2 polynomial by keeping only the terms up to \( x^2 \). This means taking the constant, linear, and quadratic terms to form:\[ p_2(x) = 6 + 3x - 4x^2 \]

Function Derivatives

Understanding a **function's derivatives** is critical in constructing Taylor and Maclaurin series. Each derivative at a particular point tells us how the function behaves:- **First Derivative** \( f'(x) \) describes the slope or rate of change of the function. It's like asking how quickly the function's value is increasing or decreasing.- **Second Derivative** \( f''(x) \) indicates the curvature or how the slope of the function changes. It helps us understand the "bend" in the graph.- **Higher Derivatives** explain more complex changes in the function's behavior.When building, say, a Maclaurin polynomial, these derivatives are essential. We evaluate them at zero (since it's Maclaurin) to fit the function's properties at that point. The more derivatives we use, the better the approximation. In our simplification from the original degree 4 polynomial to degree 2, we're prioritizing only the relevant derivatives that inform the constant, linear, and quadratic parts (viz., \( f(0) \), \( f'(0) \), and \( f''(0) \)).