Problem 9 - Calculus: One and Several Variables

Question

Determine \(P_{0}(x), P_{1}(x), P_{2}(x), P_{3}(x)\) for $$ f(x)=1-x+3 x^{2} \cdot 5 x^{3} $$

Step-by-Step Solution

Answer

The coefficients for the power series expansion of the given function, \(1-x+3x^{2}-5x^{3}\), are \(P_{0}(x)=1, P_{1}(x)=-1, P_{2}(x)=3, P_{3}(x)=-5\).

Step 1: Identify the Coefficients

The function \(f(x)\) is a polynomial and each term of the polynomial will contribute to the power series. The polynomial terms are aligned with the power of \(x\) in each term. Specifically, the coefficient is the numeric factor before the x term. So, in our function, the coefficients \(P_0, P_1, P_2, P_3\) are as follows: \(P_0\) is the coefficient of \(x^0\), here 1. \(P_1\) is the coefficient of \(x^1\), here -1. \(P_2\) is the coefficient of \(x^2\), in this case 3. \(P_3\) is the coefficient of \(x^3\), which is -5 in this case.

Step 2: Identify the power series representation

The function \(f(x)\) can be written as a power series as follows: \[f(x)=P_0+P_1x+P_2x^2+P_3x^3\] Each \(P_i\) corresponds to the coefficient of \(x^i\) in the power series expansion.

Key Concepts

Power Series

A power series is a way of expressing a function as an infinite sum of terms. Each term is made up of a coefficient and a variable raised to an increasing power. It looks like this: \[P_0 + P_1x + P_2x^2 + P_3x^3 + \ldots\] Here, each term involves a power of the variable \(x\), accompanied by a coefficient \(P_i\).
The power series is useful because it can represent functions in a way that makes complex problems easier to solve, especially when approximating values or analyzing functions near a specific point.

• It breaks down a function into simpler, easily handled pieces.
• This representation is not limited to just polynomials, it can approximate more complex functions too.

Understanding power series helps in analytically expressing polynomials, where you can identify the contribution of each degree or power of the function.

Polynomial Function

Polynomial functions are mathematical expressions involving a sum of powers of a variable, where each power has a coefficient. They could range from simple expressions like \(2x + 3\), to more complex ones like \(f(x) = x^3 - 2x^2 + x - 5\).
Polynomials are quite versatile and come with a variety of interesting properties:

• They are defined for every input, making them continuous and smooth.
• Basic arithmetic operations (addition, subtraction, multiplication) result in another polynomial.
• Polynomials of different degrees have unique graphs. For instance, a linear polynomial forms a line, a quadratic curves into a parabola, and cubic polynomials make an 'S' shape.

While each polynomial function has its structure, their representation through power series helps understand their behavior and usefulness in various mathematical areas.

Coefficients Identification

Identifying coefficients in a polynomial or power series is crucial for understanding its structure. Coefficients tell you how much impact each power of \(x\) has on the function's behavior.
In polynomial functions, each term like \(ax^n\) has a coefficient \(a\). To identify these:

• Find the term with the desired power of \(x\).
• Extract the numeric factor in front of \(x\), which is your coefficient.

For instance, in the function \(f(x) = 1 - x + 3x^2 - 5x^3\), we can see:

• The coefficient for \(x^0\) (no \(x\) at all) is 1.
• The coefficient for \(x^1\) is -1.
• For \(x^2\), it's 3, and for \(x^3\), it's -5.

Identifying coefficients this way ensures we understand the influence of each term in a series expansion, providing a clear view of its mathematical composition.