Problem 35 - Calculus

Question

(a) Find and multiply the Taylor polynomials of degree 1 near \(x=0\) for the two functions \(f(x)=1 /(1-x)\) and \(g(x)=1 /(1-2 x)\) (b) Find the Taylor polynomial of degree 2 near \(x=0\) for the function \(h(x)=f(x) g(x)\) (c) Is the product of the Taylor polynomials for \(f(x)\) and \(g(x)\) equal to the Taylor polynomial for the function \(h(x) ?\)

Step-by-Step Solution

Answer

No, the product of the Taylor polynomials is not equal to the Taylor polynomial for \( h(x) \).

Step 1: Write the Taylor Polynomial for f(x)

The function given is \( f(x) = \frac{1}{1-x} \). The Taylor series expansion for \( f(x) \) is: \[ f(x) = 1 + x + x^2 + x^3 + \ldots \]For degree 1, the polynomial is: \[ T_1(f) = 1 + x \]

Step 2: Write the Taylor Polynomial for g(x)

The function given is \( g(x) = \frac{1}{1-2x} \). The Taylor series expansion for \( g(x) \) is: \[ g(x) = 1 + 2x + (2x)^2 + (2x)^3 + \ldots \]For degree 1, the polynomial is: \[ T_1(g) = 1 + 2x \]

Step 3: Multiply the Taylor Polynomials of f(x) and g(x)

Now, multiply the degree 1 Taylor polynomials obtained:\[ T_1(f) \times T_1(g) = (1 + x)(1 + 2x) \]Expand the expression:\[ = 1 + 2x + x + 2x^2 = 1 + 3x + 2x^2 \]

Step 4: Write Taylor Polynomial for h(x) = f(x)g(x)

Now, find the Taylor polynomial of degree 2 for \( h(x) = f(x)g(x) \). Since \( f(x) = \frac{1}{1-x} \) and \( g(x) = \frac{1}{1-2x} \), we multiply their expansions:- \( f(x) \) up to degree 2: \( 1 + x + x^2 \)- \( g(x) \) up to degree 2: \( 1 + 2x + 4x^2 \)Multiply these:\[ h(x) = (1 + x + x^2)(1 + 2x + 4x^2) \]Multiply each term and collect terms up to \(x^2\):\[ = 1 + 2x + 4x^2 + x + 2x^2 + 4x^3 + x^2 + 4x^3 + x^3 \]Simplify:\[ = 1 + 3x + 7x^2 \]

Step 5: Compare the Products

The degree 2 polynomial for \( h(x) = f(x)g(x) \) is \( 1 + 3x + 7x^2 \), while the polynomial obtained by multiplying the degree 1 polynomials of \( f(x) \) and \( g(x) \) is \( 1 + 3x + 2x^2 \). Clearly, these two polynomials are not equal.

Key Concepts

Polynomial Multiplication

Polynomial multiplication is an important technique used to combine polynomials by multiplying each term of one polynomial by each term of another. Imagine you have two simple polynomials, like our Taylor polynomials from the exercise: you want to multiply

• \( T_1(f) = 1 + x \)
• \( T_1(g) = 1 + 2x \)

To do this, apply the distributive property:

• First, multiply \(1\) from \(T_1(f)\) with each term in \(T_1(g)\): \( 1 imes 1 + 1 imes 2x \).
• Next, multiply \(x\) from \(T_1(f)\) with each term in \(T_1(g)\): \( x imes 1 + x imes 2x \).

You end up with: \[ 1 + 2x + x + 2x^2 \] Combine like terms to simplify: \[ = 1 + 3x + 2x^2 \] This process allowed you to easily handle and multiply these small polynomials, leading to new and combined terms.

Degree of Polynomials

In polynomial expressions, the degree of a polynomial is the highest power of the variable within the expression. For instance, in the polynomial \( 1 + 3x + 7x^2 \), the degree is \(2\) because it is the highest exponent on the variable \(x\).
To determine the degree, simply look for the term with the largest exponent:

• \( 1 + x \) has a degree of \(1\), since \(x\) is the highest power.
• \( 1 + 3x + 7x^2 \) has a degree of \(2\), as we previously noted.

Understanding polynomial degrees is crucial in simplifying expressions and predicting the behavior of functions as their independent variables change. It's equally important when dealing with Taylor polynomials, which aim to approximate functions over intervals by using polynomials of determined degrees.

Taylor Series Expansion

The Taylor series expansion is a method to represent functions as infinite sums of terms calculated from the values of their derivatives at a single point. It's a powerful tool to approximate functions, particularly around a point like \( x = 0 \), which is often referred to as expansion around "a = 0."
For example,

• \( f(x) = \frac{1}{1-x} \) can be expanded to \( 1 + x + x^2 + x^3 + \cdots \).
• \( g(x) = \frac{1}{1-2x} \) can be expanded to \( 1 + 2x + (2x)^2 + (2x)^3 + \cdots \).

A Taylor series of a specific degree, such as degree \(1\) or \(2\), means you take only the first one or two terms from this expansion, respectively. This truncation helps in simplifying the computation but often at the cost of reduced accuracy. In the given exercise, you would look at the Taylor polynomials of degree \(1\) for functions \(f(x)\) and \(g(x)\) to perform polynomial multiplication, while a degree \(2\) polynomial for \(h(x) = f(x)g(x)\) approached with more expansion correcting terms.