Problem 44 - Thomas Calculus

Question

Approximation properties of Taylor polynomials Suppose that \(f(x)\) is differentiable on an interval centered at \(x=a\) and that \(g(x)=b_{0}+b_{1}(x-a)+\cdots+b_{n}(x-a)^{n}\) is a polynomial of degree \(n\) with constant coefficients \(b_{0}, \ldots, b_{n}\) . Let \(E(x)=\) \(f(x)-g(x) .\) Show that if we impose on \(g\) the conditions i) \(E(a)=0\) ii) $$\lim _{x \rightarrow a} \frac{E(x)}{(x-a)^{n}}=0$$ then $$\begin{array}{r}{g(x)=f(a)+f^{\prime}(a)(x-a)+\frac{f^{\prime \prime}(a)}{2 !}(x-a)^{2}+\cdots} \\ {+\frac{f^{(n)}(a)}{n !}(x-a)^{n}}\end{array}.$$ Thus, the Taylor polynomial \(P_{n}(x)\) is the only polynomial of degree less than or equal to \(n\) whose error is both zero at \(x=a\) and negligible when compared with \((x-a)^{n}.\)

Step-by-Step Solution

Answer

The polynomial \( g(x) \) is the Taylor polynomial \( P_n(x) \).

Step 1: Define the Error Function

We define the error function as \( E(x) = f(x) - g(x) \). This represents the difference between the function \( f(x) \) and the polynomial \( g(x) \), which needs to satisfy certain conditions.

Step 2: Evaluate the First Condition: \( E(a) = 0 \)

Substitute \( x = a \) into the error function \( E(x) = f(x) - g(x) \), resulting in \( E(a) = f(a) - g(a) = 0 \). This condition implies that \( g(a) = f(a) \), meaning the polynomial matches the function at \( x = a \).

Step 3: Evaluate the Second Condition

The condition \( \lim_{x \to a} \frac{E(x)}{(x-a)^{n}} = 0 \) means that near \( x = a \), the difference between \( f(x) \) and \( g(x) \) must behave like a term of order higher than \( n \). This implies that all terms in the Taylor expansion of \( f(x) \) up to degree \( n \) must match those in \( g(x) \).

Step 4: Apply the Conditions to \( g(x) \)

Since condition (i) \( E(a) = 0 \) gives us \( b_0 = f(a) \), we need to ensure that the derivatives of \( E(x) \) up to \( (n-1) \)-th order are zero at \( x = a \), and the \( n \)-th derivative matches to satisfy condition (ii).

Step 5: Derive Coefficients of \( g(x) \) Using Taylor Series

Considering \( g(x) = b_0 + b_1(x-a) + \cdots + b_n(x-a)^n \), compare derivatives at \( x=a \) with those of \( f(x) \). \( b_1 = f'(a) \), \( b_2 = \frac{f''(a)}{2!} \), ... \( b_n = \frac{f^{(n)}(a)}{n!} \). Thus, \( g(x) \) is expressed as the Taylor series.

Key Concepts

Approximation Theory

Approximation theory is a branch of mathematics that focuses on how functions can be expressed or approximated by simpler ones, such as polynomials. This is incredibly useful because complicated functions, like those involving transcendentals or other complex components, can be hard to work with.

• We often use polynomial approximations for these functions because polynomials are easier to calculate and work with.
• The goal of approximation theory is to find a function (an approximant) that is close to the original function as possible, with an error (the difference between the functions) that's as small as possible.
• An important concept in approximation theory is the Taylor Polynomial, which uses derivatives of the function to construct polynomials that approximate the function around a certain point.

By understanding the foundation of approximation theory, you can better grasp how tools like Taylor Polynomials provide practical solutions in both theoretical and applied mathematics scenarios.

Error Function

The error function in this context is defined as the difference between the actual function and its polynomial approximation: \[ E(x) = f(x) - g(x) \]

• In order to have a good approximation, we want our error function, \( E(x) \), to be small or tend to zero as specified.
• Specifically, in the approximation using Taylor Polynomials, the function \( f(x) \) and the polynomial \( g(x) \) must satisfy two conditions:
• \( E(a) = 0 \): This means that the polynomial and the function are equal at the center of approximation, \( x = a \).
• \( \lim_{x \to a} \frac{E(x)}{(x-a)^n} = 0 \): This requirement ensures that the error behaves like terms of degree higher than \( n \) near \( x = a \).

The error function helps us determine how well our polynomial approximates a given function and highlights the differences to refine our approaches.

Differentiable Functions

Differentiable functions are those functions that have a derivative at each point in their domain. This is a key requirement for constructing Taylor Polynomials:

• In approximation theory, we rely on differentiable functions because the derivatives provide the coefficients of the Taylor polynomial.
• Differentiability ensures smooth transitions between the values of the function, allowing for polynomial expressions without jumps or breaks.
• For a function \( f(x) \) to be approximated by a Taylor polynomial, it must possess derivatives up to at least the \( n \)-th order, where \( n \) is the degree of the polynomial used for approximation.

Using differentiable functions as a basis ensures a robust framework for creating reliable and accurate polynomial approximations.

Polynomial Approximation

Polynomial approximation involves expressing a function as a sum of polynomial terms that closely mimic the behavior of that function. The Taylor Polynomial is a specific type of polynomial approximation that uses derivatives at a particular point, \( x = a \), to build the approximation:

• The Taylor polynomial of degree \( n \), denoted \( P_n(x) \), has the form: \[ P_n(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x - a)^n \]
• This polynomial is designed in such a way that it agrees with \( f(x) \) up to the \( n \)-th derivative at the point \( x = a \).
• By satisfying the conditions outlined, the error between the function and the polynomial approximation becomes negligible for higher-order terms, making the approximation effectively accurate around \( x = a \).

Polynomial approximation proves to be a powerful tool in mathematics, offering simplicity and ease of computation while maintaining accuracy close to the point of interest.