Problem 40 - 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 \(P_n(x)\) for \(f(x)\).

Step 1: Understanding the Problem

We want to prove that the polynomial of degree \(n\) \(g(x)\) is in fact the Taylor series \(P_n(x)\) centered at \(x=a\) for \(f(x)\), using the conditions on error \(E(x)\). In mathematical terms, we want to show \(g(x) = P_n(x)\).

Step 2: Expanding the Taylor Polynomial

The Taylor polynomial of degree \(n\) for the function \(f(x)\) around \(x=a\) is given by: \[ P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n. \]

Step 3: Defining the Error Function

The error function is defined as the difference between \(f(x)\) and \(g(x)\), i.e., \(E(x) = f(x) - g(x)\). For a polynomial \(g(x)\) approximating \(f(x)\), we want \(E(a) = 0\) and the limit of \(\frac{E(x)}{(x-a)^n}\) as \(x\) approaches \(a\) to be zero.

Step 4: Applying Condition i: \(E(a) = 0\)

Substituting \(x = a\) in the error function, we get \(E(a) = f(a) - g(a) = 0\). This implies that \(f(a) = g(a)\), so the constant term \(b_0 = f(a)\).

Step 5: Using Taylor Remainder Theorem

By the Taylor Remainder Theorem, the error term beyond degree \(n\) is negligible in the limit. We want the limit \(\lim_{x \to a} \frac{E(x)}{(x-a)^n} = 0\) to ensure that each derivative up to \(n\) matches in \(g(x)\) and \(P_n(x)\).

Step 6: Deriving Condition ii

For the limit condition to hold, the following must be true for \(g(x)\): - \(f'(a) = g'(a)\), - \(f''(a) = g''(a)\), - up to - \(f^{(n)}(a) = g^{(n)}(a)\).This ensures each corresponding coefficient of \(g(x)\) matches the Taylor coefficients, \(b_i = \frac{f^{(i)}(a)}{i!}\).

Step 7: Conclusion and Verification

Since \(g(x)\) matches all conditions of the Taylor polynomial, \(g(x) = P_n(x)\). It is therefore the unique polynomial of degree \(\leq n\) which satisfies both \(E(a) = 0\) and \(\lim_{x \to a} \frac{E(x)}{(x-a)^n} = 0\).

Key Concepts

Approximation theory

The concept of approximation theory involves finding simpler mathematical expressions that approximate more complex functions. In this exercise, the goal is to show that the Taylor polynomial is the best polynomial approximation of a function near a certain point. Taylor polynomials are powerful tools because they can mimic a given function with great accuracy within a specific range.

Imagine a smooth curve. We can use a polynomial, which is just a combination of simple terms, to trace the original curve closely near a specific point. This technique is valuable, as polynomials are generally easier to work with compared to many complex functions.

In this scenario, Taylor polynomials serve to approximate the function’s behavior around the point of interest (often labeled as 'a'). They evolve by using the function's value and its derivatives at that point. This makes them very suitable for scenarios where you need a reliable and concise representation of the behavior of functions.

Error analysis

Error analysis investigates how close our approximation is to the actual value. With Taylor polynomials, we want the error between the actual function and our polynomial to be as small as possible, especially near the chosen point of expansion.

In this exercise, the error function is defined as the difference, \( E(x) = f(x) - g(x) \), where \( f(x) \) is the original function and \( g(x) \) is the Taylor polynomial.

Two important conditions are applied to control this error:

• The error at the chosen point, \( a \), must be zero. This ensures that both the function and polynomial agree perfectly at this point.
• The error divided by \( (x-a)^n \) should approach zero as \( x \) approaches \( a \). This condition ensures that the error becomes negligible compared to the polynomial's degree.

These criteria help ensure that the polynomial closely matches the function not just at one point but in the neighborhood around it. The smaller the error function, the better our polynomial approximation is.

Differentiable functions

Differentiable functions are those that have derivatives. This property is crucial for Taylor polynomials because these polynomials rely on the derivatives of a function.

The Taylor polynomial uses derivatives up to the \( n^{th} \) degree. These derivatives provide a way of capturing the rates at which the function changes. In simpler terms, each derivative tells us how the function behaves:

• The first derivative (\( f'(a) \)) reflects the slope or the rate of change.
• The second derivative (\( f''(a) \)) tells us about the curvature, whether the graph curves upwards or downwards.
• Higher derivatives tell us more about the function's intricacies, like twists and turns.

This is why a differentiable function is necessary for forming a Taylor polynomial: each derivative provides essential information used to 'build' the polynomial so that it precisely approximates the original function. Without these derivatives, the polynomial cannot accurately follow the function's path.