Question

Suppose you use a Taylor polynomial with \(n=2\) centered at 0 to approximate a function \(f\). What matching conditions are satisfied by the polynomial?

Short answer

#Short Answer# When a Taylor polynomial is of degree 2 (n=2) and centered at 0, it takes the form \(P_2(x) = f(0) + f'(0)x + \frac{1}{2}f''(0)x^2\). The matching conditions satisfied by this polynomial are: it matches the function's value at 0 (\(P_2(0)=f(0)\)), its first derivative at 0 (\(P_2'(0)=f'(0)\)), and its second derivative at 0 (\(P_2''(0)=f''(0)\)). These matching conditions allow the polynomial to approximate the function and its first two derivatives at the center 0.

Step-by-step solution

Define the Taylor polynomial

A Taylor polynomial is an approximation of a function using a polynomial that matches the function's derivatives at a certain point, called the center. For a function \(f\), the Taylor polynomial of degree \(n\) centered at \(\displaystyle a\) is given by: $$ P_n(x)=f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 +\cdots +\frac{f^n(a)}{n!}(x-a)^n $$

Set the given values

In this case, we have \(\displaystyle n=2\) and the center \(\displaystyle a=0\). Therefore, our Taylor polynomial should be of degree 2 and centered at 0.

Identify the Taylor polynomial for n=2 and centered at 0

Substituting the given values, we get the Taylor polynomial as: $$ P_2(x)=f(0) + \frac{f'(0)}{1!}x + \frac{f''(0)}{2!}x^2 $$

Identify the matching conditions

The matching conditions refer to the values of the function and its derivatives at the center that the polynomial approximates. In this case, the following matching conditions are satisfied by the polynomial: 1. The value of the function at the center: \(\displaystyle P_2(0)=f(0)\) 2. The first derivative of the function at the center: \(\displaystyle P_2'(0)=f'(0)\) 3. The second derivative of the function at the center: \(\displaystyle P_2''(0)=f''(0)\) These matching conditions ensure that the polynomial approximates the function and its first two derivatives at the center 0, thus providing a good local approximation.

Key concepts

Function Approximation

When working with complicated functions, it is often useful to approximate them with simpler expressions, especially for calculations or predictions close to a particular point. This is where Taylor polynomials come in handy, as they provide a polynomial approximation of a given function. The basic idea is to construct a polynomial that closely follows the behavior of the function near a specified point, known as the center. By matching not just the value of the function but also its derivatives at that point, a Taylor polynomial offers a highly accurate way to imitate the function's behavior locally.

In the exercise, a Taylor polynomial of degree 2 is used to approximate a function centered at 0. This means that the polynomial will mimic the function's value and its first two derivatives at this point. The higher the degree, the closer the approximation is at the center, but the complexity of the polynomial also increases. Taylor polynomials are particularly powerful in various fields like physics and engineering, providing manageable expressions for otherwise complex functions.

Matching Conditions

The concept of matching conditions is essential in using Taylor polynomials effectively. When we say that a polynomial matches a function, we are referring to certain key conditions at the center of approximation. These include the values of the function and its derivatives.

For this exercise with a Taylor polynomial of degree 2 centered at 0, the matching conditions are:

• The polynomial's value at the center equals the function's value: \(P_2(0) = f(0)\).
• The first derivative of the polynomial at the center equals the function's first derivative: \(P_2'(0) = f'(0)\).
• The second derivative of the polynomial at the center equals the function's second derivative: \(P_2''(0) = f''(0)\).

These conditions ensure that the polynomial not only touches the function at the center but also follows its slope and curvature closely, making it an excellent local approximation.

Derivatives

Derivatives play a central role in Taylor polynomials as they provide information about a function's rate of change, slope, and curvature. In constructing a Taylor polynomial, derivatives are used to match the behavior of the function at a specific point. Known as matching conditions, these ensure that the polynomial accurately reflects the function's characteristics.

In our exercise, the derivatives at the center (0) defined the coefficients of the Taylor polynomial of degree 2. Specifically, the first derivative informs us about the slope of the tangent line at the center, and the second derivative describes the curvature. These derivatives help create a polynomial that isn't just a loose fit but aligns with the function's behavior, making it meaningful for local approximation.

• The 0th derivative (just the function's value) describes the starting point of the polynomial.
• The 1st derivative provides the slope, contributing the linear term.
• The 2nd derivative describes the curvature, contributing the quadratic term.

Together, these derivatives ensure that the Taylor polynomial effectively approximates the function.