Question

An elementary function is approximated by a polynomial. In your own words, describe what is meant by saying that the polynomial is expanded about \(c\) or centered at \(c\)

Short answer

In mathematical terms, when a polynomial is expanded about or centered at a point \(c\), it's written in relation to that point in the form of powers of \((x-c)\). This is essentially an approximation of the function, representing how it behaves around the point \(c\), and is often applied in the form of a Taylor series (or Maclaurin series for \(c = 0\)). This point \(c\) is where the approximation is most accurate.

Step-by-step solution

Understanding the Concept of Centered At or Expanded About

In mathematics, a polynomial is said to be expanded about or centered at a number \(c\) if it is written in terms of powers of \((x - c)\). This essentially means that the polynomial is being expressed in relation to the point \(c\). This is often related to Taylor series, where function is approximated in the vicinity of a specific point \(c\). A Taylor series is typically represented as \(f(x) = f(c) + f'(c) \cdot (x - c) + f''(c) \cdot (x - c)^2 / 2! + ... \)

Polynomial Centered at Zero

In case of \(c = 0\), the approximation has the name of Maclaurin series, which is a special case of the Taylor series. The Taylor series then appears as \(f(x) = f(0) + f'(0) \cdot x + f''(0) \cdot (x^2) / 2! + ... \) In these instances, 'centered at \(c\)' refers to the point where the approximation is most accurate.

Key concepts

Polynomial Approximation

Imagine you're trying to sketch the shape of a complex curve, but you only have a ruler—no fancy curves allowed. Polynomial approximation is the mathematical equivalent of using straight lines (or simple curves you can make with a ruler) to mimic the curve as closely as possible.

Think of polynomials as multi-talented actors who can impersonate a variety of functions by tweaking their coefficients and degrees. When a polynomial plays the role of another function, it’s striving to match the function's value (and often its derivatives) at certain key points. The better it matches, the more convincingly it “approximates” the true function.

In technical terms, a polynomial approximation constructs a polynomial that has similar behavior to the function we're interested in, over a particular interval. The quality of the approximation depends on the degree of the polynomial—the higher the degree, the more points of contact, and the better the mimicry. However, with increased complexity comes increased computational cost, so finding a balance between accuracy and simplicity is key.

Maclaurin Series

Let's put the spotlight on a 'local celebrity' in the world of polynomials—the Maclaurin series. Think of it as a Taylor series that loves the neighborhood around zero so much it never wanders away.

This series is the go-to approximation when we're dealing with functions around the point zero. It’s like taking a Taylor series and telling it, 'Today, we’re only exploring your behavior right where you live—at zero'. Consequently, the Maclaurin series takes the form \( f(x) = f(0) + f'(0) \cdot x + f''(0) \cdot \frac{x^2}{2!} + ... \) and so on.

The Maclaurin series is particularly cherished because it simplifies calculations. Functions often have known values and derivatives at zero, making it easier to construct the series. In other words, if you want to predict a function's behavior without straying far from home, the Maclaurin series is your trusty companion.

Centered at c

Now imagine you're an artist who wants to paint a landscape. You'd set up your easel at a point where you have the best view. In the realm of polynomials, the point \(c\) is where we set up our mathematical 'easel'.

When we say a polynomial is centered at \(c\), we're basically saying we have chosen the best vantage point—\(c\)—from which to view and approximate our function. Here, the Taylor series showcases its adaptability, allowing us to expand it not just at zero, but at any point \(c\) on the function's domain. The expanded polynomial then takes the form \(f(x) = f(c) + f'(c) \cdot (x - c) + f''(c) \cdot \frac{(x - c)^2}{2!} + ...\)

This is extremely handy because some functions behave more 'nicely' around certain points than others. By choosing a suitable \(c\), we tailor our approximation to where the function's action is—capturing the essence of the function's behavior in that neighborhood. So the choice of \(c\) is akin to choosing the perfect angle for a photograph—it sets the stage for the best possible representation.