Taylor Polynomial

What Is a Taylor Polynomial?

A Taylor polynomial is a special type of polynomial used to approximate more complicated functions near a specific point. Imagine you have a very curvy and complex road, but you want to describe a small piece of it using simple, straight sections of track. A Taylor polynomial does something similar for functions. It builds a simple, predictable polynomial function (like $y=2x^2-3x+5$) that acts just like a more complex function (like $y=\cos (x)$) as long as you stay close to a chosen starting point.

The core idea is to "match" the behavior of the complex function at that specific point. We start by matching the function's value. Then, for a better approximation, we match its slope (how steep it is). For an even better fit, we match its curvature (how much it bends). Each time we add a new term to our polynomial, we match another level of behavior, making our approximation more and more accurate around our chosen point.

These approximations are incredibly useful in science, engineering, and computer science because they allow us to replace difficult calculations with simpler arithmetic that computers can perform at lightning speed.

Why Do We Even Need to Approximate Functions?

It might seem strange to replace a perfectly good function with an approximation, but there are very important reasons to do so. Many functions that are essential in the real world are surprisingly difficult to calculate directly.

Consider functions like $f(x)=\sin (x)$, $g(x)=\ln (x)$, or $h(x)=e^x$. There is no simple arithmetic recipe (using just addition, subtraction, multiplication, and division) to find that $\sin ({15}^{\circ })$ is exactly $0.258819...$. Your calculator doesn't have a giant list of every possible value. Instead, it uses a very accurate polynomial approximation to compute the value on the spot.

Here are a few key reasons why approximations are critical:

• Speed: Polynomials only involve basic arithmetic. Computers are incredibly fast at these operations, allowing for rapid calculations in everything from video games to weather forecasting.
• Simplicity: In many physics and engineering problems, using the exact function makes the equations impossible to solve. Replacing it with a Taylor polynomial can make the problem manageable while still giving a very accurate answer.
• Predicting Behavior: Sometimes we only need to understand how a function behaves near a certain point. A simple linear or quadratic approximation can tell us if the function is increasing, decreasing, or at a peak or valley, which is often all the information we need.

How Do We Build an Approximation by 'Matching'?

The magic of a Taylor polynomial is in how it systematically matches a function's properties at a specific point, which we'll call $x=a$. Let's call our original, complicated function $f(x)$ and our polynomial approximation $P(x)$. We want to make $P(x)$ look and act like $f(x)$ near $x=a$.

We build the approximation in layers:

1. Level 0 Match (Value): The most basic requirement is that our approximation has the same value as the function at our chosen point. We force $P(a)=f(a)$. This is called a 0th-degree approximation, which is just a constant. It's not very good, but it's a start!
2. Level 1 Match (Slope): To improve the fit, we should also match the steepness, or slope, of the function at $x=a$. A line that has the same value and the same slope as a curve at a point is called the tangent line. This gives us a 1st-degree (linear) approximation, which is much better.
3. Level 2 Match (Curvature): A line is straight, but our original function is probably curved. The next step is to match the bend, or curvature, of the function at $x=a$. This requires adding an $x^2$ term, giving us a 2nd-degree (quadratic) approximation. This parabola will "hug" the original function's curve much more closely than a straight line could.

By continuing this process—matching higher and higher orders of change—we can create polynomials of any degree that become increasingly accurate approximations of the original function near our point $a$.

How to Build a First-Degree Taylor Polynomial (A Linear Approximation)

A first-degree Taylor polynomial is simply a straight line that best approximates a function at a point. This line is formally known as the tangent line. To build it, we need two pieces of information about our function $f(x)$ at our chosen center point $x=a$:

1. The value of the function: $f(a)$.
2. The slope of the function: Let's call this $m$. In calculus, this is written as $f^{\prime }(a)$.

The formula for the linear approximation, which we'll call $P_1(x)$, is based on the point-slope form of a line:

This formula guarantees that at $x=a$, the value of $P_1(a)$ is $f(a)$ and the slope of $P_1(x)$ is $m$, perfectly matching our function's value and slope at that one point.

How to Build a Second-Degree Taylor Polynomial (A Quadratic Approximation)

A linear approximation is good, but we can do better by matching the curvature of the function. This requires a second-degree polynomial, which is a parabola. To build it, we start with our linear approximation and add one more term. We need three pieces of information about our function $f(x)$ at $x=a$:

1. The value: $f(a)$.
2. The slope: $m_1$ (which is $f^{\prime }(a)$ in calculus).
3. The curvature factor: $c_2$. This value tells us how the function is bending. In calculus, it's calculated as $\frac{f^{″}(a)}{2}$, where $f^{″}(a)$ represents the concavity.

The formula for the quadratic approximation, $P_2(x)$, is:

This parabola has the same value, same slope, AND same curvature as our function $f(x)$ at the point $x=a$, making it a much closer fit.

What Is the General Formula for a Taylor Polynomial?

We can continue adding terms to our polynomial to match more and more properties of the function, making the approximation better and better. The Taylor polynomial of degree $n$ for a function $f(x)$ centered at $x=a$ is given by a general formula. It looks a bit intimidating, but it's just an extension of the pattern we've already seen.

The formula requires values from calculus called derivatives. The first derivative, $f^{\prime }(a)$, is the slope. The second derivative, $f^{″}(a)$, is related to curvature, and so on. The general formula is:

Let's break down the new parts:

• $f^{(n)}(a)$: This is the $n$-th derivative of the function evaluated at $a$. It represents the $n$-th level of change in the function. For our purposes, these values will be given.
• $n!$: This is "n factorial," which means $n\times (n-1)\times \cdots \times 2\times 1$. For example, $3!=3\times 2\times 1=6$ and $4!=4\times 3\times 2\times 1=24$. This factorial term is necessary to make sure the properties match up correctly.

What Are Common Mistakes to Avoid?

When first learning about Taylor polynomials, it's easy to make a few common errors. Being aware of them can help you get the right answer more consistently.

• Forgetting the $(x-a)$ Term: A frequent mistake is to write the coefficients correctly but forget to multiply them by the powers of $(x-a)$. Remember, it's not just $f^{\prime }(a)$, it's $f^{\prime }(a)(x-a)$.
• Using $x$ Instead of $a$: The coefficients $f(a),f^{\prime }(a),f^{″}(a)$, etc., are constants calculated at the specific center point $a$. Don't plug the variable $x$ into the derivative terms.
• Forgetting the Factorials: The denominators $2!,3!,4!$, etc., are crucial. Forgetting them, especially for degrees higher than 1, will give you the wrong polynomial. Remember that $0!=1$ and $1!=1$, so they are often omitted in the first two terms.
• Assuming the Approximation is Global: A Taylor polynomial is an excellent approximation near its center $a$. The farther you get from $a$, the more the polynomial will drift away from the original function. It is a local, not a global, approximation.
• Using Degrees for Trig Functions: When creating or using Taylor polynomials for trigonometric functions like $\sin (x)$ or $\cos (x)$, the variable $x$ must always be in radians. The formulas for their derivatives are only valid in radians.

Quick Summary and Reference

A Taylor polynomial is a tool for approximating a complex function $f(x)$ near a specific point $x=a$ using a simpler polynomial function. The key is to match the function's properties (value, slope, curvature, etc.) at that point. The more properties you match, the higher the degree of the polynomial and the better the approximation.

Here is a summary of the first few levels of approximation:

The Maclaurin Polynomial is simply a special case of the Taylor polynomial where the center point is $a=0$. The examples for $\cos (x)$ and $e^x$ in this lesson were both Maclaurin polynomials.