Question

What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than \(10^{-3}\) ? (The answer depends on your choice of a center.) $$\sin 0.2$$

Short answer

Answer: The minimum order of the Taylor polynomial is 3.

Step-by-step solution

Taylor series of the sine function

Recall the Taylor series for the sine function, centered at \(0\): $$\sin x = \sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)!}x^{2n+1} = x - \frac{1}{3!}x^{3} + \frac{1}{5!}x^{5} - \cdots$$

Remainder and error bound using Taylor's Remainder Theorem

Taylor's Remainder Theorem states that for a function \(f\) with sufficiently many continuous derivatives in the considered interval, the error of approximating \(f(x)\) with an \(n\)-th degree Taylor polynomial is given by: $$R_n(x) = f^{(n+1)}(c) \frac{(x-a)^{n+1}}{(n+1)!}$$ where \(c\) is a number between \(x\) and \(a\), the center of the expansion. Let's use Taylor's Remainder Theorem to find an error bound on our approximation of \(\sin(0.2)\), setting \(x = 0.2\) and \(a = 0\).

Find derivatives and error bound for each term

Since we are dealing with sine function whose derivatives are sine and cosine repeatedly, let's calculate the bound of the first few derivatives for \(x \in [0, 0.2]\), and find the term corresponding to an error bound of \(10^{-3}\). Notice that, on the interval \([0, 0.2]\), the maximum value of the sine function and its odd-order derivatives will be less than or equal to \(0.2\), and the maximum value of the cosine function and its even-order derivatives will be less than or equal to \(1\). Now, we will find an error bound for each term in the expansion: 1. \(n=0\): The first term is \(x\), and the first derivative is \(1\). The error bound is \(|R_0(0.2)| \leq \frac{1}{2!}(0.2)^2 = 0.02\) 2. \(n=2\): The third term is \(\frac{1}{3!}x^3\), and the third derivative is \(-x\). The error bound is \(|R_2(0.2)| \leq \frac{0.2}{4!}(0.2)^4 = 0.00002667\) Since the error bound for the third term is less than \(10^{-3}\), a minimum order of the Taylor polynomial required to approximate \(\sin 0.2\) with an absolute error no greater than \(10^{-3}\) is \(3\).

Key concepts

Taylor Polynomial

The Taylor Polynomial is an essential mathematical tool that helps to approximate functions with a polynomial expression that has a finite number of terms. Imagine you are trying to simulate a complex wave, like a sine wave, using just algebraic expressions. This is where Taylor polynomials come into play. They allow you to express a function in terms of a series expansion about a specific point, known as the center, which in our example is the origin, 0.
For sine, the Taylor polynomial series looks something like this:

• The first term, when centered at 0, is just the value of the function. For sine, it's simply \( x \).
• Subsequent terms involve higher powers of x, divided by factorial terms, such as \( -\frac{1}{3!}x^3 + \frac{1}{5!}x^5 \), and so on.
• The pattern involves alternating signs and each term contributes less as the power of x increases.

This truncation makes calculations more manageable, especially for evaluating functions like sine near their center. The real magic happens when we examine how accurate such an approximation can be, leading us to the concept of "Error Bound."

Error Bound

An "Error Bound" is essentially a safety net for mathematicians. It tells us the maximum possible error when approximating a function with a Taylor polynomial. This is crucial because when you make approximations, you also want to know how close you are to the actual function.
Taylor's Remainder Theorem gives us a formula to estimate the error—also known as the remainder—which can arise from truncating the series:

• The error is expressed as \(R_n(x) = f^{(n+1)}(c) \frac{(x-a)^{n+1}}{(n+1)!}\) where c is some value between the center a and x you're approximating.
• It's the next term in the polynomial series that helps gauge this error, hence the importance of derivatives of the function, like sine or cosine, and their values within the interval considered.

More derivatives typically mean smaller and more precise error bounds, allowing you to say with confidence that your approximate value isn't "off" by more than a specific amount. This is particularly useful in programming algorithms and in scenarios where mathematical precision is required.
With an appropriate error bound, you can determine how many terms you genuinely need to keep while making approximations, and this is how we end up choosing a polynomial of a certain degree when approximating something like \( \sin(0.2) \).

Sine Function

The sine function is a fundamental aspect of trigonometry, waving its way across many applications in physics and engineering. Defined elegantly as the ratio of the opposite over the hypotenuse in a right triangle, it also transcends into being a periodic function with a wave-like graph.
In mathematical expressions, \( \sin(x) \) is often used, and becomes crucial when periodic motion needs to be modeled, such as waves or oscillations. The sine function is continuous and differentiable, meaning it can be expanded into a Taylor series, which gives us a powerful tool for approximation.

• When centered around zero, its Taylor series starts as \( x - \frac{1}{3!}x^3 + \frac{1}{5!}x^5 - \cdots \).
• This series helps mimic the actual shape of the sine curve, especially around small x values, where its characteristic wave shape is less pronounced.

Utilizing a Taylor series for \( \sin(0.2) \) allows mathematicians to simplify complex calculations by focusing on just a few terms while maintaining reasonable accuracy through carefully calculated error bounds. By understanding the inherent pattern and repetitive nature of the sine function and how closely it can be approximated with polynomial expressions, we can more easily perform tasks ranging from simple mathematics to intricate signal processing.