Problem 27 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Compute the sixth degree Taylor polynomial generated by \(\sin x\) about \(x=\pi\).

Step-by-Step Solution

Answer

So, the sixth degree Taylor polynomial for \(\sin x\) centered at \(x =\pi\) is \[0 + (x - \pi) - \dfrac{{(x - \pi)^3}}{3!} + \dfrac{{(x - \pi)^5}}{5!}\]

Step 1: Determine the general formula for the Taylor series of \(\sin x\)

The general formula for the Taylor series of \(\sin x\) about \(x = a\) is given by \[ \sin a + \sum_{n = 1}^\infty (-1)^n \dfrac{{(x - a)^{2n + 1}} \cos a}{(2n + 1)!}\] for \(n = 0, 1, 2, 3, ...\).

Step 2: Substitute \(x = \pi\)

Now, substitute \(x = \pi\) in the Taylor equation for \(\sin x\) to obtain the Taylor polynomial. Here, \(a=\pi\), so the Taylor series simplifies to \[\sin \pi + \sum_{n = 1}^\infty (-1)^n\dfrac{{(x - \pi)^{2n + 1}} \cos \pi}{(2n + 1)!}\]. Given \(\sin \pi = 0\) and \(\cos \pi = -1\), it simplifies further to \[\sum_{n = 1}^\infty (-1)^{n+1} \dfrac{{(x - \pi)^{2n + 1}}}{(2n + 1)!}\].

Step 3: Write the terms of the Taylor polynomial up to degree six

We want only the terms up to degree six. The terms beyond this degree won’t contribute to the sixth degree Taylor polynomial, and thus they are discarded. After substituting the values we found for the coefficients into the Taylor series and simplifying, we find the following terms up to degree six: \[0 + (x - \pi) - \dfrac{{(x - \pi)^3}}{3!} + \dfrac{{(x - \pi)^5}}{5!}\]. Notice that even powers of \((x - \pi)\) were omitted because the coefficients for those terms were zero in the Taylor series.

Key Concepts

Taylor series

The Taylor series is an essential concept in calculus used to approximate functions with a sum of polynomial terms. These terms are derived based on the derivatives of a function at a particular point. Imagine a function that you want to represent as a sum of simpler components. The Taylor series does precisely this by expressing a function as an infinite sum of terms calculated from the values of its derivatives at a single point.

The general formula for a Taylor series of a function about a point \(a\) is:

• \(f(a) + (x-a)f'(a) + \frac{(x-a)^2}{2!}f''(a) + \frac{(x-a)^3}{3!}f'''(a) + \cdots\)
• Where \(f'(a)\), \(f''(a)\), \(f'''(a)\), etc., are the first, second, third, and so on, derivatives of \(f\) at \(a\).

The beauty of the Taylor series is that it allows us to approximate complex functions around any point with potentially very high accuracy, as we include more terms. However, in practical scenarios, we often limit the number of terms, leading to Taylor polynomials of certain degrees.

Maclaurin series

The Maclaurin series is a special case of the Taylor series where the point of expansion, \(a\), is 0. This simplification helps in expressing functions around the origin in a series form. Many functions that are difficult to manipulate analytically can become much simpler when expressed as a Maclaurin series.

Here's the general formula for a Maclaurin series:

• \(f(0) + x f'(0) + \frac{x^2}{2!}f''(0) + \frac{x^3}{3!}f'''(0) + \cdots\)

An advantage of Maclaurin series over Taylor series is the reduced complexity in calculation, which makes it immensely useful for theoretical work and computational purposes. Many standard functions in mathematics, like \(\exp(x)\), \(\sin(x)\), and \(\cos(x)\), have well-known Maclaurin series expansions. For functions centered at zero, the Maclaurin series provides a neat and convenient tool for approximation.

sin function

The sine function, \(\sin(x)\), is a fundamental trigonometric function that describes the ratio of the opposite side to the hypotenuse in a right triangle. It's one of the primary periodic functions in trigonometry, repeating every \(2\pi\) radians. The sine function is defined for all real numbers and is particularly central to understanding oscillatory behaviors, such as waves.

When expanded using Taylor or Maclaurin series, the sine function can be approximated around a point as a sum of polynomial terms. Expanding it about zero gives us the well-known Maclaurin series:

• \(\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\)

These series let us work with \(\sin(x)\) in a polynomial form, which is easier to manipulate in many mathematical contexts. This approximation is particularly useful in calculators and computer algorithms to evaluate the sine of an angle closely.

degree of polynomial

In any polynomial, the degree refers to the highest power of the variable in its terms. For example, in the polynomial \(x^5 - 4x^3 + 3x - 1\), the degree is 5 because the highest exponent is 5. The concept of degree is crucial when working with Taylor polynomials, as it directly influences the polynomial's accuracy in approximating a function.

When we create a Taylor polynomial for a function, we choose up to which degree we want to approximate the function. The higher the degree, the more accurate the approximation typically is, assuming the function behaves well across its domain. However, being a polynomial, there are diminishing returns in approximation quality after a certain degree, depending on the function's behavior.

In practical applications, like the Taylor polynomial for \(\sin(x)\), choosing a degree—such as the sixth degree as in the original exercise—balances accuracy with computational simplicity. Each term and degree bring the polynomial closer to the function's actual behavior, within the vicinity of the expansion point.