Question

Find the Maclaurin polynomial of degree \(n\) for the given function. $$f(x)=\sin x, \quad n=8$$

Short answer

The Maclaurin polynomial of degree 8 for \(\sin x\) is: \[ P_8(x) = x - \frac{x^3}{6} + \frac{x^5}{120} - \frac{x^7}{5040} \].

Step-by-step solution

Understand the Maclaurin Series

The Maclaurin series is a Taylor series expansion of a function about 0, given by the formula \( f(x) = f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \frac{f'''(0)x^3}{3!} + \cdots \). For the Maclaurin polynomial of degree \(n\), we stop at the \(n\)-th derivative term.

Identify the Function and Needed Derivatives

For \( f(x) = \sin x \), we need to calculate the first few derivatives: - \( f(x) = \sin x \), - \( f'(x) = \cos x \), - \( f''(x) = -\sin x \), - \( f'''(x) = -\cos x \), - \( f^{(4)}(x) = \sin x \). Notice the pattern repeats every four derivatives.

Evaluate the Derivatives at Zero

Evaluate these derivatives at \(x = 0\):- \( f(0) = \sin 0 = 0 \),- \( f'(0) = \cos 0 = 1 \),- \( f''(0) = -\sin 0 = 0 \),- \( f'''(0) = -\cos 0 = -1 \),- \( f^{(4)}(0) = \sin 0 = 0 \). \Continue this process for derivatives up to the 8th degree.

Write the Polynomial Using Maclaurin Series Formula

Substitute the evaluated derivatives into the Maclaurin series formula:\[ f(x) = 0 + (1)x + \frac{0 \cdot x^2}{2!} + \frac{-1 \cdot x^3}{3!} + \frac{0 \cdot x^4}{4!} + \frac{1 \cdot x^5}{5!} + \frac{0 \cdot x^6}{6!} + \frac{-1 \cdot x^7}{7!} + \frac{0 \cdot x^8}{8!} \].

Simplify the Polynomial

Remove the terms multiplying to zero and write the non-zero terms:\[ P_8(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} \].

Conclude the Maclaurin Polynomial

The Maclaurin polynomial of degree 8 for \(\sin x\) is: \[ P_8(x) = x - \frac{x^3}{6} + \frac{x^5}{120} - \frac{x^7}{5040} \].

Key concepts

Taylor Series

The Taylor series is a way of representing a function as an infinite sum of its derivatives at a single point. It can be seen as a method to approximate functions using a series of polynomial terms. This concept is essential in calculus and mathematical analysis, offering a powerful tool to understand and approximate complex functions.
For any real-valued function, the Taylor series centered at a point \(a\) is expressed as:

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

A special case of the Taylor series is the Maclaurin series, where the expansion point is \(a = 0\). This simplifies our calculations, especially when working with functions like trigonometric functions, exponentials, and logarithms.

Sin Function

The sine function, denoted as \( \sin x \), is one of the fundamental trigonometric functions. It is periodic, with a period of \(2\pi\), and it describes the vertical component of a point on the unit circle as it rotates around the origin. In practical terms, it is used for modeling waves, oscillations, and circular motion, among other applications.

When dealing with its Taylor or Maclaurin series, a remarkable pattern emerges. The function \( \sin x \) becomes particularly interesting because of its derivatives. This cyclic nature of its derivatives makes expanding \( \sin x \) into a series relatively straightforward, following a recognizable pattern.

Derivatives

A derivative in mathematics signifies the rate of change of a function with respect to a variable. For the function \( f(x)=\sin x \), we can compute several derivatives, which form a repeating cycle:

• \( f'(x) = \cos x \)
• \( f''(x) = -\sin x \)
• \( f'''(x) = -\cos x \)
• \( f^{(4)}(x) = \sin x \)

This cyclic repetition every four derivatives is helpful when constructing a polynomial approximation using the Maclaurin series. Calculating derivatives at \(x = 0\) simplifies the process even further, since values like \( \sin 0 \) and \( \cos 0 \) are easy to evaluate.

Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients combined using addition, subtraction, multiplication, and non-negative integer exponents of variables. They are one of the simplest types of mathematical expressions and often appear in calculus, algebra, and applied mathematics.

When creating polynomial approximations for functions such as \( \sin x \) using Maclaurin series, we retain a specific number of terms, determined by the degree \(n\) of the polynomial. Each term in the series corresponds to a derivative of the function evaluated at zero, with increasing powers of \(x\) divided by the factorial of the power's degree. For example, the Maclaurin polynomial of degree 8 for \( \sin x \) is:

• \( P_8(x) = x - \frac{x^3}{6} + \frac{x^5}{120} - \frac{x^7}{5040} \)

This polynomial provides an approximation of the sine function, which becomes more accurate as more terms are included.