Problem 23 - Essential Calculus

Question

Find the Maclaurin series for the function. (Use the table of power series for elementary functions.) $$ g(x)=\sin 3 x $$

Step-by-Step Solution

Answer

The Maclaurin series for \(g(x)=\sin 3x\) is \(\sin 3x = 3x - \frac{27x^3}{3!} + \frac{729x^5}{5!} - \frac{19683x^7}{7!} + \frac{531441x^9}{9!} - \cdots\)

Step 1: Recognize basic Maclaurin series for sine

The Maclaurin series for \(\sin x\) is \(\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \frac{x^9}{9!} - \cdots\). This is an infinite series, alternately subtracting and adding terms of higher powers of x divided by the factorial of the exponent.

Step 2: Substitute x for 3x in the basic Maclaurin series for sine

To find the Maclaurin series for \(\sin 3x\), replace every instance of x in the Maclaurin series for \(\sin x\) with 3x. Following this substitution, the series becomes \(\sin 3x = 3x - \frac{(3x)^3}{3!} + \frac{(3x)^5}{5!} - \frac{(3x)^7}{7!} + \frac{(3x)^9}{9!} - \cdots\). This substitution simply applies the function transformation rule to the power series.

Step 3: Simplify the series

Further simplify the series to make it more readable. The series becomes \(\sin 3x = 3x - \frac{27x^3}{3!} + \frac{729x^5}{5!} - \frac{19683x^7}{7!} + \frac{531441x^9}{9!} - \cdots\)

Key Concepts

Power Series

A power series is a type of infinite series that involves powers of a variable, typically denoted as \( x \). Each term in a power series is of the form \( a_n x^n \), where \( a_n \) is a coefficient and \( n \) is a non-negative integer. When these terms are summed together, they create a function expressed as \[ f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \,\ldots \]Power series are incredibly useful because they allow complicated functions to be expressed in terms of polynomial-like expressions, which are often easier to work with.

• The series can converge to a function value for certain values of \( x \).
• They are used to approximate functions, especially in calculus.
• Simple and classical functions such as exponentials, sines, and cosines have well-known power series expansions.

In the context of the given problem, the power series helps in creating a polynomial form for the trigonometric function \( \sin 3x \), making it easier to perform calculations or approximations over a specified range of \( x \).

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are fundamental in mathematics, especially in geometry, calculus, and engineering. These functions help relate angles to side lengths of a triangle in a circle with radius 1.

• \( \sin x \) and \( \cos x \) are periodic functions, meaning their values repeat at regular intervals.
• They are essential in describing oscillatory motion, like waves.

For the function \( g(x) = \sin 3x \), we're essentially dealing with a scaled sine wave.
By scaling the input \( x \) by a factor of 3, the sine wave's frequency increases, resulting in more oscillations over the same interval of \( x \). When converting such functions into a power series, we often turn to the Maclaurin series, which provides a way to express trigonometric functions as polynomial approximations, making complex problems more manageable.
The process typically involves substituting values and using known series for basic trigonometric functions.

Series Expansion

Series expansion is a technique used to represent functions as sums of terms for approximation or analysis. The Maclaurin series, a type of series expansion, expresses a function as an infinite sum of its derivatives at zero.This series is given by:\[ f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots \] In the exercise provided, finding the Maclaurin series for \( \sin 3x \) involves substituting \( x \) with \( 3x \) in the standard Maclaurin series for \( \sin x \). This substitution helps adapt the standard series to represent \( \sin 3x \), which is effectively a scaled version of \( \sin x \).The process includes:

• Identifying the basic series you are starting with, in this case, the series for \( \sin x \).
• Performing the substitution to reflect the function's scaling factor.
• Simplifying the resulting terms for clarity.

This methodical approach makes it easier for students to approximate trigonometric functions across different scenarios.