Problem 27 - Calculus Single Variable

Question

In each of Exercises 23-34, derive the Maclaurin series of the given function \(f(x)\) by using a known Maclaurin series. $$ f(x)=5 \cos (2 x)-4 \sin (3 x) $$

Step-by-Step Solution

Answer

The Maclaurin series for \( f(x) = 5 \cos(2x) - 4 \sin(3x) \) starts as: \( 5 - 12x - 10x^2 + \frac{108x^3}{6} - \frac{80x^4}{24} + \cdots \)

Step 1: Identify Known Maclaurin Series

Recall the Maclaurin series for the cosine function: \( \cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \) and for the sine function: \( \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \). These will be used to express \( \cos(2x) \) and \( \sin(3x) \).

Step 2: Substitute Argument into Known Series

Substitute \(2x\) into the cosine series and \(3x\) into the sine series. This gives: \( \cos(2x) = 1 - \frac{(2x)^2}{2!} + \frac{(2x)^4}{4!} - \cdots \) and \( \sin(3x) = (3x) - \frac{(3x)^3}{3!} + \frac{(3x)^5}{5!} - \cdots \).

Step 3: Simplify the Series

Simplify these series: \( \cos(2x) = 1 - \frac{4x^2}{2!} + \frac{16x^4}{4!} - \cdots \) and \( \sin(3x) = 3x - \frac{27x^3}{3!} + \cdots \).

Step 4: Apply Function Coefficients

Multiply the series \( \cos(2x)\) by 5 and the series \( \sin(3x)\) by 4: \(5 \cos(2x) = 5 - 10x^2 + \frac{80x^4}{4!} - \cdots \) and \(4 \sin(3x) = 12x - \frac{108x^3}{3!} + \cdots \).

Step 5: Construct the Maclaurin Series for f(x)

Combine the adjusted series: \( f(x) = 5 - 10x^2 + \frac{80x^4}{4!} - \cdots - (12x - \frac{108x^3}{3!} + \cdots) \). Simplify by combining like terms.

Step 6: Simplify the Derived Series

Combine like terms to clean up the expression: \(f(x) = 5 - 12x - 10x^2 + \frac{108x^3}{3!} - \frac{80x^4}{4!} + \cdots \). Continue this pattern for higher-order terms if needed.

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos(x) \), is a fundamental trigonometric function that is used to represent the horizontal component of the motion of a point moving around a circle. It is a periodic function with a period of \( 2\pi \), meaning that \( \cos(x) = \cos(x + 2\pi) \) for any value of \( x \). In the Maclaurin series for the cosine function, we express \( \cos(x) \) as an infinite power series. The general formula for this series is:

• \( \cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \)

This series is derived from the Taylor series expansion of \( \cos(x) \) centered at zero. By using only even-powered terms, this series showcases the nature of how the cosine function behaves symmetrically around the y-axis.

Sine Function

The sine function, denoted as \( \sin(x) \), is another core trigonometric function that describes the vertical component of the circular motion. Similar to cosine, sine is also periodic with the same period, \( 2\pi \). The sine function, however, starts at zero and reaches its maximum at \( \frac{\pi}{2} \). Yeah, it has that smooth wave to it. Its Maclaurin series representation, like cosine, uses an infinite sum of terms:

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

This series involves only odd-powered terms and is generated from the Taylor series about zero. The alternating signs in both sine and cosine series reflect the oscillatory nature of these functions.

Power Series

A power series is an infinite sum of terms in the form \( a_n x^n \), where \( n \) is a non-negative integer, and \( a_n \) denotes the coefficient of each term. It essentially forms the foundation of methods like the Maclaurin and Taylor series, used in series expansion for functions.
To see this in action, look at the Maclaurin series for \( \cos(x) \) and \( \sin(x) \). Here, each function is written as an infinite sum of polynomial terms, tailored to approximate the function values:

• They converge to the function values as the number of terms increases.
• The more terms you include, the closer you get to the actual function value.

The beauty of power series lies in their ability to turn complex functions, like trigonometric ones, into manageable polynomials for specific intervals of \( x \).

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in mathematics, especially in areas dealing with angles and waves. These functions help us capture and describe cyclical phenomena, such as tides, sound waves, and even light.
While there are six principal trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—the most widely used are sine and cosine due to their simplicity and frequent occurrence in mathematical problems.

• Sine and cosine functions can easily model any periodic wave.
• They have applications in physics, engineering, signal processing, and many other fields.

By representing these functions as series, like in the Maclaurin series, we can calculate function values for all kinds of real-number inputs using polynomial approximations. This extends their usability in numerical methods and simulations.