Problem 17 - Thomas Calculus

Question

Find the Maclaurin series for the functions. 7 \(\cos (-x)\)

Step-by-Step Solution

Answer

The Maclaurin series for the function is: \(7 - \frac{7x^2}{2} + \frac{7x^4}{24} - \frac{7x^6}{720} + \cdots\).

Step 1: Identify the function and its transformations

The given function is \(7 \cos(-x)\). Since cosine is an even function, we know that \(\cos(-x) = \cos(x)\). So, the function simplifies to \(7 \cos(x)\).

Step 2: Recall Maclaurin series for \(\cos(x)\)

The Maclaurin series for \(\cos(x)\) is given by the expansion: \[ \cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \]

Step 3: Multiply the series by 7

Since the function is \(7 \cos(x)\), we need to multiply each term of the Maclaurin series for \(\cos(x)\) by 7. Thus, the Maclaurin series for \(7 \cos(x)\) is: \[ 7 \cos(x) = 7 \left( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \right) \]

Step 4: Simplify the series

The series becomes: \[ 7 - \frac{7x^2}{2} + \frac{7x^4}{24} - \frac{7x^6}{720} + \cdots \] Here, each term of the cosine series is multiplied by 7.

Step 5: Write the general term of the series

Observing the pattern, the general term of the series is \((-1)^n \frac{7x^{2n}}{(2n)!}\), where \(n\) starts from 0 and increases by 1 for each subsequent term.

Key Concepts

Even Function

An even function is a type of mathematical function where the function fulfills the condition:

• For every value of \(x\), the function satisfies \(f(-x) = f(x)\).

In simpler terms, the graph of an even function is symmetrical about the vertical y-axis.
It means that if you fold the graph along the y-axis, both sides will match perfectly. A classic example of an even function is the cosine function, \(\cos(x)\).
For \(\cos(x)\), \(\cos(-x) = \cos(x)\), which illustrates this symmetry property clearly. This property is often useful in simplifying calculations and solving mathematical problems.

Series Expansion

Series expansion is a powerful tool used in mathematics to express functions as the sum of simpler terms. This technique is particularly helpful when you need to approximate complex functions using polynomials.
The simpler terms, often called the terms of the series, are usually derived from the derivatives of the function at a single point. For instance, the Maclaurin series is a type of series expansion that represents functions as a sum of an infinite series of terms calculated from the values of the function's derivatives at zero.

• The Maclaurin series can approximate a variety of functions, including cosine \(f(x)\) by \(\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots\).

By approximating a function through its series expansion, mathematicians are able to analyze its behavior near a particular point with greater ease.

Cosine Function

The cosine function, denoted as \(\cos(x)\), is one of the fundamental trigonometric functions. It comes from the projection of a unit circle along the x-axis.
Some attributes of the cosine function include:

• It is periodic with period \(2\pi\), meaning it repeats its values every \(2\pi\) radians.
• It is an even function, implying \(\cos(-x) = \cos(x)\).

In mathematics, the cosine function is essential because it smoothly oscillates between -1 and 1.
This oscillation is key in modeling periodic phenomena such as sound waves, light waves, and alternating current in circuits. For functions like \(7\cos(x)\), the Maclaurin series expansion can approximate its behavior around zero by expanding it into a power series.

General Term of Series

The general term of a series describes the rule for finding any term in the sequence of products. In the context of a Maclaurin series expansion for a function, like \(7\cos(x)\), identifying the general term is crucial for predicting any subsequent term without needing to start from scratch each time.
The general term format for the series is typically written in terms of \(n\), where \(n\) represents the term number. For the Maclaurin series of \(7\cos(x)\), the general term is: \((-1)^n \frac{7x^{2n}}{(2n)!}\).

• This expression indicates that each term alternates in sign, thanks to \((-1)^n\).
• The powers of \(x\) increase by even numbers: \(2n\).
• Finally, the factorial in the denominator \((2n)!\) grows significantly with each additional term in the series.

Understanding the general term helps in recognizing the structure of the series and in extending the series further if needed.