Problem 29 - Essential Calculus

Question

Find the Maclaurin series for the function. (Use the table of power series for elementary functions.) $$ f(x)=\cos ^{2} x $$

Step-by-Step Solution

Answer

The Maclaurin series for \( cos^2(x) \) is \( 1/2 + 1/2 - x^2/2 + 2x^4/4! - (4x)^6/6! + ...\).

Step 1: Understanding the function and the power reduction identity

The given function is \( f(x) = cos^2(x) \). We can rewrite this equation using the power reduction identity which is \( cos^2(x) = 1/2 + 1/2cos(2x) \) . This identity helps to simplify the problem and it's crucial for finding the Maclaurin series for \( cos^2(x) \).

Step 2: Find the Maclaurin series for the simpler function

Now, we will find the Maclaurin series for \( cos(2x) \) . The Maclaurin series for \( cos(x) \) is \( 1 - x^2/2! + x^4/4! - x^6/6! + ... \). Substituting \( x \) with \( 2x \) will give us the Maclaurin series for \( cos(2x) = 1 - (2x)^2/2! + (2x)^4/4! - (2x)^6/6! + ...\).

Step 3: Substituting the Maclaurin series to the reduced function

We can now replace \( cos(2x) \) by its Maclaurin series in the power reduction identity equation. So, \( cos^2(x) = 1/2 + 1/2(1 - (2x)^2/2! + (2x)^4/4! - (2x)^6/6! + ...) \). By distributing \( 1/2 \) to the series we get the final Maclaurin series for \( cos^2(x) \).

Key Concepts

Power Reduction Identity

To find the Maclaurin series for functions like \( \cos^2(x) \), it's helpful to use the power reduction identity. The power reduction identity allows us to express trigonometric functions in a simpler form that is easier to manipulate with series expansions.
For the cosine function, the identity \( \cos^2(x) = \frac{1}{2} + \frac{1}{2}\cos(2x) \) is particularly useful. It breaks down \( \cos^2(x) \) into a sum involving a constant and another cosine term.
This transformation is significant because it converts a squared trig function into a format that's better suited for applying series expansion techniques, such as those used for deriving a Maclaurin series.

Cosine Function

The cosine function, \( \cos(x) \), is a fundamental periodic function used extensively in calculus, physics, and engineering. It provides an essential example for understanding how functions can be expanded into infinite series.

• A key feature of \( \cos(x) \) is its periodicity, repeating every \( 2 \pi \). This property makes the function particularly predictable and useful in Fourier expansions and other series-based analyses.
• Cosine is an even function, meaning it satisfies \( \cos(-x) = \cos(x) \). This symmetry simplifies series calculations and aids in finding both even and odd functions' series expansions.

In the context of series expansions, such as the Maclaurin series, the cosine function is expanded into powers of \( x \). This powerful approach translates cosine’s wave-like behavior into an algebraic form, useful for approximations and calculus operations.

Series Expansion

Series expansion is a mathematical tool used to express functions as sums of infinite series. It is particularly useful in calculus for approximating functions and solving complex mathematical problems.
The Maclaurin series, a special case of the Taylor series around zero, is an approach to expand a function into an infinite series based on its derivatives at \( x = 0 \): \[f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \ldots \]
For the cosine function, the Maclaurin series takes the form: \[\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \ldots \]
This series provides an accurate approximation of \( \cos(x) \) near \( x = 0 \) and can be adapted for \( \cos(2x) \) or other variations by simply substituting \( 2x \) in place of \( x \).
By substituting the expansion, it becomes possible to find the series for different functions like \( \cos^2(x) \), by utilizing their identities, like the power reduction identity. This results in a manageable expression that can be used for various calculus applications, from integrals to differential equations.