Problem 18 - Thomas Calculus

Question

Find the Maclaurin series for the functions 5 \(\cos \pi x\)

Step-by-Step Solution

Answer

The Maclaurin series for \(5\cos(\pi x)\) is \(5 - \frac{5\pi^2 x^2}{2} + \frac{5\pi^4 x^4}{24} - \frac{5\pi^6 x^6}{720} + \ldots\).

Step 1: Understand the Maclaurin Series

A Maclaurin series is a Taylor series expansion of a function about 0. The general form is:\[f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \,\ldots\] for the function \(f(x)\).

Step 2: Recall the Maclaurin Series for Cosine

The known Maclaurin series for \(\cos(x)\) is:\[ \cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \,\ldots\] This series represents \( \cos(x) \) expanded about 0.

Step 3: Substitute \(\pi x\) into the Series

Substitute \(x\) in the cosine series with \(\pi x\) to find the series for \(\cos(\pi x)\):\[ \cos(\pi x) = 1 - \frac{(\pi x)^2}{2!} + \frac{(\pi x)^4}{4!} - \frac{(\pi x)^6}{6!} + \,\ldots\] which simplifies to \[ \cos(\pi x) = 1 - \frac{\pi^2 x^2}{2!} + \frac{\pi^4 x^4}{4!} - \frac{\pi^6 x^6}{6!} + \,\ldots\]

Step 4: Multiply by 5

Multiply the entire series by 5 to find the Maclaurin series for \(5\cos(\pi x)\):\[ 5\cos(\pi x) = 5 \cdot \left(1 - \frac{\pi^2 x^2}{2!} + \frac{\pi^4 x^4}{4!} - \frac{\pi^6 x^6}{6!} + \,\ldots\right)\] Which gives the series:\[ 5 - \frac{5\pi^2 x^2}{2!} + \frac{5\pi^4 x^4}{4!} - \frac{5\pi^6 x^6}{6!} + \,\ldots\]

Step 5: Write the Final Maclaurin Series

The final Maclaurin series for \(5\cos(\pi x)\) is:\[ 5 - \frac{5\pi^2 x^2}{2} + \frac{5\pi^4 x^4}{24} - \frac{5\pi^6 x^6}{720} + \,\ldots\]

Key Concepts

Taylor Series

The Taylor Series provides a way to express a wide variety of functions as infinite sums of their derivatives at a particular point. It's a powerful tool in calculus that helps in approximating functions that may not be easily computable otherwise. When a function is expanded about the point 0, it is specifically known as a Maclaurin Series. In general, a Taylor Series for a function \( f(x) \) about a point \( a \) can be written as:

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

By selecting \( a = 0 \), it becomes a Maclaurin Series. This means that only values of the function and its derivatives at 0 are used, simplifying calculations and making it an essential concept when working with polynomials in physics and engineering.
Using the Taylor Series allows us to find polynomial approximations to complicated functions, making it easier to analyze and compute values, especially for small \( x \) close to the point of expansion.

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It is periodic and serves various practical applications across different fields, including physics and engineering. Its graph is a wave and can be described using the unit circle or geometrical relations in right-angled triangles.

• The function \( \cos(x) \) is especially interesting because it is an even function. This means \( \cos(-x) = \cos(x) \).
• Being periodic with period \( 2\pi \), it repeats its values in cycles.

The cosine function can also be expressed as a series expansion, which is particularly useful for calculations at small values of \( x \). Its Maclaurin series, an expansion about 0, is:

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

This alternating series converges to \( \cos(x) \) as more terms are added. When substituting \( \pi x \) for \( x \) to find \( \cos(\pi x) \), the series adapts by modifying each term based on \( \pi \). These expansions are highly useful in scenarios where direct computation of the cosine might be difficult or impractical.

Series Expansion

Series expansions transform functions into an infinite sum of terms. This can simplify calculations and allow for complex functions to be expressed in a polynomial form, especially useful in mathematical and engineering applications.

• The ability to write a function as a series allows approximation and analysis using simpler polynomial forms.
• A series expansion enables errors to be minimized and the continuation of calculations by truncating the series to a finite number of terms.

The Maclaurin series, a specific type of series expansion, for a function \( f(x) \) involves using the derivatives of the function at 0. For instance, **5\( \cos(\pi x) \)** was expanded by first finding the series for \( \cos(x) \) and substituting \( x \) with \( \pi x \), then multiplying the whole series by 5. Such strategic manipulation simplifies the function into a form that is easier to work with and interpret. Often, only a few terms are needed to achieve a good approximation. This makes the Maclaurin series a versatile tool in solving differential equations and modeling real-world phenomena.