Problem 17 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

The Maclaurin series for 7 \(\cos(-x)\) is \( \sum_{n=0}^{\infty} \frac{7(-1)^n}{(2n)!} x^{2n} \).

Step 1: Recognize the Function

We are given the function \(7 \cos(-x)\). Notice that \(\cos(-x)\) is an even function, implying \(\cos(-x) = \cos(x)\). Thus the function simplifies to \(7 \cos(x)\).

Step 2: Recall the Maclaurin Series Formula

The Maclaurin series is a special case of the Taylor series centered at 0. The general formula for the Maclaurin series of \(f(x)\) is \(\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n\).

Step 3: Derive the Known Series for \(\cos(x)\)

The Maclaurin series for \(\cos(x)\) is \(\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n}\), which comes from evaluating the even derivatives of \( \cos(x) \) at zero.

Step 4: Apply the Coefficient to the Series

Since our function is \(7 \cos(x)\), multiply each term in the Maclaurin series of \(\cos(x)\) by 7. This gives the series: \(7 \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n}\).

Step 5: Write the Final Maclaurin Series

The Maclaurin series for \(7 \cos(x)\) is \(\sum_{n=0}^{\infty} \frac{7(-1)^n}{(2n)!} x^{2n}\). This represents the expansion of the function about 0.

Key Concepts

Taylor series

The Taylor series is a fundamental concept in calculus used to approximate functions using polynomials. It provides a way to represent complex functions as the sum of simpler polynomial terms. These terms are derived from the function's derivatives at a specific point. The Taylor series for a function, \( f(x) \), centered at \( a \) is given by:

• \( \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n \)

For the Maclaurin series, a special type of Taylor series, the center is at 0. Thus, the formula becomes:

• \( \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n \)

This enables us to express functions as infinite sums, which is particularly useful because it lets us simplify complex functions into calculations that are easier to work with. The Taylor series is especially powerful near the center, as its approximations become more accurate the closer \( x \) is to \( a \).

cosine function

The cosine function, \( \cos(x) \), is a periodic function that arises frequently in mathematics. It's an even function, meaning that \( \cos(-x) = \cos(x) \). Cosine is part of the trigonometric functions describing the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Some key properties of \( \cos(x) \) are:

• Periodicity: \( \cos(x + 2\pi) = \cos(x) \)
• Amplitude of 1: Maximum value is 1, minimum is -1
• Symmetrical: Even function, symmetric about the y-axis

The cosine function's Maclaurin series expansion provides a powerful method for calculating its value for any \( x \), converting the trigonometric function into an infinite sum of polynomial terms. This allows us to approximate \( \cos(x) \) to any desired accuracy depending on the number of terms used in the series.

series expansion

Series expansion refers to the method of expressing a function as the sum of a sequence of terms. By doing so, complex functions can be represented as simpler polynomial forms. The Maclaurin series is a primary example of such an expansion, being a specialized Taylor series at 0.In mathematics, series expansions help in:

• Approximating functions that are otherwise difficult to calculate directly
• Analyzing and studying properties of functions more easily
• Performing numerical calculations and developing algorithms

The process involves calculating the derivatives of a function and evaluating them at a particular point to generate coefficients for the polynomial terms in the series.For \( \cos(x) \), the series is:

• \( \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n} \)

This is due to its even nature. By translating the function of \( \cos(x) \) into a polynomial format, the series offers a representation that is infinitely long but practically useful for approximations. Series expansion is a foundational tool in calculus and numerical analysis.