Problem 17 - Thomas Calculus

Question

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(11-28 .\) $$\cos ^{2} x\left(\operatorname{Hin} t \cdot \cos ^{2} x=(1+\cos 2 x) / 2\right)$$

Step-by-Step Solution

Answer

Expand and simplify the terms from the cosine identity.

Step 1: Start with the given substitution

Given: \[\cos^2 x = \frac{1+\cos 2x}{2}\]This substitution simplifies \(\cos^2 x\) using a trigonometric identity.

Step 2: Expand using the Taylor series for \(\cos 2x\)

The Taylor series for \(\cos x\) around \(x = 0\) is:\[\cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} + \cdots\]Thus, substitute \(2x\) to find \(\cos 2x\):\[\cos 2x = 1 - \frac{(2x)^2}{2} + \frac{(2x)^4}{24} - \frac{(2x)^6}{720} + \cdots\]Simplify:\[\cos 2x = 1 - 2x^2 + \frac{4x^4}{6} - \frac{8x^6}{720} + \cdots\]

Key Concepts

Power Series

Power series are mathematical expressions that help us represent complex functions in simpler terms. They are similar to polynomials, but with an infinite number of terms. The general form of a power series is:

• \[ a_0 + a_1(x - c) + a_2(x - c)^2 + a_3(x - c)^3 + \cdots \]

Here, \(a_0, a_1, a_2,\) etc. are constants and \(c\) is the center of the series.Power series are useful because they can approximate functions to a high degree of accuracy, especially near the center \(c\). This makes calculations easier, particularly for functions that are difficult or impossible to compute directly. In calculus, power series often make it simpler to derive integrals and derivatives.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the involved variables. These identities are instrumental in simplifying expressions and solving trigonometric equations. One such identity used in the solution is:

• \(\cos^2 x = \frac{1 + \cos 2x}{2}\)

This identity transforms the square of cosine into a more manageable form using the cosine of double angles. Such identities help in expanding expressions, finding simplifications, and tackling integrals or derivatives in calculus.

Series Expansion

Series expansion is a method used to express a function as a sum of terms, typically involving powers of a variable. The Taylor series is a renowned type of series expansion.When we talk about a Taylor series, we are expanding a function around a specific point, usually 0, similar to our current problem. For instance, the Taylor series for \(\cos x\) around \(x=0\) is:

• \[\cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots \]

This makes it possible to calculate approximate values by taking only a few terms, rather than the entire infinite series. Expansions like these are crucial when dealing with complex expressions, making them more accessible.

Calculus

Calculus is a significant branch of mathematics focusing on change. It incorporates both differentiation and integration, essential for analyzing dynamic systems. In our exercise, calculus principles are applied to find the Taylor series, a process that involves infinitely differentiable functions. Familiarity with derivatives is important because each term in a Taylor series relates to derivatives of the function at a specific point. Understanding these concepts is crucial:

• Differentiation: Helps find the rate of change of quantities.
• Integration: Helps determine the accumulation of quantities.

Calculus, combined with series and trigonometric identities, enables us to simplify and solve complex expressions efficiently.