Question

Key Idea 8.8 .1 gives the \(n^{\text {th }}\) term of the Taylor series of common functions. In Exercises \(3-6,\) verify the formula given in the Key Idea by finding the first few terms of the Taylor series of the given function and identifying a pattern. $$f(x)=\sin x ; \quad c=0$$

Short answer

The Taylor series for \( \sin x \) around zero is \( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \).

Step-by-step solution

Understanding the Taylor Series Formula

The Taylor series for a function \( f(x) \) about \( c = 0 \) is given by \( f(x) = \sum_{n=0}^{fty} \frac{f^{(n)}(0)}{n!} x^n \). We need to find the Taylor series for \( \sin x \) around \( c = 0 \), also known as the Maclaurin series.

Find the Derivatives of the Function

First, we need to find the first few derivatives of \( \sin x \):- \( f(x) = \sin x \)- \( f'(x) = \cos x \)- \( f''(x) = -\sin x \)- \( f'''(x) = -\cos x \)- \( f^{(4)}(x) = \sin x \)Repeat this cycle as needed.

Evaluate Derivatives at Zero

Next, evaluate each of these derivatives at \( x = 0 \):- \( f(0) = \sin(0) = 0 \)- \( f'(0) = \cos(0) = 1 \)- \( f''(0) = -\sin(0) = 0 \)- \( f'''(0) = -\cos(0) = -1 \)- \( f^{(4)}(0) = \sin(0) = 0 \)

Construct the Terms of the Series

Using the values found, the Taylor series terms are:- First term: \( \frac{0}{0!} x^0 = 0 \)- Second term: \( \frac{1}{1!} x^1 = x \)- Third term: \( \frac{0}{2!} x^2 = 0 \)- Fourth term: \( \frac{-1}{3!} x^3 = -\frac{x^3}{6} \)- Fifth term: \( \frac{0}{4!} x^4 = 0 \)

Identify the Pattern

The pattern of non-zero terms appears to alternate signs and involves only odd-power terms:\[ f(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \] This matches the known Taylor series for \( \sin x \) centered at zero.

Key concepts

Maclaurin Series

The Maclaurin series is a special kind of Taylor series, where the function is expanded about zero, i.e., where the center, denoted as \( c \), is 0. In essence, the Maclaurin series simplifies the Taylor series formula by assuming \( c = 0 \). This series efficiently allows us to express functions as infinite sums of terms calculated from the derivatives of the function evaluated at zero.

• It is expressed as: \[ f(x) = f(0) + \frac{f'(0)}{1!}x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots \]
• This form is particularly useful for approximating functions near the origin.
• Highlighted by the simplicity in substitution where all terms involving \( c \) cancel out.

In the case of the \( \sin x \) function, employing the Maclaurin Series allows for a simplified power series that accurately represents the behavior of \( \sin x \) around zero, leading to a more straightforward calculation.

Derivatives

Derivatives are a fundamental concept in calculus that measure how a function changes as its input changes. For constructing the Taylor or Maclaurin series, derivatives at a point are crucial, as they provide the coefficients for each term in the series.

• The first derivative, \( f'(x) \), provides information about the slope of the tangent line to the function at a given point.
• The second derivative, \( f''(x) \), indicates the function's concavity, or how it bends.
• Higher-order derivatives continue to offer additional details on the function's behavior.

For the \( \sin x \) function, calculating the derivatives and evaluating them at zero shows a repeating pattern: alternating between \( \sin \) and \( \cos \), and switching signs. Recognizing these patterns in derivatives helps in predicting the terms of the series, as derivatives' behavior governs the expansion.

Sin Function

The sine function is a trigonometric function that is periodic and oscillatory, with a range between -1 and 1. This function is pivotal in describing waves and circular motion. Understanding its derivatives and their values at certain points is essential in series expansions.

• The function \( \sin x \) starts at zero when \( x = 0 \).
• When expanded using a Maclaurin series, \( \sin x \) only includes odd powers of \( x \), such as \( x^1, x^3, x^5, \ldots \).
• Each term of this expansion alternates in sign, reflecting the oscillatory nature of the function.

The Maclaurin series for \( \sin x \) helps in approximating the sine function especially near zero, where it behaves most predictably, capturing the wave-like pattern efficiently through its power series terms.

Power Series Expansion

A power series is an infinite sum of terms in the form \( a_n x^n \) where \( a_n \) indicates the coefficient of each term, and \( n \) is a non-negative integer. Power series expansions can represent functions as sums of more manageable polynomial terms, providing approximations that are particularly effective near specific points.

• The general form is: \( f(x) = \sum_{n=0}^{\infty} a_n x^n \).
• Each coefficient, \( a_n \), is derived from evaluating derivatives of the function at a specific point.
• Can be used to approximate complex functions with simpler polynomial expressions.

In the context of \( \sin x \), the power series expansion results in a sequence that captures the essence of the sine function's oscillatory behavior. The terms highlighted involve divisions by factorials, resulting in manageable coefficients that taper off smoothly. This property allows for precise approximations with just a few terms, ideal for practical calculations.