Problem 21 - Calculus Early Transcendentals

Question

a. Find the first four nonzero terms of the Taylor series for the given function centered at \(a\) b. Write the power series using summation notation. $$f(x)=\sin x, a=\pi / 2$$

Step-by-Step Solution

Answer

Answer: The Taylor series for the first four nonzero terms is given by: $$f(x) = 1 - \frac{(x - \pi/2)^2}{2!} + \frac{(x - \pi/2)^4}{4!} + \cdots$$ Expressed in summation notation, the power series is: $$f(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!}(x - \pi/2)^{2n}$$

Step 1: Calculate the derivatives of the function

We need to find the first four derivatives of the function \(f(x) = \sin x\). First derivative: \(f'(x) = \cos x\) Second derivative: \(f''(x) = -\sin x\) Third derivative: \(f^{(3)}(x) = -\cos x\) Fourth derivative: \(f^{(4)}(x) = \sin x\)

Step 2: Evaluate the derivatives at the center point \(a = \pi/2\)

Now we need to evaluate each derivative at the center point \(a = \pi/2\). \(f(\pi/2) = \sin(\pi/2) = 1\) \(f'(\pi/2) = \cos(\pi/2) = 0\) \(f''(\pi/2) = -\sin(\pi/2) = -1\) \(f^{(3)}(\pi/2) = -\cos(\pi/2) = 0\) \(f^{(4)}(\pi/2) = \sin(\pi/2) = 1\)

Step 3: Build the Taylor series using the formula

We can now use the information from Steps 1 and 2 to build the Taylor series for the function \(f(x) = \sin x\). The formula for Taylor series is: $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$$ Using the derivatives and center point evaluations, we have: $$f(x) = 1 - \frac{(x - \pi/2)^2}{2!} + \frac{(x - \pi/2)^4}{4!} + \cdots$$ Now we have found the first four nonzero terms of the Taylor series for the given function.

Step 4: Write the power series using summation notation

To express the Taylor series found in Step 3 in summation notation, we can use the following notation: $$f(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!}(x - \pi/2)^{2n}$$ This is the power series in summation notation for the function \(f(x) = \sin x\) centered at \(a = \pi/2\).

Key Concepts

Derivatives

Derivatives are a fundamental concept in calculus that describe how a function changes as its input changes. In the context of Taylor series, derivatives help us approximate a function around a specific point, known as the center. Taylor series use derivatives to form polynomial expressions that closely represent the original function.

To start with a Taylor series, you need to first calculate the derivatives of the function at the center, which, in this exercise, is \( a = \pi/2 \). For the function \( f(x) = \sin x \), we compute several derivatives:

• First derivative: \( f'(x) = \cos x \)
• Second derivative: \( f''(x) = -\sin x \)
• Third derivative: \( f^{(3)}(x) = -\cos x \)
• Fourth derivative: \( f^{(4)}(x) = \sin x \)

Each derivative gives us insight into how the slope and curvature of the function behave at specific points. For a Taylor series, these derivatives are evaluated at the center to determine the coefficients of the polynomial terms.

Power Series

A power series is an infinite sum of terms that takes the form of polynomials. They are valuable because they provide a way to represent more complex functions using simpler polynomial terms. Taylor series are specific types of power series centered around a particular point, enhancing our understanding of the function's behavior near that point.

Power series are represented as:\[ f(x) = a_0 + a_1(x-a) + a_2(x-a)^2 + a_3(x-a)^3 + \cdots \]where \( a \) is the point of expansion, and \( a_n \) are the coefficients determined using the derivatives. The goal is to approximate the function at and near \( a \). In this exercise, the power series for \( f(x) = \sin x \) centered at \( a = \pi/2 \) is:\[ f(x) = 1 - \frac{(x - \pi/2)^2}{2!} + \frac{(x - \pi/2)^4}{4!} + \cdots \]This expression uses polynomials like \( (x-\pi/2)^n \), helping us find an approximation at any point close to \( a \). These terms make power series a versatile tool in math, especially for solving problems involving infinite series.

Summation Notation

Summation notation is a compact way to express the sum of a series. It is frequently used in mathematics to simplify expressions where terms follow a specific pattern. In the context of Taylor and power series, summation allows us to write infinite sums concisely.

The generic form of a summation is:\( \sum_{n=0}^{\infty} a_n \), where \( a_n \) are the terms of the sequence. This notation is especially useful for expressing power series, capturing an infinite progression of terms without listing each one individually.

For the power series in this exercise, after evaluating the derivatives at the center point and building out the terms, the summation notation can be used:\[ f(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!}(x - \pi/2)^{2n} \]This expression indicates that the series includes terms for all even powers, with coefficients calculated from the derivatives. By using summation notation, mathematicians and students are able to convey complex infinite series in a clear and understandable manner, aiding their comprehension and application in various problems.