/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 43 Use the identity \(\sin ^{2} x=(... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 43

Use the identity \(\sin ^{2} x=(1-\cos 2 x) / 2\) to obtain the Maclaurin series for \(\sin ^{2} x .\) Then differentiate this series to obtain the Maclaurin series for 2 \(\sin x \cos x .\) Check that this is the series for \(\sin 2 x .\)

Short Answer

Expert verified
The Maclaurin series for \( \sin 2x \) is the same as the series obtained by differentiating \( \sin^2 x \).

Step by step solution

01

Recall the Identity and Expand

The given identity is \( \sin^2 x = \frac{1 - \cos 2x}{2} \). First, we need to find the Maclaurin series for \( \cos 2x \). The Maclaurin series for \( \cos x \) is \( 1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} + \cdots\). We substitute \( 2x \) in place of \( x \) to find the series for \( \cos 2x \), resulting in \( 1 - \frac{(2x)^2}{2} + \frac{(2x)^4}{24} - \frac{(2x)^6}{720} + \cdots \). Simplifying this, we get \( 1 - 2x^2 + \frac{4x^4}{3} - \frac{8x^6}{45} + \cdots \).
02

Use the Identity to Write the Maclaurin Series for \( \sin^2 x \)

Apply the identity: \( \sin^2 x = \frac{1 - \cos 2x}{2} \). Substitute the series for \( \cos 2x \) into this identity. This gives: \( \sin^2 x = \frac{1 - (1 - 2x^2 + \frac{4x^4}{3} - \frac{8x^6}{45} + \cdots)}{2} \). Simplifying yields: \( \sin^2 x = x^2 - \frac{2x^4}{3} + \frac{4x^6}{45} + \cdots \).
03

Differentiate to Find Series for \( 2 \sin x \cos x \)

Next, differentiate the series obtained for \( \sin^2 x \), which is \( x^2 - \frac{2x^4}{3} + \frac{4x^6}{45} \), with respect to \( x \). The derivative is: \( 2x - \frac{8x^3}{3} + \frac{24x^5}{45} + \cdots \). This is the series for \( 2 \sin x \cos x \).
04

Check if it Matches Series for \( \sin 2x \)

The Maclaurin series for \( \sin x \) is \( x - \frac{x^3}{6} + \frac{x^5}{120} - \cdots \). For \( \sin 2x \), we replace \( x \) with \( 2x \), resulting in: \( 2x - \frac{(2x)^3}{6} + \frac{(2x)^5}{120} - \cdots \), which simplifies to: \( 2x - \frac{8x^3}{6} + \frac{32x^5}{120} - \cdots \), or equivalently \( 2x - \frac{8x^3}{3} + \frac{24x^5}{45} + \cdots \). The differentiated series in Step 3 matches this series.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Identities
Trigonometric identities are mathematical equations that express relationships between trigonometric functions. These identities are essential as they allow us to simplify expressions and solve equations involving trigonometric functions.

In the provided exercise, the identity \( \sin^2 x = \frac{1 - \cos 2x}{2} \) is used to transform a trigonometric expression into a form that can be expanded into a Maclaurin series. This particular identity involves double angle formulas, which are crucial for reducing complex expressions and are widely used in calculus and algebra.

Using trigonometric identities effectively requires recognizing relationships between different functions and understanding how these identities can simplify processes like integration or series expansion. This foundational tool is not only essential for calculating derivatives but also for solving trigonometric equations in various fields of study including physics and engineering.
Differentiation
Differentiation is a key concept in calculus that involves finding the derivative of a function, which represents the function's rate of change. This process is fundamental in understanding how quantities change in relation to one another.

In the context of the exercise, differentiation is used to obtain the Maclaurin series for \( 2 \sin x \cos x \) by differentiating the series for \( \sin^2 x \). The process of finding derivatives allows us to examine behavior of functions and their approximations accurately. The derivative of a power series is found term by term, allowing us to expand our range of solutions and applications.

The process itself involves applying rules such as the power rule, product rule, or chain rule, depending on the complexity of the function involved. Mastery of differentiation is vital for interpreting graphs, solving problems involving motion, and for more advanced studies in fields such as economics or any other areas requiring the analysis of dynamic systems.
Power Series Expansion
Power series expansion is a method of expressing a function as an infinite sum of terms calculated from the values of its derivatives at a single point. This technique is particularly useful in approximating functions that are otherwise difficult to handle analytically.

In the exercise example, we encounter the Maclaurin series, a specific type of power series where all derivatives are evaluated at zero. The challenge involves expanding trigonometric functions like \( \cos 2x \) and \( \sin 2x \) into power series to facilitate their manipulation and comprehension.

The power series allows us not only to approximate functions but also to discover properties like continuity and convergence. By substituting into these series, complicated expressions can be reduced to polynomial forms that are easier to work with. Moreover, power series are the cornerstone of solving differential equations and integral computations, extending their utility across numerous scientific and mathematical applications.

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Most popular questions from this chapter

How many terms of the Taylor series for \(\ln (1+x)\) should you add to be sure of calculating ln \((1.1)\) with an error of magnitude less than \(10^{-8} ?\) Give reasons for your answer.

Is it true that if \(\sum_{n=1}^{\infty} a_{n}\) is a divergent series of positive numbers, then there is also a divergent series \(\sum_{n=1}^{\infty} b_{n}\) of positive numbers with \(b_{n}<\) \(a_{n}\) for every \(n ?\) Is there a smallest divergent series of positive numbers? Give reasons for your answers.

Series for tan \(^{-1} x\) for \(|x|>1\) Derive the series \begin{equation} \tan ^{-1} x=\frac{\pi}{2}-\frac{1}{x}+\frac{1}{3 x^{3}}-\frac{1}{5 x^{5}}+\cdots, \quad x>1 \end{equation} \begin{equation} \tan ^{-1} x=-\frac{\pi}{2}-\frac{1}{x}+\frac{1}{3 x^{3}}-\frac{1}{5 x^{5}}+\cdots, \quad x<-1 \end{equation} by integrating the series \begin{equation} \frac{1}{1+t^{2}}=\frac{1}{t^{2}} \cdot \frac{1}{1+\left(1 / t^{2}\right)}=\frac{1}{t^{2}}-\frac{1}{t^{4}}+\frac{1}{t^{6}}-\frac{1}{t^{8}}+\cdots \end{equation} in the first case from \(x\) to \(\infty\) and in the second case from \(-\infty\) to \(x\)

Computer Explorations Taylor's formula with \(n=1\) and \(a=0\) gives the linearization of a function at \(x=0 .\) With \(n=2\) and \(n=3\) we obtain the standard quadratic and cubic approximations. In these exercises we explore the errors associated with these approximations. We seek answers to two questions: \begin{equation} \begin{array}{l}{\text { a. For what values of } x \text { can the function be replaced by each }} \\ {\text { approximation with an error less than } 10^{-2} \text { ? }} \\ {\text { b. What is the maximum error we could expect if we replace the }} \\ {\text { function by each approximation over the specified interval? }}\end{array} \end{equation} Using a CAS, perform the following steps to aid in answering questions (a) and (b) for the functions and intervals in Exercises \(53-58 .\) \begin{equation} \begin{array}{l}{\text { Step } 1 : \text { Plot the function over the specified interval. }} \\ {\text { Step } 2 : \text { Find the Taylor polynomials } P_{1}(x), P_{2}(x), \text { and } P_{3}(x) \text { at }} \\\ {x=0 .}\\\\{\text { Step } 3 : \text { Calculate the }(n+1) \text { st derivative } f^{(n+1)}(c) \text { associated }} \\ {\text { with the remainder term for each Taylor polynomial. }} \\ {\text { Plot the derivative as a function of } c \text { over the specified interval }} \\ {\text { and estimate its maximum absolute value, } M .}\\\\{\text { Step } 4 : \text { Calculate the remainder } R_{n}(x) \text { for each polynomial. }} \\ {\text { Using the estimate } M \text { from Step } 3 \text { in place of } f^{(n+1)}(c), \text { plot }} \\ {R_{n}(x) \text { over the specified interval. Then estimate the values of }} \\ {x \text { that answer question (a). }}\\\\{\text { Step } 5 : \text { Compare your estimated error with the actual error }} \\ {E_{n}(x)=\left|f(x)-P_{n}(x)\right| \text { by plotting } E_{n}(x) \text { over the specified }} \\ {\text { interval. This will help answer question (b). }} \\ {\text { Step } 6 : \text { Graph the function and its three Taylor approximations }} \\ {\text { together. Discuss the graphs in relation to the information }} \\ {\text { discovered in Steps } 4 \text { and } 5 .}\end{array} \end{equation} $$f(x)=(1+x)^{3 / 2},-\frac{1}{2} \leq x \leq 2$$

A cubic approximation Use Taylor's formula with \(a=0\) and \(n=3\) to find the standard cubic approximation of \(f(x)=\) 1\(/(1-x)\) at \(x=0 .\) Give an upper bound for the magnitude of the error in the approximation when \(|x| \leq 0.1\)
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