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Chapter 30: Problem 26

Find the first two nonzero terms of the Maclaurin expansion of the given functions. $$f(x)=x e^{\sin x}$$

Short Answer

Expert verified
The first two nonzero terms are \( x + x^2 \).

Step by step solution

01

Understanding the Maclaurin Series

A Maclaurin series is a special case of a Taylor series, centered at x = 0. Its general form is given by \( f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots \). Our goal is to find the first two nonzero terms of this series for the function \( f(x) = x e^{\sin x} \).
02

Series Expansion of \( e^{\sin x} \)

First, we'll find the series expansion for \( e^{\sin x} \), using the fact that \( e^y = 1 + y + \frac{y^2}{2!} + \frac{y^3}{3!} + \cdots \). Substituting \( y = \sin x \) into this expansion, we have: - Since \( \sin x = x - \frac{x^3}{3!} + \cdots \), substitute in to get \( e^{\sin x} = 1 + \sin x + \frac{\sin^2 x}{2!} \).Regarding \( \sin x \), up to the first two nonzero powers of x, we use \( \sin x \approx x \), and then it begins with \( x - \frac{x^3}{6} \). Thus, we start with:\[ e^{\sin x} \approx 1 + x + \frac{(x - \frac{x^3}{6})^2}{2!} \]Thus, ignoring higher terms, \( e^{\sin x} \approx 1 + x \).
03

Multiply by x for \( f(x) \)

Now consider \( f(x) = x e^{\sin x} \). Substitute the approximation of \( e^{\sin x} \) into the function:\[ f(x) = x \times (1 + x) \]This simplifies to:\[ f(x) = x + x^2 \]
04

Identify Nonzero Terms

The Maclaurin series obtained from the multiplication gives us: \( x + x^2 \). These correspond to the first two nonzero terms.

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Key Concepts

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

Taylor series
A Taylor series is a fundamental concept in calculus that allows us to express a function as an infinite sum of terms calculated from the function's derivatives at a single point. The Taylor series formula is:
  • Centered around a point "a": \( f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots \)
  • The Maclaurin series is a specific case where the center is at \( a = 0 \), simplifying the formula to: \( f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots \)
This series is particularly useful for estimating functions that are complex or non-linear around the point \( a = 0 \), providing insights into the behavior of the function. The power of the Maclaurin and Taylor series lies in their ability to convert difficult functions into potentially simpler polynomial forms.
power series expansion
Power series expansion is a method to represent a function as a sum of powers, i.e., as an infinite series. A power series about zero is represented as:
  • \( f(x) = c_0 + c_1x + c_2x^2 + c_3x^3 + \cdots \)
  • Here, \( c_n \) are the coefficients of the series.
By using such an expansion, functions that may seem complex at first glance can be decomposed into a series of simpler components, each being a contribution proportional to a power of \( x \). This approach is widely used in mathematical analyses and helps in simplifying mathematical models for practical computations, especially in engineering and physical sciences.
exponential function
The exponential function \( e^x \) is one of the most significant functions in mathematics, characterized by the unique property that it is its own derivative. Its power series expansion is particularly neat and is given by:
  • \( e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \)
When dealing with variants like \( e^{\sin x} \), the power series expansion of the exponential function can be combined with the power series of \( \sin x \), making complex functions approachable by expressing them as sums of straightforward integer powers. Its flexibility and ubiquity make the exponential function fundamental in growth processes like compound interest, population growth, and radioactive decay.
differentiation
Differentiation is a basic operation in calculus that involves finding the derivative of a function. The derivative reflects how a function changes as its input changes, providing critical insights into the function's rate of change or slope.
  • The derivative of \( f(x) \) at a point \( x \) is formulated as \( f'(x) = \frac{df(x)}{dx} \).
  • For exponential functions, like \( e^x \), the derivative is particularly simple: \( \frac{d}{dx}e^x = e^x \).
In our context, differentiation is a key step in determining the terms of the Taylor and Maclaurin series, as it helps in identifying the coefficients for each degree term in the power series. Understanding this concept is crucial for applying calculus effectively across various disciplines.

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Most popular questions from this chapter

Solve the given problems. Find the first three nonzero terms of the Maclaurin expansion for (a) \(f(x)=e^{x}\) and \(\left(\text { b) } f(x)=e^{x^{2}} .\) Compare these expansions. \right.

Solve the given problems as indicated. Referring to Chapter \(19,\) we see that the sum of the first \(n\) terms of a geometric sequence is \(S_{n}=\frac{a_{1}\left(1-r^{n}\right)}{1-r} \quad(r \neq 1)\) $$(19.6)$$ where \(a_{1}\) is the first term and \(r\) is the common ratio. We can visualize the corresponding infinite series by graphing the function \(f(x)=a_{1}\left(1-r^{x}\right) /(1-r)(r \neq 1)\) (or using a calculator that can graph a sequence). The graph represents the sequence of partial sums for values where \(x=n,\) since \(f(n)=S_{n}\) Use a calculator to visualize the first five partial sums of the series \(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots\) What value does the infinite series approach? (Remember: Only points for which \(x\) is an integer have real meaning.)

Use a calculator to display (a) the given function and (b) the first three series approximations of the function in the same display. Each display will be similar to that in Fig. 30.9 for the function \(y=\sin x\) and its first three approximations. Be careful in choosing the appropriate window values. $$y=\ln (1+x) \quad(|x| < 1)$$

Determine whether the Fourier series of the given functions will include only sine terms, only cosine terms, or both sine terms and cosine terms. $$f(x)=\cos (\sin x) \quad-\pi \leq x<\pi$$

Calculate the value of each of the given functions. Use the expansion for \(\sqrt{1+x},\) and in Exercises 23 and \(24,\) use the expansion for \(\sqrt[3]{1+x} .\) Use three terms. $$\sqrt[3]{0.9628}$$
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