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Magazine Chapter 9: Problem 13
Find the first four nonzero terms of the Maclaurin series for the function by multiplying the Maclaurin series of the factors. $$\begin{array}{lll}{\text { (a) } e^{x} \sin x} & {\text { (b) } \sqrt{1+x} \ln (1+x)} & {}\end{array}$$
Short Answer
Expert verified
(a) The first four nonzero terms are \(x + x^2 + \frac{x^3}{3}\). (b) The first four nonzero terms are \(x - \frac{x^3}{8}\).
Step by step solution
01
Write the Maclaurin Series for Each Factor
The Maclaurin series for \( e^x \) is given by \( 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \). The Maclaurin series for \( \sin x \) is \( x - \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots \). We will use these series to find the product for (a).
02
Multiply the Series for \( e^x \) and \( \sin x \)
To multiply the Maclaurin series of \( e^x \) and \( \sin x \), we will multiply each term in the series for \( e^x \) by each term in the series for \( \sin x \). Collect the first four non-zero terms:- \( (1)(x) = x \) - \( (x)(x) = x^2 \) - \( \left( \frac{x^2}{2} \right)(x) = \frac{x^3}{2} \) - \( (1)(-\frac{x^3}{6}) = -\frac{x^3}{6} \)Combine similar terms: the \( x^3 \) terms are \( \frac{x^3}{2} - \frac{x^3}{6} = \frac{x^3}{3} \). So, the first four non-zero terms are \( x + x^2 + \frac{x^3}{3} + \cdots \)
03
Write the Maclaurin Series for Each Factor in Part (b)
The Maclaurin series for \( \sqrt{1+x} \) is \( 1 + \frac{x}{2} - \frac{x^2}{8} + \cdots \). The Maclaurin series for \( \ln(1+x) \) is \( x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots \). We will use these series to find the product for (b).
04
Multiply the Series for \( \sqrt{1+x} \) and \( \ln(1+x) \)
Multiply each term in the series for \( \sqrt{1+x} \) by each term in the series for \( \ln(1+x) \):- \( (1)(x) = x \)- \( (\frac{x}{2})(x) = \frac{x^2}{2} \)- \( (-\frac{x^2}{8})(x) = -\frac{x^3}{8} \)- \( (1)(-\frac{x^2}{2}) = -\frac{x^2}{2} \)Combine and simplify terms: the \( x^2 \) terms are zero when combined, resulting in \( x - \frac{x^3}{8} \).Thus, the first four non-zero terms are \( x + 0 - \frac{x^3}{8} + \cdots \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exponential Functions
An exponential function is a mathematical function of the form \( e^x \), where \( e \) is a constant approximately equal to 2.71828. This function is a cornerstone in mathematics, particularly because of its unique properties in calculus and its ability to model growth processes.
### Characteristics of Exponential Functions:- **Base Constant**: The base \( e \) is irrational, meaning its decimal representation is infinite and non-repeating.
- **Rate of Increase**: The function grows extremely fast, as the rate of increase itself increases with \( x \). In calculus, the derivative of \( e^x \) is also \( e^x \), which means the slope at any point is the same as its value at that point.
- **Usage**: Exponential functions are widely used to model scenarios with constant relative growth rates, such as population growth, radioactive decay, and compound interest.
### Maclaurin Series Representation:The Maclaurin series for \( e^x \) is an infinite sum given by\[1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \]This representation allows us to approximate the value of \( e^x \) for any small \( x \). When working with Maclaurin series in problems like finding the series for \( e^x \sin x \), each term of the exponential series contributes to the result in a straightforward manner.
### Characteristics of Exponential Functions:- **Base Constant**: The base \( e \) is irrational, meaning its decimal representation is infinite and non-repeating.
- **Rate of Increase**: The function grows extremely fast, as the rate of increase itself increases with \( x \). In calculus, the derivative of \( e^x \) is also \( e^x \), which means the slope at any point is the same as its value at that point.
- **Usage**: Exponential functions are widely used to model scenarios with constant relative growth rates, such as population growth, radioactive decay, and compound interest.
### Maclaurin Series Representation:The Maclaurin series for \( e^x \) is an infinite sum given by\[1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \]This representation allows us to approximate the value of \( e^x \) for any small \( x \). When working with Maclaurin series in problems like finding the series for \( e^x \sin x \), each term of the exponential series contributes to the result in a straightforward manner.
Trigonometric Functions
Trigonometric functions like \( \sin x \) and \( \cos x \) are foundational in understanding waves, circles, and oscillations. Among the 6 trigonometric functions, \( \sin x \) is unique for its periodic, wave-like behavior.
### Key Features:- **Periodicity**: \( \sin x \) is periodic with a period of \( 2\pi \), meaning its values repeat every \( 2\pi \) units along the x-axis.
- **Unit Circle Relation**: In the unit circle, \( \sin x \) represents the y-coordinate of a point at an angle \( x \). It ranges from -1 to 1.
- **Derivative and Integral**: In calculus, the derivative of \( \sin x \) is \( \cos x \), and its integral is \(-\cos x \) (plus a constant).
### Maclaurin Series for \( \sin x \):The Maclaurin series of \( \sin x \) is helpful in approximating values for small angles and is given by:\[x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \]This series alternates in sign and includes only odd powers of \( x \). When multiplying with another series, like that of \( e^x \), each term in \( \sin x \) combines with each term in the other series, carefully contributing to the overall series expansion for the product, such as in the problem \( e^x \sin x \).
### Key Features:- **Periodicity**: \( \sin x \) is periodic with a period of \( 2\pi \), meaning its values repeat every \( 2\pi \) units along the x-axis.
- **Unit Circle Relation**: In the unit circle, \( \sin x \) represents the y-coordinate of a point at an angle \( x \). It ranges from -1 to 1.
- **Derivative and Integral**: In calculus, the derivative of \( \sin x \) is \( \cos x \), and its integral is \(-\cos x \) (plus a constant).
### Maclaurin Series for \( \sin x \):The Maclaurin series of \( \sin x \) is helpful in approximating values for small angles and is given by:\[x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \]This series alternates in sign and includes only odd powers of \( x \). When multiplying with another series, like that of \( e^x \), each term in \( \sin x \) combines with each term in the other series, carefully contributing to the overall series expansion for the product, such as in the problem \( e^x \sin x \).
Logarithmic Functions
Logarithmic functions are the inverses of exponential functions. The natural logarithm \( \ln(x) \) plays an especially important role due to its base \( e \), seamlessly connecting with calculus and growth models.
### Properties of Logarithmic Functions:- **Inverse Function**: \( \ln(x) \) is the inverse of \( e^x \). For any positive number \( y \), \( x = e^y \) has its logarithm \( y = \ln(x) \).
- **Derivative**: The derivative of \( \ln(x) \) is \( \frac{1}{x} \). This is valuable for solving growth and decay problems in calculus.
- **Behaviour**: \( \ln(x) \) is defined only for \( x > 0 \), and it approaches negative infinity as \( x \) approaches zero from the positive side.
### Maclaurin Series for \( \ln(1+x) \):The Maclaurin series for \( \ln(1+x) \) is used to approximate logarithmic values near \( x = 0 \):\[x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots\]This series also alternates in sign, and each term's contribution becomes progressively smaller. When combined with another series, like \( \sqrt{1+x} \), it requires careful term multiplication to uncover meaningful series expansions. For instance, solving \( \sqrt{1+x}\ln(1+x) \) through multiplication of their series showcases how logarithmic expansions are integrated.
### Properties of Logarithmic Functions:- **Inverse Function**: \( \ln(x) \) is the inverse of \( e^x \). For any positive number \( y \), \( x = e^y \) has its logarithm \( y = \ln(x) \).
- **Derivative**: The derivative of \( \ln(x) \) is \( \frac{1}{x} \). This is valuable for solving growth and decay problems in calculus.
- **Behaviour**: \( \ln(x) \) is defined only for \( x > 0 \), and it approaches negative infinity as \( x \) approaches zero from the positive side.
### Maclaurin Series for \( \ln(1+x) \):The Maclaurin series for \( \ln(1+x) \) is used to approximate logarithmic values near \( x = 0 \):\[x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots\]This series also alternates in sign, and each term's contribution becomes progressively smaller. When combined with another series, like \( \sqrt{1+x} \), it requires careful term multiplication to uncover meaningful series expansions. For instance, solving \( \sqrt{1+x}\ln(1+x) \) through multiplication of their series showcases how logarithmic expansions are integrated.
