Problem 35 - Thomas Calculus

Question

Find the first four nonzero terms in the Maclaurin series for the functions in Exercises \(31-38 .\) $$e^{\sin x}$$

Step-by-Step Solution

Answer

The first four nonzero terms are: \( 1 + x + \frac{x^2}{2} - \frac{x^3}{6} \).

Step 1: Understanding the Maclaurin Series

The Maclaurin series is a type of Taylor series expansion of a function about 0 (i.e., \( a = 0 \)). It is given by \( \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n \), where \( f^{(n)}(0) \) is the \( n \)-th derivative of the function evaluated at 0.

Step 2: Find the Maclaurin Series of \( e^x \)

The Maclaurin series expansion for \( e^x \) is given by \( 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \), or more generally, \( \sum_{n=0}^{\infty} \frac{x^n}{n!} \).

Step 3: Find the Maclaurin Series of \( \sin x \)

The Maclaurin series expansion for \( \sin x \) is \( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \), or in general, \( \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!} \).

Step 4: Substitute \( \sin x \) Maclaurin series into \( e^{\sin x} \) series

Substitute the \( \sin x \) Maclaurin series into \( e^{\sin x} \)'s expansion:\[e^{\sin x} = e^{x - \frac{x^3}{3!} + \cdots} = 1 + (x - \frac{x^3}{6} + \cdots) + \frac{(x - \frac{x^3}{6} + \cdots)^2}{2!} + \cdots \].

Step 5: Compute and Combine First Four Nonzero Terms

Expand the expression and collect terms up to first four nonzero terms. From the previous expansion:1. The first term is \( 1 \).2. The second term involves \( x \).3. The third term involves \( -\frac{x^3}{6} \).4. Next, we expand to bring a \( \frac{x^2}{2!} \) term from the square of \( x \ and thus include it entirely to get: \ 1 + x + \frac{x^2}{2} - \frac{x^3}{6} + \cdots \).

Step 6: Write the First Four Nonzero Terms

Collecting the first four non-zero terms, we have \[ e^{\sin x} \approx 1 + x + \frac{x^2}{2} - \frac{x^3}{6} \].

Key Concepts

Taylor series

The Taylor series is a powerful mathematical tool used to approximate transcendental functions. This series expands a function into an infinite sum of its derivatives at a single point.

• A Taylor series centered at a point \(a\) is expressed as \( \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n \).
• The Maclaurin series is a special case of the Taylor series where \(a = 0\).
• This means we expand the function around the origin, simplifying calculations.

Understanding how to apply Taylor series is critical for approximating complex functions in fields such as engineering and physics. A good example is using a Maclaurin series to approximate \(e^{\sin x}\), requiring expansions for both \(e^x\) and \(\sin x\) to compose these functions efficiently.By transforming functions into polynomials, Taylor series allow simple arithmetic replacements for complex calculations. They are particularly useful when a small number of terms are sufficient, providing a balance between accurate modeling and computational simplicity.

exponential functions

Exponential functions are a key concept in calculus and algebra. They are defined by the function \(f(x) = a^x\), where \(a\) is a positive constant. The most common base is \(e\), where \(e\) is the natural number approximately equal to 2.71828.

• Exponential functions, such as \(e^x\), grow rapidly because their rate of growth is proportional to their value.
• The derivative of an exponential function is particularly simple: \(f'(x) = e^x\), the same as the original function.
• This property makes \(e^x\) exceptionally useful when analyzing continuous growth or decay.

The Maclaurin series for \(e^x\) is an infinite sum: \(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\). This series illustrates a direct application of exponential functions in calculation of functional approximations. By substituting \(x\) with other expressions, such as \(\sin x\), we can use exponential functions to engage more complex operations like in the expression \(e^{\sin x}\). This reveals the versatility and significance of the exponential function across different mathematical contexts.

trigonometric functions

Trigonometric functions like \( \sin x \), \( \cos x \), and \( \tan x \) describe relationships between angles and lengths in right-angled triangles. These functions also play a significant role in periodic phenomena, including waves and circular motion.

• The Maclaurin series for \( \sin x \) is \( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \).
• Such series allow for the approximation of trigonometric functions near zero, turning them into manageable polynomials.
• By substituting into other functions using their series, we can analyze more complex compositions like \( e^{\sin x} \).

Understanding trigonometric functions is essential for handling oscillations, frequency, and harmonic analysis. These functions are not only pivotal in math but are crucial to fields like physics, engineering, and computer science. The series expansions enable practical calculations and models, simplifying tasks such as deriving solutions of differential equations or performing Fourier analysis.