Problem 25 - Essential Calculus

Question

Find the Maclaurin series for the function. (Use the table of power series for elementary functions.) $$ f(x)=\cos x^{3 / 2} $$

Step-by-Step Solution

Answer

The Maclaurin series for the function \( f(x) = \cos(x^{3/2}) \) is: \( f(x) = \cos(x^{3/2}) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{3n} \).

Step 1: Recall the Maclaurin series for \( \cos x \)

Recall that the Maclaurin series for \( \cos x \) is given by: \( \cos x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n} = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + \dots \). This is the foundational formula this problem will be built upon.

Step 2: Substitute \( x^{3/2} \) into the Maclaurin series formula

We know that \( f(x) = \cos(x^{3/2}) \) . So, we should substitute \( x^{3/2} \) everywhere we see \( x \) in the Maclaurin series formula for \( \cos x \). So we obtain: \( f(x) = \cos(x^{3/2}) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} (x^{3/2})^{2n} \). This gives the Maclaurin series for the given function.

Step 3: Simplify the expression

We simplify the series expression to make it easier to understand. To simplify, we need to handle the expression \( (x^{3/2})^{2n} \). This simplifies as follows: \( (x^{3/2})^{2n} = x^{3n} \). Thus, the series becomes: \( f(x) = \cos(x^{3/2}) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{3n} \). This is the Maclaurin series for the function \( f(x) = \cos(x^{3/2}) \).

Key Concepts

Calculus and Maclaurin Series

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. A powerful application of calculus is the use of Maclaurin series, which are a type of power series that allow us to express complex functions as an infinite sum of terms based on the derivatives of the function at a single point (usually zero).

The Maclaurin series for a function is found by taking the function's derivatives, evaluating them at zero, and then constructing a series from these values. This series is crucial for approximating the values of functions and for performing calculations in various fields of science and engineering where exact solutions are not feasible or necessary. The basic formula for a Maclaurin series is given by: \
\[ f(x) = \frac{f(0)}{0!} + \frac{f'(0)}{1!}x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \text{...} \]
Where \( f^{(n)}(0) \) denotes the \(n\)-th derivative of the function evaluated at zero, and \( n! \) is the factorial of \(n\). Knowing how to apply and manipulate these series is a fundamental skill in calculus, as it intertwines concepts of differentiation and series.

Understanding Power Series

A power series is an infinite series of the form \
\[ \text{Power Series: } \[ \text{Power Series: } \sum_{n=0}^{\text{\infty}} a_n (x - c)^n \]\]
where \(a_n\) are coefficients and \(c\) is the center of the series. In practical terms, power series are like polynomials with infinitely many terms and can be used to represent a wide range of functions, including exponential, logarithmic, and trigonometric functions.

When the center \(c\) is zero, the series is specifically called a Maclaurin series. One of the primary reasons for using power series in calculus is their simplicity in calculation. Additionally, they can converge to the function's value within a certain interval or radius of convergence around the center point, providing an efficient tool for function approximation and analysis.

Power series can be added, subtracted, and multiplied in ways similar to polynomials, making them incredibly versatile. They also form the foundation for more complex concepts like Taylor series, which are power series centered at points other than zero.

Trigonometric Functions and Their Series

Trigonometric functions are fundamental in mathematics, and understanding their behavior through series expansions extends their applicability. By expressing trigonometric functions as infinite series, we can calculate their values at various points and apply them in contexts where direct computation isn't possible. For instance, the Maclaurin series for \(\cos x\) is: \
\[ \cos x = \sum_{n=0}^{\infty} \frac{{(-1)^n}}{{(2n)!}} x^{2n} = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + \dots \]
Similarly, other trigonometric functions like \(\sin x\) and \(\tan x\) have their own series representations.

These series not only serve computational purposes but also help in the analysis of trigonometric functions, such as finding limits, derivatives, and integrals. When dealing with complicated compositions of trigonometric functions, like \(\cos(x^{3/2})\) in the exercise, these series become invaluable tools for simplifying and understanding the behavior of the functions over complex domains.