Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-18, find the terms through \(x^{5}\) in the Maclaurin series for \(f(x)\). Hint: It may be easiest to use known Maclaurin series and then perform multiplications, divisions, and so on. For example, \(\tan x=(\sin x) /(\cos x)\). $$ f(x)=\left(1-x^{2}\right)^{2 / 3} $$

Step-by-Step Solution

Answer

The Maclaurin series for \(f(x) = (1-x^2)^{2/3}\) is approximately \(1 - \frac{2}{3}x^2 + \frac{2}{9}x^4\) through \(x^5\).

Step 1: Identify the known Maclaurin series

First, recall the Maclaurin series for the basic functions. For example, the Maclaurin series for \((1-x)^{a}\) (where \(a\) can be any exponent) is given by:\[(1-x)^{a} = 1 + ax + \frac{a(a-1)}{2!}x^2 + \frac{a(a-1)(a-2)}{3!}x^3 + \cdots\]This series expansion can help in finding the Maclaurin series for \((1-x^2)^{2/3}\).

Step 2: Adjust series for the specific function

We need to adjust the known Maclaurin series for \((1-x)^{2/3}\) to match our function \((1-x^2)^{2/3}\). Substitute \(x^2\) in place of \(x\) in the series:\[(1-x^2)^{2/3} = 1 - \frac{2}{3}x^2 + \frac{2}{9}x^4 - \cdots\]This substitution changes the power of each term by replacing \(x\) with \(x^2\).

Step 3: Calculate terms up to \(x^5\)

To find terms through \(x^5\), calculate the first few terms from the series:- The term for \(x^0\) is 1.- The term for \(x^2\) is \(-\frac{2}{3}x^2\).- The term for \(x^4\) is \(\frac{2}{9}x^4\).Combining these terms gives:\[(1-x^2)^{2/3} \approx 1 - \frac{2}{3}x^2 + \frac{2}{9}x^4\]No \(x^1, x^3,\) or \(x^5\) terms appear in this expansion.

Key Concepts

Series Expansion

When tackling calculus problems, especially when dealing with complex functions, one approach that simplifies computations is using series expansion. Series expansion involves expressing a complicated function as a sum of terms from a simpler sequence of functions, usually polynomials. These polynomials are often easier to manage and apply for calculations.

In the context of Maclaurin series, which is a type of Taylor series centered at zero, the function is expressed as a power series. This makes it possible to approximate functions by considering polynomial expansions. For example, the original function \( \(1-x^{2}\)^{2/3} \) is approximated using the Maclaurin series into simpler polynomial terms like \[ 1 - \frac{2}{3}x^2 + \frac{2}{9}x^4 \cdots \] through a systematic transformation.

By expanding functions into a series, it becomes easier to compute derivatives, integrals, and solve differential equations. For students learning calculus or needing calculus tutoring, series expansions are indispensable in breaking down difficult expressions, providing a foundation for further advanced topics.

Power Series

A power series is an infinite sum of terms in the form \[ a_0 + a_1x + a_2x^2 + a_3x^3 + \cdots \] where each term is a power of \( x \). Power series are at the heart of series expansions and are used to represent functions as the sum of infinitely many terms. In calculus, they provide a powerful tool for analyzing and approximating functions.

Exploiting the properties of power series, we manipulate functions into a format where they can be easily differentiated, integrated, or evaluated. For instance, to handle the function \( \(1-x^2\)^{2/3} \), we adjusted the known Maclaurin series of \( \(1-x\)^{a} \) to plug in \( x^2 \) wherever \( x \) appears. By doing this, we maintain the simplicity of working with powers of \( x \), allowing us to isolate and compute specific terms like those needed up to \( x^5 \).

Power series expansions reduce complex expressions, enabling educators and learners to address otherwise challenging mathematical operations with greater ease.

Calculus Tutoring

Understanding calculus deeply calls for a sturdy grasp on fundamental principles such as series expansions and power series. In calculus tutoring, using techniques like series expansions transforms intimidating calculus problems into manageable pieces. Through tutelage, students encounter various strategies, such as substituting terms or adjusting existing series to fit new functions.

Tutors often use real-world examples to demonstrate these concepts. They might show how the Maclaurin series gives a better grasp of polynomial behavior using graphical illustrations of function approximations. This step-by-step guidance helps learners visualize how approximations are achieved by piecing together the components of the series.

By focusing on incremental practice, tutors encourage learners to discover the utility of these series in solving differential equations, evaluating limits, and estimating values. Regular practice ensures students build the self-confidence needed to apply these concepts independently. Tutoring sessions might include solving progressively harder problems, interactive discussions, and using computational tools to dynamically explore the implications of these mathematical principles.