Problem 44 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use a CAS to find the first four nonzero terms in the Maclaurin series for each of the following. Check Problems \(43-48\) to see that you get the same answers using the methods of Section \(9.7 .\) \(\exp \left(x^{2}\right)\)

Step-by-Step Solution

Answer

The first four nonzero terms are 1, \(x^2\), and \(\frac{1}{2}x^4\).

Step 1: Understand the Maclaurin Series

The Maclaurin series for a function \( f(x) \) is given by \( f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots \). This series is an expansion of the function about 0.

Step 2: Define the Function and Compute Derivatives

Our function is \( f(x) = e^{x^2} \). We need to find the first four derivatives and evaluate them at 0:- \( f(x) = e^{x^2} \)- \( f'(x) = 2xe^{x^2} \)- \( f''(x) = (4x^2 + 2)e^{x^2} \)- \( f'''(x) = (8x^3 + 12x)e^{x^2} \)- \( f^{(4)}(x) = (16x^4 + 48x^2 + 12)e^{x^2} \).

Step 3: Evaluate Derivatives at x = 0

Substitute \( x = 0 \) into each derivative to find:- \( f(0) = e^{0^2} = 1 \)- \( f'(0) = 2(0)e^{0^2} = 0 \)- \( f''(0) = (4(0)^2 + 2)e^{0^2} = 2 \)- \( f'''(0) = (8(0)^3 + 12(0))e^{0^2} = 0 \)- \( f^{(4)}(0) = (16(0)^4 + 48(0)^2 + 12)e^{0^2} = 12 \).

Step 4: Construct the Maclaurin Series

Using the derivatives, build the series:- The zeroth term is \( 1 \).- The first term is \( 0 \).- The second term is \( \frac{2}{2!}x^2 = x^2 \).- The third term is \( 0 \).- The fourth term is \( \frac{12}{4!}x^4 = \frac{1}{2}x^4 \).- Therefore, the Maclaurin series is \( 1 + x^2 + \frac{1}{2}x^4 + \cdots \).

Key Concepts

Taylor Series

The Taylor series is a foundational idea in calculus that allows us to represent complex functions with simpler polynomial forms. It expands a function at a point, providing an infinite sum of terms particularly useful in approximations. The general formula for the Taylor series of a function \( f(x) \) at a point \( a \) is given by:

• \( f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots \)

The usefulness of the Taylor series lies in its ability to convert intricate functions into polynomial expressions that are easier to manipulate. If this point \( a \) is zero, the Taylor series becomes a Maclaurin series. The terms in these series are derived from the function's derivatives calculated at the specific point. This allows for a focused expansion that simplifies many calculus problems.

Calculus

Calculus, the mathematical study of change, is divided into two main branches: differential calculus and integral calculus. Differential calculus concerns the concept of the derivative, which is the measure of how a function changes as its input changes. This branch of calculus is essential for constructing Taylor series.

• It provides the tools needed to create derivatives, which are pivotal in determining the polynomial form in a Taylor series.
• Calculus aids in understanding the behavior and rate of change in functions, which directly influences how functions are represented in series.

Whether dealing with acceleration, growth rates, or change over time, calculus provides the framework to analyze and simplify these concepts using derivatives and integrals.

Derivatives

Derivatives are a core concept in calculus, serving as the building blocks for functions like the Taylor series. They represent the rate of change of a quantity and are fundamental in understanding how a function behaves. Deriving the Maclaurin series requires calculating derivatives at a specific point (often zero).

• The first derivative, \( f'(x) \), represents the slope of the function at a point.
• The second derivative, \( f''(x) \), indicates the curvature of the function.
• Higher-order derivatives continue to reveal more about the function's behavior.

When constructing a Taylor or Maclaurin series, these derivatives help form each term of the series, making them vital for accurate mathematical expansion.

Mathematical Expansion

Mathematical expansion, like the Maclaurin series, allows a function to be expressed as an infinite sum of terms. This provides a simplified polynomial approximation of the function that is easier to work with. The expansion process breaks down complex functions into a series of simple, easy-to-calculate terms based on derivatives.

• In a Maclaurin series, which is a specific type of Taylor expansion centered at zero, each term is built using the derivatives of the function evaluated at zero.
• These expansions lead to series like \( 1 + x^2 + \frac{1}{2}x^4 + \cdots \), replacing the original function with an infinite sum of polynomial terms.

Using this method, a mathematical expansion transforms complex functions into familiar polynomial expressions, making them considerably more manageable.