Problem 15 - Essential Calculus

Question

In Exercises \(7-18\), find the Maclaurin polynomial of degree \(n\) for the function. $$ f(x)=\frac{1}{x+1}, \quad n=4 $$

Step-by-Step Solution

Answer

The Maclaurin polynomial of degree 4 for the function \(f(x) = \frac{1}{x+1}\) is \(1 -x + x^2 - x^3 + x^4\)

Step 1: Compute the First Few Derivatives

Start by computing the value of the function and its first four derivatives at \(x = 0\). These will provide the coefficients needed for the polynomial. Below are the computations:For \(f(x) = \frac{1}{x+1}\):At \(x=0\), \(f(0) = 1\)First derivative, \(f'(x) = \frac{-1}{(x+1)^2}\):At \(x=0\),\(f'(0) = -1\)Second derivative, \(f''(x) = \frac{2}{(x+1)^3}\):At \(x=0\), \(f''(0) = 2\)Third derivative, \(f'''(x) = \frac{-6}{(x+1)^4}\):At \(x=0\), \(f'''(0) = -6\)Fourth derivative, \(f''''(x) = \frac{24}{(x+1)^5}\):At \(x=0\), \(f''''(0) = 24\)

Step 2: Generate the Maclaurin Polynomial Using the Coefficients

Now that all the coefficients for the polynomial have been calculated, substitute these values into the Maclaurin series formula:\(f(x) ≈ f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \frac{f''''(0)}{4!}x^4\)Making these substitutions gives the Maclaurin polynomial of degree 4 for the function \(f(x)\):\(f(x) ≈ 1 -x + x^2 - x^3 + x^4\)

Key Concepts

Calculus

Calculus is a branch of mathematics that involves the study of rates of change and the accumulation of quantities. It is built upon two foundational concepts: differentiation, which deals with rates of change, and integration, concerning accumulation. Both concepts make extensive use of limits, a fundamental idea that considers the behavior of functions as inputs approach certain points or infinity.

When studying calculus, one important application is to understand how functions behave and can be approximated near specific points. This understanding is essential for fields such as engineering, economics, and the physical sciences, where we often use mathematical models to describe real-world phenomena and need to make accurate predictions.

Derivatives

Derivatives are a cornerstone of differential calculus. They represent the rate at which a function's output changes as its input changes. In more straightforward terms, the derivative of a function at a particular point describes the slope of the line tangent to the function's graph at that point.

In the process of finding a Maclaurin polynomial, calculating the derivatives at the point of expansion (usually at x=0) is crucial. These derivatives give us the coefficients for each term in the polynomial. Understanding their computation is vital for constructing accurate series representations of functions.

Series Expansion

Series expansion is a mathematical method used to represent functions as the sum of an infinite series of terms. This is particularly useful when dealing with complex functions that are difficult to calculate. Through series expansion, we can approximate these functions using simpler polynomials that are more manageable.

There are several kinds of series expansions, including Taylor and Maclaurin series, each useful in different contexts. They are powerful tools since they allow us to approximate functions near a point to a high degree of accuracy, depending on the number of terms included in the series.

Maclaurin Series

The Maclaurin series is a special case of the Taylor series, centered at zero. It's an expansion of a function into an infinite sum of terms calculated from the derivatives of a function at a single point. The general form of a Maclaurin series for a function f(x) is given by:
\[ f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \frac{f''''(0)}{4!}x^4 + \cdots \]
Each term in the series has a coefficient derived from evaluating the function’s nth derivative at zero, divided by n! (n factorial). As the degree of the polynomial increases, the approximation becomes more accurate. The exercise involving the function \( f(x) = \frac{1}{x+1} \) and finding its Maclaurin polynomial up to degree 4 demonstrates this series expansion in action, approximating the function nearby the origin.