Problem 40 - Calculus: One and Several Variables

Question

Find the Lagrange form of the remainder \(R_{n}(x)\). $$f(x)=\frac{1}{1+x} ; \quad n=4$$

Step-by-Step Solution

Answer

The Lagrange form of the remainder for the function \(f(x) = \frac{1}{1+x}\) with \(n=4\) is: \[ R_4(x) ≤ \frac{x^5}{5} \]

Step 1: Recall the Lagrange form of the remainder

The Lagrange form of the remainder is given by the following formula: \[ R_n(x) = \frac{f^{(n+1)}(\xi)}{(n + 1)!} (x - a)^{n + 1} \] Here, \(f^{(n+1)}(\xi)\) is the \((n + 1)^{\text{th}}\) derivative of the function evaluated at some point \(\xi\) between \(a\) and \(x\). The goal is to find an estimate for the error when using the Taylor polynomial of degree \(n\).

Step 2: Differentiate the function \(f(x)\) up to the \(4^{th}\) time

In order to find the Lagrange form of the remainder, we first need to find the derivatives of the function \(f(x) = \frac{1}{1+x}\) up to the \(4^{th}\) order. 1st derivative: \(f'(x) = -\frac{1}{(1+x)^2}\) 2nd derivative: \(f''(x) = \frac{2}{(1+x)^3}\) 3rd derivative: \(f'''(x) = -\frac{6}{(1+x)^4}\) 4th derivative: \(f^{(4)}(x) = \frac{24}{(1+x)^5}\) We will use this \(4^{th}\) derivative in the next step to find an estimate for the error bound.

Step 3: Find the maximum of the \(4^{th}\) derivative

Since the error depends on the maximum value of the \((n + 1)^{\text{th}}\) derivative in the given interval, we should find the maximum of this derivative. In this case, the \(4^{th}\) derivative is given by: \[ f^{(4)}(x) = \frac{24}{(1+x)^5} \] This is a monotonically decreasing function for any \(x > -1\), as the denominator increases. So, the maximum value of the \(4^{th}\) derivative is obtained at the leftmost point \(x = a\). For our purposes, let's consider \(a=0\), so we've got: \[ f^{(4)}(0) = \frac{24}{(1+0)^5} = 24 \]

Step 4: Write the formula for the Lagrange form of the remainder

Now that we have the maximum value of the 4th derivative, we can write the formula for the remainder, i.e., the error estimate: \[ R_4(x) = \frac{f^{(4)}(\xi)}{5!}(x - a)^{5} \] Since \(f^{(4)}(\xi) ≤ 24\), the error estimate can be given by: \[ R_4(x) ≤ \frac{24}{5!}(x - a)^{5} = \frac{24}{120}x^{5} = \frac{x^5}{5} \] So, the Lagrange form of the remainder for this function is: \[ R_4(x) ≤ \frac{x^5}{5} \]

Key Concepts

Taylor polynomial

A Taylor polynomial is a mathematical tool that approximates a function using a finite sum of terms derived from the function's derivatives at a single point. Imagine expanding a function into a polynomial but stopping after a certain number of terms. This approximation becomes particularly useful for computational purposes as it simplifies complex functions into more manageable expressions.

• Each term in the polynomial is associated with a derivative of the function, taken at a specific point called the center, usually denoted as \(a\).
• The more terms you include, the better the polynomial approximates the function around that point.
• This tool becomes particularly important when dealing with functions that are challenging to compute directly.

For instance, in this exercise, we are looking into the Taylor polynomial of degree 4 for the function \(f(x) = \frac{1}{1+x}\). This means we are using up to the 4th derivative of \(f(x)\) to form the polynomial. The sum of these terms offers an approximation of the original function.

n-th derivative

Derivatives are fundamental in calculus and they represent the rate at which a function changes. The first derivative tells you how fast the function's value is changing. But, what if you want to know how that rate itself is changing? That's where higher-order derivatives come in.
The \(n\)-th derivative of a function is a further extension of this concept. When we call a derivative the \(n\)-th, it implies that we differentiated the function \(n\) times. Here's a breakdown:

• The first derivative gives the slope of the tangent line at a point.
• The second derivative gives information about the curvature of the function.
• Continuing this process gives us the \(n\)-th derivative.

In our given problem, the function \(f(x) = \frac{1}{1+x}\) has its derivatives calculated up to the 4th degree. These derivatives help form the terms of the Taylor polynomial. Calculating these derivatives help us find the Lagrange remainder, which is crucial in estimating how good our approximation is.

error estimate

Whenever you approximate a function using a Taylor polynomial, knowing how close your approximation is to the actual function is essential. This difference or potential discrepancy is expressed as the error.
The Lagrange Remainder Theorem comes into play here. It provides a formula to estimate the maximum possible error when a Taylor polynomial is used for approximation:
\[ R_n(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{n+1} \]
In this formula:

• \(f^{(n+1)}(\xi)\) is the \((n+1)^{th}\) derivative evaluated at some point \(\xi\) within the interval of approximation.
• \((n+1)!\) is the factorial of \(n+1\) and serves to scale the derivative appropriately.
• \((x-a)^{n+1}\) indicates how far you are from the center point \(a\).

Using the fourth derivative in our example, we estimated the error as \(\frac{x^5}{5}\). This tells us just how confident we can be that our polynomial is a good approximation of the original function within a specific interval.

monotonically decreasing function

A function is "monotonically decreasing" if, as you move along the x-axis from left to right, the function either steadily decreases or remains constant. These functions are always falling or flat, never rising. Recognizing a function of this sort is important, especially when estimating maximum error bounds.
In our example, the fourth derivative \(f^{(4)}(x) = \frac{24}{(1+x)^5}\) is a monotonically decreasing function for \(x > -1\). But, why is that? As \(x\) increases, the denominator \((1+x)^5\) grows, leading to a smaller value of \(f^{(4)}(x)\).

• This characteristic signifies that the highest value occurs early in the interval.
• For our scenario of approximation, we need this derivative to find maximum possible error bounds.

Knowing where a function reaches its peak within an interval gives us the worst-case scenario for the error. This knowledge is utilized to establish confidence in the accuracy of the Taylor polynomial.