Problem 59 - Thomas Calculus

Question

\begin{equation} \begin{array}{l}{\text { Using a CAS, perform the following steps to aid in answering }} \\ {\text { questions (a) and (b) for the functions and intervals in Exercises }} \\ {57-62 .}\\\\{\text { Step } I : \text { Plot the function over the specified interval. }} \\ {\text { Step } 2 : \text { Find the Taylor polynomials } P_{1}(x), P_{2}(x), \text { and } P_{3}(x) \text { at }} \\ {x=0}\\\\{\text { Step } 3 : \text { Calculate the }(n+1) \text { st derivative } \int^{(n+1)}(c) \text { associat- }} \\ {\text { ed with the remainder term for each Taylor polynomial. Plot }} \\ {\text { the derivative as a function of } c \text { over the specified interval }} \\ {\text { and estimate its maximum absolute value, } M .}\\\\{\text { Step } 4 : \text { Calculate the remainder } R_{n}(x) \text { for each polynomial. }} \\ {\text { Using the estimate } M \text { from Step } 3 \text { in place of } f^{(n+1)}(c), \text { plot }} \\ {R_{n}(x) \text { over the specified interval. Then estimate the values of }} \\ {x \text { that answer question (a). }}\\\\{\text { Step } 5 : \text { Compare your estimated error with the actual error }} \\ {E_{n}(x)=\left|f(x)-P_{n}(x)\right| \text { by plotting } E_{n}(x) \text { over the specified }} \\ {\text { interval. This will help answer question (b). }} \\ {\text { Step } 6 : \text { Graph the function and its three Taylor approxima- }} \\ {\text { tions together. Discuss the graphs in relation to the informa- }} \\ {\text { tion discovered in Steps } 4 \text { and } 5 .}\end{array} \end{equation} $$f(x)=\frac{x}{x^{2}+1}, \quad|x| \leq 2$$

Step-by-Step Solution

Answer

Plot the function, find Taylor polynomials, estimate the errors, and compare the function with its approximations.

Step 1: Examining the Problem

The function we are considering is \( f(x) = \frac{x}{x^2 + 1} \) over the interval \([-2, 2]\). We need to follow six steps involving calculus and Taylor polynomials to analyze the function using a Computer Algebra System (CAS).

Step I: Plotting the Function

Using a CAS, plot the function \( f(x) = \frac{x}{x^2 + 1} \) over the interval \([-2, 2]\) to visualize its behavior. Make sure to observe any noticeable characteristics of the curve within this range.

Step II: Finding Taylor Polynomials

Compute the first three Taylor polynomials \( P_1(x), P_2(x), \) and \( P_3(x) \) centered at \( x = 0 \). These can be determined using Taylor series formula, where \( P_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(0)}{k!} x^k \). Derive \( f'(x), f''(x), \text{ and } f'''(x) \) at \( x = 0 \) to find the coefficients.

Step III: Calculating the Derivative for the Remainder Term

For each polynomial, compute the \((n+1)\)st derivative which in this case includes finding \( f^{(2)}(c) \text{ and } f^{(3)}(c) \) for different \( c \) in the interval. Plot these functions over \([-2, 2]\) and estimate the maximum absolute value \( M \).

Step IV: Calculating and Plotting the Remainders

Using the estimates \( M \) from Step 3, calculate the remainder terms \( R_n(x) = \frac{M}{(n+1)!}|x^{n+1}| \) for each polynomial. Plot these \( R_n(x) \) over the interval to identify where the estimates align closest to \( f(x) \), answering question (a).

Step V: Comparing Estimated and Actual Error

Calculate the actual error \( E_n(x) = |f(x) - P_n(x)| \) for each polynomial and plot them over \([-2, 2]\). This comparison will provide insight into the accuracy of \( P_n(x) \), which helps in answering question (b).

Step VI: Graphing the Function and Taylor Approximations

Graph \( f(x) \) along with its Taylor approximations \( P_1(x), P_2(x), \) and \( P_3(x) \) on the same axes. Compare how well each polynomial matches \( f(x) \) by visually inspecting the plots in light of findings from Steps 4 and 5.

Key Concepts

Taylor Series

Taylor Series are an essential concept in mathematics, particularly in calculus. They allow us to approximate complicated functions with polynomials that are relatively simple to work with. A Taylor Series for a function, centered at a point, is an infinite sum of terms calculated from the derivatives of that function evaluated at the center point.

The general formula for a Taylor series of a function \( f(x) \) centered at \( x=a \) is: \[ f(x) \approx P_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!} (x-a)^k \]
In practice, we usually work with Taylor polynomials, \( P_n(x) \), which are finite sums and provide an approximation of the function around a particular point within a specific interval.

These approximations are useful because polynomials are much simpler to compute and analyze compared to more complex functions. However, the quality of the approximation can vary depending on how many terms are used (the degree \( n \) of the polynomial) and how far the approximation is taken from the center point \( a \).

Remainder Term Analysis

When dealing with Taylor polynomials, a key consideration is understanding how accurate the polynomial approximation is in representing the actual function. This accuracy is gauged by the Remainder Term, which tells us how far the Taylor polynomial \( P_n(x) \) is from the true function \( f(x) \) at a given point.

The Remainder Term \( R_n(x) \) is given by: \[ R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1} \]
Here, \( c \) is some value between \( a \) and \( x \). The \((n+1)\)-th derivative of \( f \) at \( c \) considers the behavior of the function beyond the degree of the polynomial.

In practice, estimating the Remainder Term provides insight into how the polynomial converges to the function over the interval. By determining the maximum value of \( f^{(n+1)}(c) \) over the interval, we can estimate an upper bound for the remainder, helping us to understand the error involved in our approximation.

Error Estimation

Error Estimation is critical when using Taylor Polynomials to approximate functions. By determining how far off our approximation is, we can decide whether these approximations are suitable for our purposes. One method of estimating error is by comparing the Remainder Term with the actual error you observe when substituting values.

The actual error \( E_n(x) \) between the function \( f(x) \) and its Taylor polynomial \( P_n(x) \) is computed as: \[ E_n(x) = |f(x) - P_n(x)| \]
This absolute difference is plotted to see where the polynomial fits well and where it diverges from the function.

By comparing this calculated error with the error estimate from the remainder term, we can assess how reliable our polynomial approximation is. This insight is crucial for applications where precision is essential, and it helps in deciding whether to increase the degree of the Taylor polynomial for a more accurate approximation.

Graphical Comparison of Approximations

Comparing the graphs of a function and its Taylor approximations is a very visual way to understand how well these polynomials approximate the original function. By plotting \( f(x) \), and its Taylor polynomials \( P_1(x), P_2(x), \) and \( P_3(x) \), one can see how the polynomials converge to the function as the number of terms increases.

This graphical comparison helps in several ways:

• Identifying the range where the polynomial provides a good approximation.
• Visualizing the error margin between \( f(x) \) and \( P_n(x) \).
• Understanding the impact of adding more terms on the accuracy of the polynomial.

Analyzing these plots gives an intuitive sense of how Taylor polynomials "fit" around the function's key behaviors and features, such as slopes and curvatures, within the specified interval. This graphical approach is particularly useful for students and practitioners needing a clear, visual representation of approximation efficiency.