Problem 58 - Essential Calculus

Question

Use a computer algebra system to find the fifth-degree Taylor polynomial (centered at \(c\) ) for the function. Graph the function and the polynomial. Use the graph to determine the largest interval on which the polynomial is a reasonable approximation of the function. $$ h(x)=\sqrt[3]{x} \arctan x, \quad c=1 $$

Step-by-Step Solution

Answer

The fifth-degree Taylor polynomial for the function \(h(x)=\sqrt[3]{x} \arctan x\) centered at \(c=1\) is obtained by calculating the derivatives of the function at \(c=1\) up to the fifth derivative and substituting these values into the Taylor series formula. The interval on which the polynomial is a reasonable approximation of the function can be visually determined by graphing both the function and the Taylor polynomial and observing the range where they overlap significantly.

Step 1: Understanding Taylor series

The Taylor series of a function can be considered as an infinite sum of terms that are calculated from the function's derivatives at a single point. A Taylor polynomial is a sum of a finite number of terms of the Taylor series. The Taylor series of a function \( f(x) \) around a point \( c \) is given by the formula: \( P(x) = f(c) + f'(c) *(x-c) + f''(c) *(x-c)^2/2! + \ldots + f^(n)(c) *(x-c)^n/n! \).

Step 2: Calculating the Taylor Series using a computer algebra system

Using a computer algebra system like Mathematica, one can calculate the Taylor series of the given function \( h(x) \) around \( x = 1 \) up to the fifth degree. This can be done using the command: \n Taylor[h[x], {x, 1, 5}].

Step 3: Plotting the function and the Taylor polynomial

Again using the computer algebra system, plot both the original function and the resulting Taylor polynomial for comparison. Plot them together using the following commands: \n \n Plot[h[x], {x, -2, 2}]\n\n Plot[TaylorPolynomial, {x, -2, 2}].

Step 4: Determine the interval for a reasonable approximation

Based on the graph obtained from the previous step, visually inspect the range for which the Taylor polynomial closely approximates the original function. It's the interval where the graphs of the original function and the Taylor polynomial overlap significantly and the difference between them is minimal.

Key Concepts

Understanding Taylor Polynomials

A Taylor polynomial is essentially an approximation of a given function by way of a polynomial. When you calculate a Taylor polynomial of a function, you are creating an estimate of the function using a few specific terms taken from its Taylor series.

The Taylor series itself is like an infinite sum of terms based on the derivatives of a function at a certain point, often referred to as the center, represented by \(c\). When these terms are limited to a certain number, say up to the fifth degree, it forms a Taylor polynomial. For example, if we have a fifth-degree Taylor polynomial, it includes terms up to \((x - c)^5\).

In the problem, our goal is to approximate the function \(h(x) = \sqrt[3]{x} \arctan x\) using a fifth-degree Taylor polynomial centered at \(c = 1\). The Taylor series in this case would provide a finite polynomial:

• \(P(x) = f(1) + f'(1)\times(x-1) + … + f^{(5)}(1)\times(x-1)^5/5!\)

Using Taylor polynomials is very helpful for approximations, especially when dealing with complex functions that are otherwise difficult to handle.

Using a Computer Algebra System for Taylor Series

Computer algebra systems (CAS), such as Mathematica or MATLAB, are excellent tools for performing complex calculations like finding Taylor polynomials. These systems can easily handle symbolic algebra and calculus, providing precise solutions that would otherwise be challenging to compute manually.

To calculate the fifth-degree Taylor polynomial for the function \(h(x)\) using a CAS, you would use specific commands designed for such operations. For example, in Mathematica, you can use the command:

• \(\text{Taylor}[h[x], \{x, 1, 5\}]\)

This command tells the system to calculate the Taylor series for \(h(x) = \sqrt[3]{x} \arctan x\), specifically around \(x = 1\) and only up to the fifth degree.

The power of CAS comes in its ability to easily handle derivatives and generate polynomials of high degrees, even for complicated functions, simplifying the process significantly.

Graphical Analysis of Functions and Polynomials

Graphical analysis is a powerful way to visualize and compare a function and its Taylor polynomial. By plotting both the original function and the Taylor polynomial, we can observe how well the polynomial approximates the function over a specific range.

Using your computer algebra system, you can generate graphs for both expressions. The command may look something like this in Mathematica:

• \(\text{Plot}[h[x], \{x, -2, 2\}]\)
• \(\text{Plot}[\text{TaylorPolynomial}, \{x, -2, 2\}]\)

These plots will display the function \(h(x)\) and the corresponding fifth-degree Taylor polynomial, giving a visual representation of how the polynomial approaches the behavior of the original function.

By analyzing this graph, you can determine the interval over which the polynomial provides a reasonable approximation. Typically, this is where the two plots overlap closely without significant deviation. This graphical comparison makes it much easier to understand the effectiveness of the polynomial approximation.