Problem 57 - 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. $$ g(x)=\sqrt{x} \ln x, \quad c=1 $$

Step-by-Step Solution

Answer

The detailed solution would typically require a computer algebra system. The largest reasonable approximation interval can be estimated from the generated graph, which we cannot provide without performing the computation.

Step 1: Computing Taylor Polynomial

The Taylor series of a function \(f\) centered at \(c\) is given by: \[ f(x) = f(c) + f'(c)(x-c) +\frac{f''(c)}{2!}(x-c)^2 +\frac{f'''(c)}{3!}(x-c)^3 + \cdots \]. The fifth-degree Taylor series therefore includes only the terms up to \(x^5\). Utilize a computer algebra system to derive the necessary function derivatives and their values at \(c = 1\).

Step 2: Graphing the Function and the Taylor Polynomial

Plot both the original function \(g(x) = \sqrt{x} \ln{x}\) and the fifth-degree Taylor polynomial obtained in Step 1. These graphs should ideally be superimposed for easier comparison.

Step 3: Determining the Interval of Reasonable Approximation

Examine the graph to identify the largest interval within which the polynomial approximates the function with reasonable accuracy. Note that the appropriateness of the approximation typically reduces when moving away from the center \(c = 1\).

Key Concepts

Taylor Series

The Taylor series is a mathematical concept used to approximate more complex functions with polynomials, making advanced calculus more approachable. The essence of this series is to expand a function into an infinite sum of terms calculated from the values of its derivatives at a single point. For a function \( f(x) \), the Taylor series about the point \( c \) is given by:

• \( f(c) + f'(c)(x-c) + \frac{f''(c)}{2!}(x-c)^2 + \frac{f'''(c)}{3!}(x-c)^3 + \cdots \)

The beauty of the Taylor series lies in its ability to approximate functions that are otherwise difficult to compute directly. The series translates a function into a polynomial centered at \( c \), which becomes more accurate as more terms are taken into account.
When using a Taylor polynomial of degree five, only terms up to \( x^5 \) are considered. This means that the polynomial derived would be used as an approximation of the function, capturing its essential behavior near the point \( c = 1 \). The Taylor polynomial becomes a powerful tool when higher degree terms (such as ninth or tenth) are not necessary for an accurate approximation. Its utility lies in simplifying computations while keeping essential features of the function intact.

Function Approximation

Function approximation is a crucial aspect of mathematics, designed to simplify complex functions using less complicated polynomial expressions. When working with real-world data or mathematical models, exact calculations can often be challenging or even impossible due to the nature of certain functions.
By using methods like the Taylor polynomial, we can approximate the output of a complex function over a specified range of input values. This is particularly useful when predicting trends or simplifying real-world phenomena.

• It makes computation faster and more manageable.
• Results are easier to interpret.
• Approximation gives insight into the behavior of a function without detailed computations.

In our exercise, the fifth-degree Taylor polynomial serves as a function approximation of \( g(x) = \sqrt{x} \ln{x} \). It is particularly effective near the center, \( c = 1 \), but its accuracy diminishes as we move further from this point. To determine how well the polynomial approximates the original function, we graph both and observe the range where they align closely.

Computer Algebra System

A computer algebra system (CAS) is a powerful tool in modern mathematics, designed to perform symbolic computations. It aids in solving complex equations, simplifying expressions, and even conducting calculus operations like differentiation and integration. These systems process symbolic mathematics, enabling users to work precisely rather than numerically approximating.
In the context of developing Taylor polynomials, a CAS is particularly useful. Computing derivatives manually, especially for functions involving multiple terms like \( g(x) = \sqrt{x} \ln{x} \), can be tedious and error-prone. However, with a CAS:

• Derivatives are computed quickly and accurately.
• Algebraic manipulations become less prone to human error.
• It offers a convenient way to generate and visualize the Taylor polynomial alongside the original function.

For students and professionals alike, leveraging a CAS provides an efficient method to not only calculate high-order derivatives at the center point but also to graphically portray the polynomial and the function. This visualization helps in identifying the interval of best approximation, affording a visual clarity that numerical data alone might not immediately convey.