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Chapter 10: Problem 30

Use a sixth-degree Taylor polynomial centered at zero to approximate the integral. Function \(\quad\) Approximation \(f(x)=\frac{1}{\sqrt[3]{1+x^{2}}} \quad \int_{0}^{1 / 2} \frac{1}{\sqrt[3]{1+x^{2}}} d x\)

Short Answer

Expert verified
The approximation of the integral of the function \( f(x)=\frac{1}{\sqrt[3]{1+x^{2}}}\) from 0 to 1/2 using a sixth-degree Taylor polynomial centered at zero can be found following the steps: derivation of the sixth-degree Taylor Polynomial, integration of the Taylor polynomial and evaluation of the integral within the given limits.

Step by step solution

01

Derivation of the Taylor Polynomial

First, we need to find the sixth-degree Taylor polynomial for the function \(f(x)=\frac{1}{\sqrt[3]{1+x^{2}}\). A Taylor polynomial is given by the formula: \(P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ... + \frac{f^(n)(a)}{n!}(x-a)^n.\) In this case, our center is at zero (a=0) and we want a sixth-degree polynomial, therefore we need to find up to the sixth derivative of our function at zero and substitute them into the formula.
02

Integration of the Taylor Polynomial

Once we have the sixth-degree Taylor polynomial, we proceed by integrating this polynomial within the limits of 0 to 1/2: \( \int_{0}^{1/2} P_n(x) dx \). This should be easier than integrating the original function.
03

Evaluation of the Integral

After the integration in the second step, we have a polynomial which we can easily evaluate within the given limits. We get the value of the integral by subtracting the value of the polynomial at 0 from the value of the polynomial at 1/2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sixth-Degree Polynomial
A sixth-degree polynomial is a mathematical expression of the form \(c_0 + c_1x + c_2x^2 + c_3x^3 + c_4x^4 + c_5x^5 + c_6x^6\), where the highest term, or the degree of the polynomial, is 6.
These polynomials can model complex situations and approximate functions more accurately than polynomials of lower degrees.
  • The goal of creating such polynomials is often to use them in approximations, like the Taylor polynomial, to simplify complex functions.
  • Finding a sixth-degree polynomial involves calculating up to and including the sixth derivative of a function.
These derivatives are used because they provide information about the rate of change of the function at a specific point, enabling a detailed representation of the function's behavior near that point.
Integration of Polynomial
Integration is a mathematical process that enables us to find the area under curves or between lines, represented mathematically as functions.
When integrating a polynomial, each term of the polynomial is integrated separately.
  • The power rule for integration states that the integral of \(x^n\) is \(\frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration.
  • For instance, if a polynomial is \(ax^n\), the integral will be \(\frac{ax^{n+1}}{n+1} + C\).
In this exercise, integrating the sixth-degree Taylor polynomial is easier than the original function. This is because polynomials have a straightforward integration process, whereas the original function might be complex or not have a known antiderivative.
Taylor Polynomial Approximation
A Taylor polynomial approximation uses derivatives of a function to create a polynomial that closely resembles the function near a specific point.
This polynomial serves as an approximation tool, especially useful when the function itself is difficult to work with.
  • The formula for the Taylor polynomial is \(P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \ldots + \frac{f^{(n)}(a)}{n!}(x-a)^n\), where \(a\) is the center of the approximation.
  • This method simplifies the calculation of values and integrals for functions that are otherwise very complex.
By using up to the sixth derivative, the Taylor polynomial gives an accurate approximation of the original function near the specified center.
Centered at Zero
When a Taylor polynomial is centered at zero, it is also known as a Maclaurin series.
This occurs specifically when the center point \(a\) in the Taylor polynomial formula is set to zero.
  • Using zero as the center point simplifies calculations because all terms involving \(x-a\) become just \(x\), eliminating any need for subtraction from \(x\).
  • This type of polynomial is especially advantageous when evaluating integrals or derivatives at or near zero, reducing complexity even further.
The Maclaurin series hence provides a streamlined method to approximate functions by focusing around the origin point, keeping the mathematics as simple as possible while retaining accuracy.

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Most popular questions from this chapter

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