Partial fraction method examples pdf

Partial fraction method examples pdf
Partial Fractions Problems and Examples Practice problems. Remember that we will only cover partial fraction decompositions where the denominator factors
Some Comments on the two methods for handling complex roots. The two previous examples have demonstrated two techniques for performing a partial fraction expansion of a term with complex roots.
The method is called “Partial Fraction Decomposition”, and goes like this: Step 1: Factor the bottom Step 2: Write one partial fraction for each of those factors
Conceptual Understanding: Your lecture described four cases for partial fraction decomposition. An example of each case occurs below. Write out the general form for the partial fraction decom-
Various methods exist for finding the coefficients for the complex poles using partial fraction expansion. The coefficients for the real poles–distinct or repeated–are found in …
What is it? The cover-up rule is a technique for streamlining the calculations when setting up partial fractions. It is simpler to apply than the usual long-winded technique and as a result makes for more confident and so more accurate work.
The method for computing partial fraction decompositions applies to all rational functions with one qualification: The degree of the numerator must be less than the degree of the denomi-nator. One can always arrange this by using polynomial long division, as we shall see in the examples. Looking at the example above (in Equation 1), the denominator of the right side is. Factoring the
method for nding the partial fraction decomposition of a rational function in the spe- cial case when the denominator is a power of a single linear or irreducible quadratic factor, that is, the denominator is either (ax + b) k or (ax 2 + bx + c) k with 4 ac > b 2 .

Partial Fractions Examples Partial fractions is the name given to a technique, used, for example, in evaluating integrals that is used to express any ratio of polynomials in a simple standard form.
Examples of the Method of Partial Fractions October 20, 2011 Question: How do we compute Z x+ 5 x2 + x 2 dx? Answer: The key is to factor the denominator to get
Find the partial fraction decomposition of this expression using the full factorization mode. In this mode, partfrac factors the denominator into linear expressions, reducing quadratic polynomials to linear expressions with complex coefficients.
The standard method of PFD is the Method of Undetermined Coefficients (MUC), with which the calculation is often tedious. However, we can formulize the process of partial fractions, and then all
Abstract—An improved Heaviside approach to compute the partial fraction expansions of proper rational functions is presented. This method involves simple substitutions and polynomial divisions only, without the use of differentiation or solution of a system of linear equations. Examples on its applications in some topics of engineering mathematics, such as indefinite integration, inverse
Section 5-5 : Partial Fractions. This section doesn’t really have a lot to do with the rest of this chapter, but since the subject needs to be covered and this was a fairly short chapter it seemed like as good a place as any to put it.
The method of partial fractions is a technique for decomposing functions like Y(s) above so that the inverse transform can be determined in a straightforward manner.

The Method of Partial Fractions Math 121 Calculus II Clark U

https://youtube.com/watch?v=4lOoOJEjyuU


2. Rational functions and partial fractions

Laplace Substitution Method for Solving Partial Differential Equations Involving Mixed Partial Derivatives we will apply it to some examples and in last section we give some conclusion. 2. Laplace Substitution Method: The aim of this section is to discuss the Laplace substitution method. We consider the general form of non homogeneous partial differential equation with initial conditions
The techniques for computing the Partial Fraction Decomposition are numerous to say the least and tend to fall into two categories, general methods which will work for any decomposition and
2. Rational functions and partial fractions 2.1. Rational functions A rational function is a function of the form f(x) = p(x) q(x) where p(x) and q(x) are polynomials in x with q ≡ 0. orF example
Partial Fractions and the Coverup Method 18.031 Haynes Miller and Jeremy Orlo *Much of this note is freely borrowed from an MIT 18.01 note written by Arthur Mattuck. 1 Heaviside Cover-up Method 1.1 Introduction The cover-up method was introduced by Oliver Heaviside as a fast way to do a decomposition into partial fractions. This is an essential step in using the Laplace transform to …
The Method of Partial Fractions Math 121 Calculus II Spring 2015 Rational functions. Recall that a rational function is a quotient of two polynomials such


Partial fraction decomposition allows complicated polynomial fractions to be written as the sum of simpler fractions. In this lesson, we used examples to showcase the rules for four cases of
S. Boyd EE102 Lecture 5 Rational functions and partial fraction expansion †(reviewof)polynomials †rationalfunctions †pole-zeroplots †partialfractionexpansion
In this paper, we introduce a new approach to compute the partial fraction decompositions of rational functions and describe the results of its trials at three secondary schools in Hong Kong.

Partial Fractions and the Coverup Method Haynes Miller and

(PDF) Teaching a new method of partial fraction

MA 114 Worksheet # 19 Integration by Partial Fractions


Partial fraction decomposition MATLAB partfrac

https://youtube.com/watch?v=paEXW4RHg2g

Partial Fraction Decomposition Rules & Examples Video


An Improved Heaviside Approach to Partial Fraction

Partial Fractions by Substitution

https://youtube.com/watch?v=2bMndF92Oqo

Cover-up Method for Partial Fractions

or how to make partial fractions easy Meikleriggs

Method of Partial Fractions Oregon State University


Lecture 5 Rational functions and partial fraction expansion

https://youtube.com/watch?v=HZTv4zCgEnA

The Method of Partial Fractions University of Michigan

The Method of Partial Fractions University of Michigan
(PDF) Teaching a new method of partial fraction

method for nding the partial fraction decomposition of a rational function in the spe- cial case when the denominator is a power of a single linear or irreducible quadratic factor, that is, the denominator is either (ax b) k or (ax 2 bx c) k with 4 ac > b 2 .
2. Rational functions and partial fractions 2.1. Rational functions A rational function is a function of the form f(x) = p(x) q(x) where p(x) and q(x) are polynomials in x with q ≡ 0. orF example
What is it? The cover-up rule is a technique for streamlining the calculations when setting up partial fractions. It is simpler to apply than the usual long-winded technique and as a result makes for more confident and so more accurate work.
The Method of Partial Fractions Math 121 Calculus II Spring 2015 Rational functions. Recall that a rational function is a quotient of two polynomials such
Partial Fractions and the Coverup Method 18.031 Haynes Miller and Jeremy Orlo *Much of this note is freely borrowed from an MIT 18.01 note written by Arthur Mattuck. 1 Heaviside Cover-up Method 1.1 Introduction The cover-up method was introduced by Oliver Heaviside as a fast way to do a decomposition into partial fractions. This is an essential step in using the Laplace transform to …

Cover-up Method for Partial Fractions
The Method of Partial Fractions University of Michigan

The method for computing partial fraction decompositions applies to all rational functions with one qualification: The degree of the numerator must be less than the degree of the denomi-nator. One can always arrange this by using polynomial long division, as we shall see in the examples. Looking at the example above (in Equation 1), the denominator of the right side is. Factoring the
Abstract—An improved Heaviside approach to compute the partial fraction expansions of proper rational functions is presented. This method involves simple substitutions and polynomial divisions only, without the use of differentiation or solution of a system of linear equations. Examples on its applications in some topics of engineering mathematics, such as indefinite integration, inverse
Partial Fractions and the Coverup Method 18.031 Haynes Miller and Jeremy Orlo *Much of this note is freely borrowed from an MIT 18.01 note written by Arthur Mattuck. 1 Heaviside Cover-up Method 1.1 Introduction The cover-up method was introduced by Oliver Heaviside as a fast way to do a decomposition into partial fractions. This is an essential step in using the Laplace transform to …
Partial Fractions Examples Partial fractions is the name given to a technique, used, for example, in evaluating integrals that is used to express any ratio of polynomials in a simple standard form.
The techniques for computing the Partial Fraction Decomposition are numerous to say the least and tend to fall into two categories, general methods which will work for any decomposition and
Find the partial fraction decomposition of this expression using the full factorization mode. In this mode, partfrac factors the denominator into linear expressions, reducing quadratic polynomials to linear expressions with complex coefficients.
Various methods exist for finding the coefficients for the complex poles using partial fraction expansion. The coefficients for the real poles–distinct or repeated–are found in …

The Method of Partial Fractions University of Michigan
Partial Fraction Decomposition Rules & Examples Video

S. Boyd EE102 Lecture 5 Rational functions and partial fraction expansion †(reviewof)polynomials †rationalfunctions †pole-zeroplots †partialfractionexpansion
Partial Fractions and the Coverup Method 18.031 Haynes Miller and Jeremy Orlo *Much of this note is freely borrowed from an MIT 18.01 note written by Arthur Mattuck. 1 Heaviside Cover-up Method 1.1 Introduction The cover-up method was introduced by Oliver Heaviside as a fast way to do a decomposition into partial fractions. This is an essential step in using the Laplace transform to …
Various methods exist for finding the coefficients for the complex poles using partial fraction expansion. The coefficients for the real poles–distinct or repeated–are found in …
The method for computing partial fraction decompositions applies to all rational functions with one qualification: The degree of the numerator must be less than the degree of the denomi-nator. One can always arrange this by using polynomial long division, as we shall see in the examples. Looking at the example above (in Equation 1), the denominator of the right side is. Factoring the
Some Comments on the two methods for handling complex roots. The two previous examples have demonstrated two techniques for performing a partial fraction expansion of a term with complex roots.
The Method of Partial Fractions Math 121 Calculus II Spring 2015 Rational functions. Recall that a rational function is a quotient of two polynomials such
What is it? The cover-up rule is a technique for streamlining the calculations when setting up partial fractions. It is simpler to apply than the usual long-winded technique and as a result makes for more confident and so more accurate work.
Partial Fractions Problems and Examples Practice problems. Remember that we will only cover partial fraction decompositions where the denominator factors

(PDF) Teaching a new method of partial fraction
Partial Fractions by Substitution

The method of partial fractions is a technique for decomposing functions like Y(s) above so that the inverse transform can be determined in a straightforward manner.
Examples of the Method of Partial Fractions October 20, 2011 Question: How do we compute Z x 5 x2 x 2 dx? Answer: The key is to factor the denominator to get
2. Rational functions and partial fractions 2.1. Rational functions A rational function is a function of the form f(x) = p(x) q(x) where p(x) and q(x) are polynomials in x with q ≡ 0. orF example
The method for computing partial fraction decompositions applies to all rational functions with one qualification: The degree of the numerator must be less than the degree of the denomi-nator. One can always arrange this by using polynomial long division, as we shall see in the examples. Looking at the example above (in Equation 1), the denominator of the right side is. Factoring the
Laplace Substitution Method for Solving Partial Differential Equations Involving Mixed Partial Derivatives we will apply it to some examples and in last section we give some conclusion. 2. Laplace Substitution Method: The aim of this section is to discuss the Laplace substitution method. We consider the general form of non homogeneous partial differential equation with initial conditions
Some Comments on the two methods for handling complex roots. The two previous examples have demonstrated two techniques for performing a partial fraction expansion of a term with complex roots.

Partial Fraction Decomposition Rules & Examples Video
The Method of Partial Fractions University of Michigan

Conceptual Understanding: Your lecture described four cases for partial fraction decomposition. An example of each case occurs below. Write out the general form for the partial fraction decom-
Section 5-5 : Partial Fractions. This section doesn’t really have a lot to do with the rest of this chapter, but since the subject needs to be covered and this was a fairly short chapter it seemed like as good a place as any to put it.
The Method of Partial Fractions Math 121 Calculus II Spring 2015 Rational functions. Recall that a rational function is a quotient of two polynomials such
What is it? The cover-up rule is a technique for streamlining the calculations when setting up partial fractions. It is simpler to apply than the usual long-winded technique and as a result makes for more confident and so more accurate work.
The standard method of PFD is the Method of Undetermined Coefficients (MUC), with which the calculation is often tedious. However, we can formulize the process of partial fractions, and then all
method for nding the partial fraction decomposition of a rational function in the spe- cial case when the denominator is a power of a single linear or irreducible quadratic factor, that is, the denominator is either (ax b) k or (ax 2 bx c) k with 4 ac > b 2 .
Abstract—An improved Heaviside approach to compute the partial fraction expansions of proper rational functions is presented. This method involves simple substitutions and polynomial divisions only, without the use of differentiation or solution of a system of linear equations. Examples on its applications in some topics of engineering mathematics, such as indefinite integration, inverse
Laplace Substitution Method for Solving Partial Differential Equations Involving Mixed Partial Derivatives we will apply it to some examples and in last section we give some conclusion. 2. Laplace Substitution Method: The aim of this section is to discuss the Laplace substitution method. We consider the general form of non homogeneous partial differential equation with initial conditions
The method is called “Partial Fraction Decomposition”, and goes like this: Step 1: Factor the bottom Step 2: Write one partial fraction for each of those factors
Various methods exist for finding the coefficients for the complex poles using partial fraction expansion. The coefficients for the real poles–distinct or repeated–are found in …
Partial Fractions and the Coverup Method 18.031 Haynes Miller and Jeremy Orlo *Much of this note is freely borrowed from an MIT 18.01 note written by Arthur Mattuck. 1 Heaviside Cover-up Method 1.1 Introduction The cover-up method was introduced by Oliver Heaviside as a fast way to do a decomposition into partial fractions. This is an essential step in using the Laplace transform to …
S. Boyd EE102 Lecture 5 Rational functions and partial fraction expansion †(reviewof)polynomials †rationalfunctions †pole-zeroplots †partialfractionexpansion

Partial Fractions by Substitution
An Improved Heaviside Approach to Partial Fraction

Partial Fractions and the Coverup Method 18.031 Haynes Miller and Jeremy Orlo *Much of this note is freely borrowed from an MIT 18.01 note written by Arthur Mattuck. 1 Heaviside Cover-up Method 1.1 Introduction The cover-up method was introduced by Oliver Heaviside as a fast way to do a decomposition into partial fractions. This is an essential step in using the Laplace transform to …
The Method of Partial Fractions Math 121 Calculus II Spring 2015 Rational functions. Recall that a rational function is a quotient of two polynomials such
Conceptual Understanding: Your lecture described four cases for partial fraction decomposition. An example of each case occurs below. Write out the general form for the partial fraction decom-
Laplace Substitution Method for Solving Partial Differential Equations Involving Mixed Partial Derivatives we will apply it to some examples and in last section we give some conclusion. 2. Laplace Substitution Method: The aim of this section is to discuss the Laplace substitution method. We consider the general form of non homogeneous partial differential equation with initial conditions
Find the partial fraction decomposition of this expression using the full factorization mode. In this mode, partfrac factors the denominator into linear expressions, reducing quadratic polynomials to linear expressions with complex coefficients.
The method for computing partial fraction decompositions applies to all rational functions with one qualification: The degree of the numerator must be less than the degree of the denomi-nator. One can always arrange this by using polynomial long division, as we shall see in the examples. Looking at the example above (in Equation 1), the denominator of the right side is. Factoring the
The method is called “Partial Fraction Decomposition”, and goes like this: Step 1: Factor the bottom Step 2: Write one partial fraction for each of those factors
The method of partial fractions is a technique for decomposing functions like Y(s) above so that the inverse transform can be determined in a straightforward manner.
In this paper, we introduce a new approach to compute the partial fraction decompositions of rational functions and describe the results of its trials at three secondary schools in Hong Kong.
S. Boyd EE102 Lecture 5 Rational functions and partial fraction expansion †(reviewof)polynomials †rationalfunctions †pole-zeroplots †partialfractionexpansion
Partial fraction decomposition allows complicated polynomial fractions to be written as the sum of simpler fractions. In this lesson, we used examples to showcase the rules for four cases of
Partial Fractions Problems and Examples Practice problems. Remember that we will only cover partial fraction decompositions where the denominator factors
Section 5-5 : Partial Fractions. This section doesn’t really have a lot to do with the rest of this chapter, but since the subject needs to be covered and this was a fairly short chapter it seemed like as good a place as any to put it.
The standard method of PFD is the Method of Undetermined Coefficients (MUC), with which the calculation is often tedious. However, we can formulize the process of partial fractions, and then all

Lecture 5 Rational functions and partial fraction expansion
Partial fraction decomposition MATLAB partfrac

Partial fraction decomposition allows complicated polynomial fractions to be written as the sum of simpler fractions. In this lesson, we used examples to showcase the rules for four cases of
The Method of Partial Fractions Math 121 Calculus II Spring 2015 Rational functions. Recall that a rational function is a quotient of two polynomials such
In this paper, we introduce a new approach to compute the partial fraction decompositions of rational functions and describe the results of its trials at three secondary schools in Hong Kong.
Partial Fractions Problems and Examples Practice problems. Remember that we will only cover partial fraction decompositions where the denominator factors
S. Boyd EE102 Lecture 5 Rational functions and partial fraction expansion †(reviewof)polynomials †rationalfunctions †pole-zeroplots †partialfractionexpansion
Partial Fractions Examples Partial fractions is the name given to a technique, used, for example, in evaluating integrals that is used to express any ratio of polynomials in a simple standard form.
Various methods exist for finding the coefficients for the complex poles using partial fraction expansion. The coefficients for the real poles–distinct or repeated–are found in …
Some Comments on the two methods for handling complex roots. The two previous examples have demonstrated two techniques for performing a partial fraction expansion of a term with complex roots.
Partial Fractions and the Coverup Method 18.031 Haynes Miller and Jeremy Orlo *Much of this note is freely borrowed from an MIT 18.01 note written by Arthur Mattuck. 1 Heaviside Cover-up Method 1.1 Introduction The cover-up method was introduced by Oliver Heaviside as a fast way to do a decomposition into partial fractions. This is an essential step in using the Laplace transform to …
The method of partial fractions is a technique for decomposing functions like Y(s) above so that the inverse transform can be determined in a straightforward manner.
The method is called “Partial Fraction Decomposition”, and goes like this: Step 1: Factor the bottom Step 2: Write one partial fraction for each of those factors

Partial fraction decomposition MATLAB partfrac
2. Rational functions and partial fractions

method for nding the partial fraction decomposition of a rational function in the spe- cial case when the denominator is a power of a single linear or irreducible quadratic factor, that is, the denominator is either (ax b) k or (ax 2 bx c) k with 4 ac > b 2 .
The method for computing partial fraction decompositions applies to all rational functions with one qualification: The degree of the numerator must be less than the degree of the denomi-nator. One can always arrange this by using polynomial long division, as we shall see in the examples. Looking at the example above (in Equation 1), the denominator of the right side is. Factoring the
Partial Fractions and the Coverup Method 18.031 Haynes Miller and Jeremy Orlo *Much of this note is freely borrowed from an MIT 18.01 note written by Arthur Mattuck. 1 Heaviside Cover-up Method 1.1 Introduction The cover-up method was introduced by Oliver Heaviside as a fast way to do a decomposition into partial fractions. This is an essential step in using the Laplace transform to …
Some Comments on the two methods for handling complex roots. The two previous examples have demonstrated two techniques for performing a partial fraction expansion of a term with complex roots.

An Improved Heaviside Approach to Partial Fraction
Cover-up Method for Partial Fractions

Laplace Substitution Method for Solving Partial Differential Equations Involving Mixed Partial Derivatives we will apply it to some examples and in last section we give some conclusion. 2. Laplace Substitution Method: The aim of this section is to discuss the Laplace substitution method. We consider the general form of non homogeneous partial differential equation with initial conditions
Conceptual Understanding: Your lecture described four cases for partial fraction decomposition. An example of each case occurs below. Write out the general form for the partial fraction decom-
Some Comments on the two methods for handling complex roots. The two previous examples have demonstrated two techniques for performing a partial fraction expansion of a term with complex roots.
method for nding the partial fraction decomposition of a rational function in the spe- cial case when the denominator is a power of a single linear or irreducible quadratic factor, that is, the denominator is either (ax b) k or (ax 2 bx c) k with 4 ac > b 2 .
S. Boyd EE102 Lecture 5 Rational functions and partial fraction expansion †(reviewof)polynomials †rationalfunctions †pole-zeroplots †partialfractionexpansion

Partial Fractions by Substitution
Lecture 5 Rational functions and partial fraction expansion

Partial Fractions Examples Partial fractions is the name given to a technique, used, for example, in evaluating integrals that is used to express any ratio of polynomials in a simple standard form.
Partial fraction decomposition allows complicated polynomial fractions to be written as the sum of simpler fractions. In this lesson, we used examples to showcase the rules for four cases of
Conceptual Understanding: Your lecture described four cases for partial fraction decomposition. An example of each case occurs below. Write out the general form for the partial fraction decom-
Section 5-5 : Partial Fractions. This section doesn’t really have a lot to do with the rest of this chapter, but since the subject needs to be covered and this was a fairly short chapter it seemed like as good a place as any to put it.
The method of partial fractions is a technique for decomposing functions like Y(s) above so that the inverse transform can be determined in a straightforward manner.
Find the partial fraction decomposition of this expression using the full factorization mode. In this mode, partfrac factors the denominator into linear expressions, reducing quadratic polynomials to linear expressions with complex coefficients.
The Method of Partial Fractions Math 121 Calculus II Spring 2015 Rational functions. Recall that a rational function is a quotient of two polynomials such
What is it? The cover-up rule is a technique for streamlining the calculations when setting up partial fractions. It is simpler to apply than the usual long-winded technique and as a result makes for more confident and so more accurate work.
Some Comments on the two methods for handling complex roots. The two previous examples have demonstrated two techniques for performing a partial fraction expansion of a term with complex roots.
2. Rational functions and partial fractions 2.1. Rational functions A rational function is a function of the form f(x) = p(x) q(x) where p(x) and q(x) are polynomials in x with q ≡ 0. orF example
Various methods exist for finding the coefficients for the complex poles using partial fraction expansion. The coefficients for the real poles–distinct or repeated–are found in …
Partial Fractions and the Coverup Method 18.031 Haynes Miller and Jeremy Orlo *Much of this note is freely borrowed from an MIT 18.01 note written by Arthur Mattuck. 1 Heaviside Cover-up Method 1.1 Introduction The cover-up method was introduced by Oliver Heaviside as a fast way to do a decomposition into partial fractions. This is an essential step in using the Laplace transform to …
Abstract—An improved Heaviside approach to compute the partial fraction expansions of proper rational functions is presented. This method involves simple substitutions and polynomial divisions only, without the use of differentiation or solution of a system of linear equations. Examples on its applications in some topics of engineering mathematics, such as indefinite integration, inverse

The Method of Partial Fractions University of Michigan
MA 114 Worksheet # 19 Integration by Partial Fractions

Partial Fractions Examples Partial fractions is the name given to a technique, used, for example, in evaluating integrals that is used to express any ratio of polynomials in a simple standard form.
Abstract—An improved Heaviside approach to compute the partial fraction expansions of proper rational functions is presented. This method involves simple substitutions and polynomial divisions only, without the use of differentiation or solution of a system of linear equations. Examples on its applications in some topics of engineering mathematics, such as indefinite integration, inverse
Partial Fractions Problems and Examples Practice problems. Remember that we will only cover partial fraction decompositions where the denominator factors
The techniques for computing the Partial Fraction Decomposition are numerous to say the least and tend to fall into two categories, general methods which will work for any decomposition and
In this paper, we introduce a new approach to compute the partial fraction decompositions of rational functions and describe the results of its trials at three secondary schools in Hong Kong.
Section 5-5 : Partial Fractions. This section doesn’t really have a lot to do with the rest of this chapter, but since the subject needs to be covered and this was a fairly short chapter it seemed like as good a place as any to put it.
2. Rational functions and partial fractions 2.1. Rational functions A rational function is a function of the form f(x) = p(x) q(x) where p(x) and q(x) are polynomials in x with q ≡ 0. orF example
Conceptual Understanding: Your lecture described four cases for partial fraction decomposition. An example of each case occurs below. Write out the general form for the partial fraction decom-
Partial fraction decomposition allows complicated polynomial fractions to be written as the sum of simpler fractions. In this lesson, we used examples to showcase the rules for four cases of

Cover-up Method for Partial Fractions
2. Rational functions and partial fractions

Laplace Substitution Method for Solving Partial Differential Equations Involving Mixed Partial Derivatives we will apply it to some examples and in last section we give some conclusion. 2. Laplace Substitution Method: The aim of this section is to discuss the Laplace substitution method. We consider the general form of non homogeneous partial differential equation with initial conditions
The techniques for computing the Partial Fraction Decomposition are numerous to say the least and tend to fall into two categories, general methods which will work for any decomposition and
Various methods exist for finding the coefficients for the complex poles using partial fraction expansion. The coefficients for the real poles–distinct or repeated–are found in …
2. Rational functions and partial fractions 2.1. Rational functions A rational function is a function of the form f(x) = p(x) q(x) where p(x) and q(x) are polynomials in x with q ≡ 0. orF example
Examples of the Method of Partial Fractions October 20, 2011 Question: How do we compute Z x 5 x2 x 2 dx? Answer: The key is to factor the denominator to get
The method is called “Partial Fraction Decomposition”, and goes like this: Step 1: Factor the bottom Step 2: Write one partial fraction for each of those factors
Section 5-5 : Partial Fractions. This section doesn’t really have a lot to do with the rest of this chapter, but since the subject needs to be covered and this was a fairly short chapter it seemed like as good a place as any to put it.