As always, you should rework my example(s) successfully BEFORE attempting the homework.

Here are the weights and abscissas of the 3-point and 4-point Gauss quadratures.

1. Use the 3 point Gauss quadrature to show that the integral of   –ln(x)   on the interval [0, 1] is approximately 0.94767. Compute the percent error of the approximation.
   The weights and abscissas of the 3 Pt GQ are in the link above. The answer is given in the lecture notes.

2. Use the 3 point Gauss quadrature to show that the integral of sin x on the interval [0, Pi] is approximately 2.00138 89136 07743. Compute the percent error of the approximation.
   The weights and abscissas of the 3 Pt GQ are in the link above. The answer is given in the lecture notes.

3. Do the problems on this worksheet.

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NOTES on Legendre Polynomials:

Certain sets of functions arise in mathematical, engineering, and scientific applications that share specific special properties — such sets (classes) are called special functions. You are already familiar with some classes of special functions:

• The trig functions form a class of special functions.
• The hyperbolic trig functions form a class of special functions.
• The exponential functions form a class of special functions.
• The logarithmic functions form a class of special functions.
• etc.

Another class of special functions is called the Legendre polynomials. Besides having many other uses, the Legendre polynomials are curiously related to Gauss quadrature:

The n abscissas of the n point Gauss quadrature integration formula are the n zeros of the nth degree Legendre polynomial.

For example, the 4 abscissas of the 4 point Gauss quadrature integration formula are the 4 zeros of the 4th degree Legendre polynomial. Since the Legendre polynomials are used in many engineering and scientific applications, Maple has the Legendre polynomials stored in one of its libraries. To load the Legendre polynomials into your Maple worksheet, do this:

[> with( orthopoly, P ) ; (this loads the Legendre polynomials into the Maple worksheet from the orthopoly library)

[> P(2,t) ; (displays the 2nd degree Legendre polynomial as a ftn of t)

[> P(3,t) ; (displays the 3rd degree Legendre polynomial as a ftn of t)

[> P(4,t) ; (displays the 4th degree Legendre polynomial as a ftn of t)

[> plot( P(4,t), t = –1.0 .. 1.0, color = blue, thickness = 6 ) ; (plots the 4th degree Legendre polynomial on its domain)

[> T := [ fsolve( P(4,t), t = –1.0 .. 1.0 ) ] ; (solves for the zeros of the 4th degree Legendre polynomial and stores them in a sequence named T)

The domain of all Legendre polynomials is [–1, 1].

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4. As in the example above, use Maple to determine the 7 abscissas of the 7-point Gauss quadrature formula.

5. As in the example above, use Maple to determine the 8 abscissas of the 8-point Gauss quadrature formula.

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