Mathematica Labs for Math 22, Calculus II - Section B, Spring 2009

Lab A :    An Introduction to Mathematica

Lab written by Helen Read providing a brief introduction to Mathematica .

Lab 1 :    Taylor Polynomials

This lab considers the first few Taylor polynomial approximations to the function f(x) = sin x near x=0, primarily in terms of how well they numerically approximate the values of sin x.

Exercises include determination of the two polynomials of smallest degree whose values can be used to compute all the values of the functions sin x and cos x valid to 11 decimal places, sufficient for most pocket calculators.

Lab 2 :    Taylor Polynomials for ex and ln (1+x)

This lab considers the first few Taylor polynomial approximations to the functions ex and ln(1+x) near x=0, primarily in terms of how well they numerically approximate the values of the two functions.

Exercises include determination of the polynomial of smallest degree whose values can be used to compute all the values of the function ex for x between -2 and +2 valid to 10 decimal places, sufficient for most pocket calculators. Another exercise shows that, unlike the Taylor polynomials for functions previously considered, the Taylor polynomials near x=0 for the function ln(1+x) cannot be used to find the numerical values of this function for x > 1.

Lab 3 :    Approximation by Trigonometric Polynomials -
                 an introduction to Fourier Series

  • Trigonometric Polynomials
  • Coefficients of Trigonometric Polynomials
  • The Fourier polynomials for f(x) = x: a "sawtooth wave"
  • The Fourier polynomials for a "square wave"
    This lab considers the problem of approximating a function by trigonometric polynomials, including a description of the formulas for computing the Fourier coefficients of the harmonic components. The functions f(x) = x (leading to the "sawtooth" waveform) and the step function (leading to the "squarewave" waveform) are considered in some detail.

    Exercises include modifying the worked examples and finding the integrals necessary for the Fourier coefficients for the function ex.

    • Lab 4 :    Numerical Integration

    1. Left, Right, and Midpoint Riemann Sums; Trapezoidal approximations
    2. Simpson's Rule
    3. Error Estimates
    This lab considers several elementary methods for computing definite integrals numerically, including some comparisons on the accuracy of each of the methods. Error estimates are used to compute numerical values for some definite integrals to known accuracy.

    Exercises include determining the numerical values of logarithms, Fresnel-type integrals, and the value of pi to known accuracy.

    Lab 5 :    Parametric Equations and Polar Coordinates

    1. Curves defined parametrically
    2. Plotting in Polar Coordinates
    This lab introduces the "ParametricPlot" command in Mathematica for plotting curves defined by parametric equations, including the plots of curves defined by equations in polar coordinates.

    Exercises include various plots, including Lissajous figures, hypocycloids and epicycloids.

    Lab 6 :    The Fourier Series for a Bernoulli Polynomial

    1. Fourier Series
    2. Computing the Fourier Coefficients of a Fourier Series
    3. Computing the Fourier Coefficients for the Bernoulli polynomial
    This lab continues the study of trigonometric polynomials begun in Lab 3 to Fourier Series, computing the Fourier series for a particular function defined by a quadratic Bernoulli polynomial.

    The exercise includes computation of the Fourier coefficients and the evaluation of this series at x=0 to determine the sum of reciprocal squares of the integers.

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