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ME205: Numerical Methods for Engineers

Unit 2: Numerical Differentiation   If you have experience with the definition of a derivative, then this unit may seem rather trivial. There are, however, several ways with which to approximate the value of the derivative of a function or set of discrete data. The details of the bookkeeping involved in computing such approximations can be very important in the use of those approximations in more advanced numerical algorithms. Hence, it is worthwhile to spend a little time understanding the details of the bookkeeping involved. 

Unit 2 Time Advisory
Time Advisory:  This unit will take you approximately 13 hours to complete.

☐    Subunit 2.1: 2 hours

☐    Subunit 2.2: 4 hours

        ☐    Web Media: 1 hour

        ☐    Reading: 2 hours

        ☐    Assessment: 1 hour

☐    Subunit 2.3: 4 hours

        ☐    Web Media: 1 hour

        ☐    Reading: 2 hours

        ☐    Assessment: 1 hour

☐    Subunit 2.4: 3 hours

Unit2 Learning Outcomes
Upon successful completion of this unit, the student will be able to:

  • Derive difference equations for first and second order derivatives.
  • Evaluate first and second order derivatives from numerical evaluations of continuous functions or table lookup of discrete data.

2.1 Review of Differentiation   - Reading: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter on Differential Calculus” Link: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter on Differential Calculus” (PDF)
 
Instructions: Please read the chapter in its entirety (32 pages).  You may skim through the chapter if it is a review for you.
 
Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

  • Web Media: University of South Florida: Holistic Numerical Methods Institute: Autar Kaw’s “Background of Differentiation” Link: University of South Florida: Holistic Numerical Methods Institute: Autar Kaw’s “Background of Differentiation” (YouTube)

    Instructions: Please view the entire 7-minute video.  Please note that the material in this subunit may be a review.  Make sure to test your knowledge by completing the multiple choice quiz before skipping this section.
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Assessment: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Background of Differential Calculus” Link: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Background of Differential Calculus” (PDF)
     
    Instructions: Attempt all questions in the multiple choice quiz.  If you are having difficulty answering any of the questions, make sure to refer back to and review the resources in subunit 2.1.  You can find the answers here (PDF).
     
    Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

2.2 Differentiation of Continuous Functions   - Reading: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter on Numerical Differentiation of Continuous Functions” Link: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter on Numerical Differentiation of Continuous Functions” (PDF)
 
Instructions: Read the entire chapter (18 pages).
 
Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

  • Web Media: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “Differentiation of Continuous Functions” Link: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “Differentiation of Continuous Functions” (YouTube)
     
    Instructions: Please view all 9 videos under the heading “Digital Audiovisual Lectures.”  You may access each video by clicking on the YouTube link after each title.  The total run time is approximately 63 minutes.
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Assessment: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Numerical Differentiation of Continuous Functions” Link: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Background of Numerical Differentiation of Continuous Functions” (PDF)

    Instructions: Complete all questions in the multiple choice quiz.  If you encounter any difficulties in answering a question, refer to the resources in subunit 2.2.  You can find the answers here (PDF).
     
    Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

2.3 Differentiation of Discrete Data   - Reading: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter on Numerical Differentiation of Discrete Functions” Link: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter on Numerical Differentiation of Discrete Functions” (PDF)
 
Instructions: Read the entire chapter (9 pages).  How does the treatment differ for discrete data and continuous functions?
 
Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

  • Web Media: University of South Florida: Holistic Numerical Methods Institute’s “Differentiation of Discrete Functions” Lectures Link: University of South Florida: Holistic Numerical Methods Institute’s “Differentiation of Discrete Functions” (YouTube)
     
    Instructions: Under the “Digital Audiovisual Lectures” heading, select the YouTube links for each video.  View all four videos in their entirety: “Divided Difference Approach,” “Polynomial Interpolation Method,” “Newton’s Divided Difference Polynomial Method: Theory,” and “Newton’s Divided Difference Polynomial Method: Example.”  The total run time is approximately 35 minutes. 
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Assessment: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Numerical Differentiation of Discrete Functions” Link: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Background of Numerical Differentiation of Discrete Functions” (PDF)
     
    Instructions: Attempt all questions in the multiple choice quiz.  If you encounter any difficulties answering a question, please refer to the resources in subunit 2.3.  You can find the answers here (PDF).
     
    Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

2.4 An Exercise in Numerical Differentiation   - Assessment: The Saylor Foundation’s “ME205: Unit 2.4 Exercise” Link: The Saylor Foundation’s “ME205: Unit 2.4 Exercise” (PDF)
 
Instructions: Please perform this exercise.  When you are done, check your work against The Saylor Foundation’s “ME304: Unit 2.4 Exercise Solution Guide" (PDF).  This exercise should require less than 3 hours to complete.