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More Multivariable Calculus: Least Squares, ODEs and Local Extrema, and Newton’s Method

More Multivariable Calculus: Least Squares, ODEs and Local Extrema, and Newton’s Method. Dr. Jeff Morgan Department of Mathematics University of Houston jmorgan@math.uh.edu. Shameless Advertisement. Houston Area Calculus Teachers Association – http://www.HoustonACT.org

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More Multivariable Calculus: Least Squares, ODEs and Local Extrema, and Newton’s Method

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  1. More Multivariable Calculus: Least Squares, ODEs and Local Extrema, and Newton’s Method Dr. Jeff Morgan Department of Mathematics University of Houston jmorgan@math.uh.edu

  2. Shameless Advertisement • Houston Area Calculus Teachers Association – http://www.HoustonACT.org • Houston Area Teachers of Statistics – http://www.HoustonATS.org • Online practice AP Calculus and Statistics Exams – April and May 2009. See the links above. • UH High School Mathematics Contest – http://mathcontest.uh.edu

  3. Technology Tool Tips • PDF Annotator • Mimio Notebook • WinPlot • Bamboo Tablet

  4. Linear Least Squares Example 1: Consider the problem of finding a line that fits the data: Question: How can calculus be used to determine how we should proceed?

  5. The General Process Consider the problem of finding a line that fits the data: Question: How can calculus be used to determine how we should proceed?

  6. Solution to Example 1 in Excel • Select ranges to write updated values. • Use the commands transpose, mmult and minverse and select the data that the commands will act on. • Press ctrl+shift+enter.

  7. Quadratic Least Squares Example 2: Consider the problem of finding a parabola that fits the data: Question: How can calculus be used to determine how we should proceed?

  8. The General Process Consider the problem of finding a parabola that fits the data: Question: How can calculus be used to determine how we should proceed?

  9. Solution to Example 2 in Excel • Select ranges to write updated values. • Use the commands transpose, mmult and minverse and select the data that the commands will act on. • Press ctrl+shift+enter.

  10. Example 3:

  11. Chain Rule, Directional Derivatives, Gradients and Differential Equations • Extending the one dimensional chain rule. • Directional derivatives and their relation to the gradient. • Level sets and their relation to the gradient. • Using ODEs to help sketch level sets in two dimensions. • Classifying the behavior of the gradient near critical points. • Using ODEs to find local extrema.

  12. Example 4: (Illustration with Winplot Implicit Plots)

  13. Example 5:

  14. Question: How can we related this to differential equations? (Illustration with Winplot and Polking’s Java)

  15. Example 6: (Illustration with both implicit plots and ODEs)

  16. Example 7: (Illustration with Winplot and Polking’s Java)

  17. What is Newton’s Method?

  18. Example 8: (Illustration with Winplot and Excel)

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