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Multivariable Calculus: Least Squares, ODEs, Extrema, and Newton's Method

This tutorial explores the applications of multivariable calculus in solving problems related to least squares regression, ordinary differential equations, local extrema, and Newton's method. It provides step-by-step examples and explanations, along with relevant technology tool tips.

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Multivariable Calculus: Least Squares, ODEs, 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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