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

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More Multivariable Calculus Least Squares, ODEs
and Local Extrema, and Newtons Method

• Dr. Jeff Morgan
• Department of Mathematics
• University of Houston
• jmorgan_at_math.uh.edu

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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

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Technology Tool Tips

• PDF Annotator
• Mimio Notebook
• WinPlot
• Bamboo Tablet

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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?
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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?
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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 ctrlshiftenter.

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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?
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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?
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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 ctrlshiftenter.

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Example 3
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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.

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Example 4
(Illustration with Winplot Implicit Plots)
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Example 5
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Question How can we related this to differential
equations?

• (Illustration with Winplot and Polkings Java)

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Example 6
(Illustration with both implicit plots and ODEs)
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Example 7
(Illustration with Winplot and Polkings Java)
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What is Newtons Method?
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Example 8
(Illustration with Winplot and Excel)