Applications of Multivariable Calculus: Least Squares, Gradient Descent, 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
• [e-mail]

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• 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

Technology Tool Tips

• PDF Annotator
• Mimio Notebook
• WinPlot
• Bamboo Tablet

Linear Least Squares

• Example 1: Consider the problem of finding a line that fits the data:

x =	0	1	2	3	4	5	6	8	9	11	12	15
y =	1	2	4	3.5	5	4	7	9	12	17	22	29

• Question: How can calculus be used to determine how we should proceed?

The General Process

• Consider the problem of finding a line that fits the data:

x =	x1	x2	x3	…	xn
y =	y1	y2	y3	…	yn

• Question: How can calculus be used to determine how we should proceed?

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.

Quadratic Least Squares

• Example 2: Consider the problem of finding a parabola that fits the data:

x =	0	1	2	3	-1	-2	-3	-4
y =	1	3.5	11	22	3	9	18	35

• Question: How can calculus be used to determine how we should proceed?

The General Process

• Consider the problem of finding a parabola that fits the data:

x =	x1	x2	x3	…	xn
y =	y1	y2	y3	…	yn

• Question: How can calculus be used to determine how we should proceed?

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.

Displacement (meters)	Force (Newtons)
.01	.21
.02	.42
.03	.63
.05	.83
.06	1.0
.08	1.3
.10	1.5
.13	1.7
.16	1.9
.18	2.1
.21	2.3
.25	2.5

• Example 3:

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.

• Example 4:

• (Illustration with Winplot Implicit Plots)

• Example 5:

Question: How can we related this to differential equations?

• (Illustration with Winplot and Polking’s Java)

• Example 6:

• (Illustration with both implicit plots and ODEs)

• Example 7:

• (Illustration with Winplot and Polking’s Java)

What is Newton’s Method?

• Example 8:

• (Illustration with Winplot and Excel)