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
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Technology Tool Tips
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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 …
y=
y1
y2
y3 … yn
xn
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 …
y=
y1
y2
y3 … yn
xn
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)
Example 3:
.01
A rubber band is stretched and some
.02
data is recorded relating force to
.03
displacement. Determine whether this
.05
.06
data is best approximated using a linear,
.08
quadratic or logarithmic least squares fit.
.10
Note: The logarithmic form is
.13
y  a  b ln  x  1 .
.16
.18
.21
.25
Force
(Newtons)
.21
.42
.63
.83
1.0
1.3
1.5
1.7
1.9
2.1
2.3
2.5
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:
Describe the level sets of
f ( x, y)  4 x  5 y  x  y  sin  xy   1.
4
4
(Illustration with Winplot Implicit Plots)
Example 5:
Use the gradient descent method to approximate the minimum
value of
f ( x , y )  4 x 4  5 y 4  x  y  sin  xy   1
 1 1
starting from a guess of   ,  .
 2 2
Question: How can we related this to
differential equations?
(Illustration with Winplot and Polking’s Java)
Example 6:
Describe the level sets of
f ( x, y)  4 x 4  5 y 4  x  y  10sin( xy)  1.
(Illustration with both implicit plots and ODEs)
Example 7:
Use differential equations to approximate the minimum
value of
f ( x, y)  4 x 4  5 y 4  x  y  10sin( xy)  1
 1 1
starting from a guess of   ,  .
 2 2
(Illustration with Winplot and Polking’s Java)
What is Newton’s Method?
Example 8:
Use Newton's method to approximate the critical
values of f ( x, y)  4 x  5 y  x  y  10sin( xy)  1.
4
4
(Illustration with Winplot and Excel)