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

2. Shameless Advertisement
Houston Area Calculus Teachers Association –
Houston Area Teachers of Statistics –
Online practice AP Calculus and Statistics Exams – April and May See the links above.
UH High School Mathematics Contest –

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

4. Question: How can calculus be used to determine how we should proceed?
Linear Least Squares
Example 1: Consider the problem of finding a line that fits the data:
x = 1 2 3 4 5 6 8 9 11 12 15
y =
3.5 7 17 22 29
Question: How can calculus be used to determine how we should proceed?

5. 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?

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:
x = 1 2 3 -1 -2 -3 -4
y =
3.5 11 22 9 18 35
Question: How can calculus be used to determine how we should proceed?

8. 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?

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. 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
2.3
.25
2.5
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. (Illustration with Winplot Implicit Plots)
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. (Illustration with both implicit plots and ODEs)
Example 6:
(Illustration with both implicit plots and ODEs)

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

17. What is Newton’s Method?

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