1. Mathematical Modeling
Unit 12
Mathematical Modeling

2. Schema of Math Modeling
Real World - external to the mind, our reality
Mathematical World - internal world of intellect, our conceptions

3. Map as a Model

4. Definition of Model
A scaled down or simplified version of a complex situation that allows one to answer important questions

5. Properties of a good model
Simplifies the phenomenon it represents
Allows for interpolation and extrapolation from the known values or relationships.
In Algebra the models are functions.

6. Real World Applications
Two types of problems
Problems modeled by a previously known relationship (Section 2.1).
Problems requiring data analysis since the relationship underlying the model is unknown (Section 2.2 & 2.3).

7. Model: Septic Tank
A septic tank is the shape of a cylinder with a length of 6 feet, with a ½ sphere on each end with a diameter of d feet. The tank must be buried 3 feet under the ground. Determine the volume of the tank if the diameter is 4?

8. DECAL Heuristic
A heuristic is an organized method of solving problems.
Stimulates reasoning leading to a plausible solution.
Use DECAL Heuristic to solve the Septic Tank Problem.

9. Step 1: Describe the Problem
Setting: Identify known relationship or law that underlies the problem.

10. Step 1: Describe the Problem
Setting: Identify known relationship or law that underlies the problem.
Commonly Known Formulas: Page 237
Solution: The setting is not a sewer problem, that is a context.
The underlying math relationship is finding the volume of a solid.
Volume of Cylinder: V = Base area X Height
Volume of Sphere:

11. Step 1: Describe the Problem
Question: Identify the unknowns and label them as variable quantities. Clearly identify the question you are answering.

12. Step 1: Describe the Problem
Question: Identify the unknowns and label them as variable quantities. Clearly identify the question you are answering.
Solution: Question is to find the volume V of the tank.
Variables: V = volume of tank
l = length of tank
r = radius of tank

13. Step 1: Describe the Problem
Facts: List and identify all key facts.

14. Step 1: Describe the Problem
Facts: List and identify all key facts.
Solution: l = 6 feet
d = 4 feet

15. Step 1: Describe the Problem
Distractors: List all extraneous information – information not needed to solve the problem.

16. Step 1: Describe the Problem
Distractors: List all extraneous information – information not needed to solve the problem.
Solution: Burying tank 3 feet.

17. Problem Formulation Real World Problems often have either:
Too much information  distractors
To little information  research
Provide students opportunity to state and clarify problem situations

18. Step 2: Explore
Critical thinking part of the heuristic. Determine how to move from Describing the problem to formulating a function model.
Employ variety of strategies (Pg. 232)
Deductive Reasoning  analogy, brainstorm
Inductive Reasoning  trial & error, generalize from cases
Analytic Reasoning  identify subgoals
Recursive Reasoning  work backwards
Visualizing

19. Step 2: Explore
Visualize the problem by drawing a sketch and labeling the parts. d
l = 6

20. Step 2: Explore Analytic Reasoning: break the problem into parts. d d

21. Step 2: Explore Apply the Known Relationship
Total Vol. = Sphere Vol. + Cylinder Vol.
Total Vol. = d d

22. Step 3: Create Model Total Vol. = Sphere Vol. + Cylinder Vol.
Function Model

23. Step 4: Apply the Model
Let d = 4

24. Step 5: Link to New Situations
Review the Problem
Cubic Feet
which seems to be a reasonable answer

25. Step 5: Link to New Situations
Extend the Problem: What would be the capacity of a tank with a diameter of d = 10 feet?

26. Mathematical Modeling – Finite Differences
Charles Babbage (England, 1821) created a forerunner of the computer called the Difference Engine.
Based his discovery on looking for constant value by taking differences
Basic premise of math is to determine what remains constant within change

27. n
F(n)
Model: Guess My Rule
An old mathematical game where one person makes up a rule and generates data, then a second person tries to guess the rule.
How can we determine if the rule is linear or curvilinear?

28. n
F(n)
Model: Guess My Rule
An old mathematical game where one person makes up a rule and generates data, then a second person tries to guess the rule.
How can we determine if the rule is linear or curvilinear?
Solution: Determine if the rate of change is constant.

29. n
F(n)
1st Difference 7
10 – 7 = 3 1 10
13 – 10 = 3 2 13
16 – 13 = 3 3 16
19 – 13 = 3 4 19
22 – 19 = 3 5 22
25 – 22 = 3 6 25
28 – 25 = 3 28
31 – 28 = 3 8 31
34 – 31 = 3 9 34
37 – 34 = 3 37
Finite Differences
Find differences between successive terms in a sequence of numbers until a common difference occurs.
If the data is modeled by a polynomial function (linear, quadratic, or cubic, etc.), then there will be a common difference.

30. n
F(n)
1st Difference 7
10 – 7 = 3 1 10
13 – 10 = 3 2 13
16 – 13 = 3 3 16
19 – 13 = 3 4 19
22 – 19 = 3 5 22
25 – 22 = 3 6 25
28 – 25 = 3 28
31 – 28 = 3 8 31
34 – 31 = 3 9 34
37 – 34 = 3 37
Finite Differences
Find differences between successive terms in a sequence of numbers until a common difference occurs.
If the data is modeled by a polynomial function (linear, quadratic, or cubic, etc.), then there will be a common difference.
Solution: Common difference in 1st difference, so model is linear

31. Finite Differences Model
Compare the table of differences for the data to the general finite differences table for the linear case f(x) = mx + b (Table 4, Pg 2-31)
Generate the Linear Case table by letting x assume values 0, 1, 2, 3, 4, 5, …..
Since the data is linear and the Linear Case table represents the general pattern for any line, the entries in the table must be equal.
Select a line of the table, set the entries equal, and solve for m and b.
Solution:

32. Finite Differences Model
General Finite Differences table for the linear case f(x) = mx + b x
F(x) = mx+b
1st Difference b 1
m+b m 2
2m+b 3
3m+b

33. Finite Differences Model
Compare the table of differences for the data to the general finite differences table for the linear case f(x) = mx + b (Table 4, Pg 260)
Generate the Linear Case table by letting x assume values 0, 1, 2, 3, 4, 5, …..
Since the data is linear and the Linear Case table represents the general pattern for any line, the entries in the table must be equal.
Select a line of the table, set the entries equal, and solve for m and b.
Solution: My rule was F(n) = 3n + 7

34. When is finite differences a good method to use?
Theoretical data with no scatter due to variation or measurement error
Example: mathematical sequences
Scientific data with little scatter due to measurement error
Distance an object falls in a given time
NOT GOOD for Social Science data which often has a lot of variation
Example: Income level by age

35. Curvilinear Case
Given n points in a plane, what is the maximum number of straight line segments (edges) that can be drawn joining them?
Gather data for n = 1, 2, 3, 4 and 5 points
Is the data linear? Why or why not?
If the data is curvilinear, should we use a quadratic or cubic polynomial to model it?

36. Data for edges problem e = 0 edges n = 1 point n = 2 points e = 1 edge
Find the number of edges for n = 4 and n = 5.

37. Data for edges problem Find the number of edges for n = 4 and n = 5.
e = edges
n = 4 points
n = 5 points
e = edges

38. n
e(n)
1st Difference 1
1 – 0 =1 2
3- 1 = 2 3
6 - 3 = 3 4 6
10 – 6 = 4 5 10
15 – 10 = 5 15
21 – 15 = 6 7 21
28 – 21 = 7 8 28 ? 9
Data for Edges Problem
Here is the data for the first n = 8 cases of the edges problem.
Use finite differences to determine if the data is linear or curvilinear.

39. n
e(n)
1st Difference 1
1 – 0 = 0 2
3- 1 = 2 3
6 - 3 = 3 4 6
10 – 6 = 4 5 10
15 – 10 = 5 15
21 – 15 = 6 7 21
28 – 21 = 7 8 28 ? 9
Data for Edges Problem
Here is the data for the first n = 8 cases of the edges problem.
Use finite differences to determine if the data is linear or curvilinear.
Solution: Data is curvilinear.

40. Ladder of Powers .
p(x) = x5
p(x) = x4
p(x) = x3
p(x) = x2
p(x) = x
p(x) = x-1
p(x) = x-2
p(x) = x-3
p(x) = x-4
p(x) = x-5
Ladder of Powers
How do we determine if the data is quadratic or cubic?
Ladder of Powers is list of power functions p(x) = Axn where A=1 and n is an integer.
Plot power functions with data to determine which power function most closely matches the steepness and curvature of the data.

41. Ladder of Powers – Which power function best matches the curvature and steepness of the data?
y=x3
y=x2
y=x
y=x-1

42. Ladder of Powers – Trial and Error to find A in
y=0.4x2

43. Finite Differences – Quadratic case
The model for the edges problem appears to be quadratic. How do we determine the model with finite differences?

44. Finite Differences – Quadratic case
The model for the edges problem appears to be quadratic. How do we determine the model with finite differences?
Find the second successive difference – difference of the first difference.
If the second difference is constant the data has a quadratic model.

45. Finite Differences -Quadratic Case
Compare the data differences table to the finite differences table for the general quadratic case (Table 5, Pg 2-33). What is the quadratic model for the edges problem?

46. Comparison of Data to General Table

47. Edges Solution 2a = 1 so a = 1/2 3a + b = 1, where a is known
a + b + c = 0, where a and b are known
1/2 + (-1/2) + c = 0, so c = 0
Model:

48. Finite Differences Summary
Useful method if the data is theoretic with no error or has little measurement error and variation.
If there is any variation we have to look for a difference which is approximately constant.
Try a finite differences problem where the model is a cubic polynomial. Which finite difference column would be constant?

49. Mathematical Modeling – Least Squares
Three Modeling Methods
Known Relationship – underlying mathematical setting is known
Finite Differences – theoretical data or hard science data with little scatter
Least Squares – modeling data with scatter

50. Model: Global Warming
Year
Cars
1940
27.5
1950
40.3
1960
61.7
1970
89.3
1980
121.6
1990
150.5
Global warming is partly the result of burning fuels, which increases the amount of carbon dioxide in the air. One of the major sources of fuel consumption are cars. Let’s examine the number of cars in the U.S. (in millions) as one variable of global warming.

51. Linear or Curvilinear?
Numeric Method – use Finite Differences method to determine if data has a near linear trend.
Is the first difference nearly constant?

52. Linear or Curvilinear?
Numeric Method – use Finite Differences method to determine if data has a near linear trend.
Is the first difference nearly constant?
Solution: Nearly so after 1960.

53. Linear or Curvilinear? Graphic Method – plot the data to see the trend
Year
Scaled
Cars 1
27.5 2
40.3 3
61.7 4
89.3 5
121.6 6
150.5
Graphic Method – plot the data to see the trend

54. Scatter Plot

55. Line of Best Fit-Approximation
Find a line that represents the trend of the data
What point should the best-fit line pass through?
Solution:
How can we use a ruler to determine a line that best fits the data?

56. Line of Best Fit-Approximation
Find a line that represents the trend of the data
What point should the best-fit line pass through?
Solution: Midpoint of the data
How can we use a ruler to determine a line that best fits the data?
Solution: Place ruler on the midpoint, vary slope of ruler to best match data trend

57. Eyeball line of fit

58. Line of Fit Estimate equation for line of best fit
Line passes through the midpoint of data set: Midpoint (3.5,81.8)
Estimate the trend and approximate slope of line with that trend, say m = 30
So line of fit is y = 30x + b, solve for b.

59. Thank you.