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5
Simplification
Most models simplify reality. Generally, models can only approximate real-world behavior. One very powerful simplifying relationship is proportionality.
Let's think of a mathematical model as a mathematical construct designed to study a particular real-world system or behavior of interest.
3
The model allows us to reach mathematical conclusions about the behavior, as illustrated in Figure 1.1.
The definition means that the graph of y versus x lies along a straight line through the origin. This graphical observation is useful in testing whether a given data collection reasonably assumes a proportionality relationship.
Mathematical Modeling
数学建模(英文版) 机械工业出版社,
北京, 2003. 5 经典原版书库, 原书名:
A First Course in Mathematical Modeling
(Third Edition) by
Frank R. Giordano, Maurice D. Weir, William
6
Definition Two variables y and x are proportional (to each other) if one is always a constant multiple of the other, that is, if
y = kx
for some nonzero constant k. We write y ? x.
Table 1.1 Spring-mass system
9
A scatterplot graph of the stretch or elongation of the spring versus the mass or weight placed on it reveals an approximate straight line passing through the origin.
Figure 1.2 Spring-mass
system
8源自文库
Mass
50 100 150 200 250
Elongation 1.000 1.875 2.750 3.250 4.375
300 350 400 450 500 550 4.875 5.675 6.500 7.250 8.000 8.750
Figure 1.3 Data from spring-mass system
10
The data appear to follow the proportionality rule that elongation e is proportional to the mass m, or symbolically,
7
Example 1
Testing for Proportionality
Consider a spring-mass system (Figure 1.2). We conduct an experiment to measure the stretch of the spring as a function of the mass (measured as weight) placed on the spring. Consider the data collected for this experiment, displayed in Table 1.1.
These conclusions can be interpreted to help a decision maker plan for the future.
In this chapter we direct our attention to modeling change.
4
Real-world data simplification
P. Fox
1
Chapter 1 Modeling Change
Introduction We often describe a particular phenomenon
mathematically (by means of a function or an equation, for instance).
Model
Verification
Analysis
Predictions/ explanations
Interpretation
Mathematical conclusions
Figure 1.1 A flow of the modeling process beginning with an examination of real-world data
Such a mathematical model is an idealization of the real-world phenomenon and never a completely accurate representation.
2
Mathematical Models
We are often interested in predicting the value of a variable at some time in the future. A mathematical model can help us understand a behavior better or aid us in planning for the future.
Simplification
Most models simplify reality. Generally, models can only approximate real-world behavior. One very powerful simplifying relationship is proportionality.
Let's think of a mathematical model as a mathematical construct designed to study a particular real-world system or behavior of interest.
3
The model allows us to reach mathematical conclusions about the behavior, as illustrated in Figure 1.1.
The definition means that the graph of y versus x lies along a straight line through the origin. This graphical observation is useful in testing whether a given data collection reasonably assumes a proportionality relationship.
Mathematical Modeling
数学建模(英文版) 机械工业出版社,
北京, 2003. 5 经典原版书库, 原书名:
A First Course in Mathematical Modeling
(Third Edition) by
Frank R. Giordano, Maurice D. Weir, William
6
Definition Two variables y and x are proportional (to each other) if one is always a constant multiple of the other, that is, if
y = kx
for some nonzero constant k. We write y ? x.
Table 1.1 Spring-mass system
9
A scatterplot graph of the stretch or elongation of the spring versus the mass or weight placed on it reveals an approximate straight line passing through the origin.
Figure 1.2 Spring-mass
system
8源自文库
Mass
50 100 150 200 250
Elongation 1.000 1.875 2.750 3.250 4.375
300 350 400 450 500 550 4.875 5.675 6.500 7.250 8.000 8.750
Figure 1.3 Data from spring-mass system
10
The data appear to follow the proportionality rule that elongation e is proportional to the mass m, or symbolically,
7
Example 1
Testing for Proportionality
Consider a spring-mass system (Figure 1.2). We conduct an experiment to measure the stretch of the spring as a function of the mass (measured as weight) placed on the spring. Consider the data collected for this experiment, displayed in Table 1.1.
These conclusions can be interpreted to help a decision maker plan for the future.
In this chapter we direct our attention to modeling change.
4
Real-world data simplification
P. Fox
1
Chapter 1 Modeling Change
Introduction We often describe a particular phenomenon
mathematically (by means of a function or an equation, for instance).
Model
Verification
Analysis
Predictions/ explanations
Interpretation
Mathematical conclusions
Figure 1.1 A flow of the modeling process beginning with an examination of real-world data
Such a mathematical model is an idealization of the real-world phenomenon and never a completely accurate representation.
2
Mathematical Models
We are often interested in predicting the value of a variable at some time in the future. A mathematical model can help us understand a behavior better or aid us in planning for the future.