What are limitations in mathematical worded equations and modeling?

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embizze | High School Teacher | (Level 1) Educator Emeritus

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This is a very broad question, but I can give a few limitations:

(1) Confusion of language or lack of precision: If I say x is less than 15, do I mean `x<15` or `x<=15` (exclusive versus inclusive)?

If I say that profits are growing exponentially, do I mean that they are growing quickly (common usage) or that the growth can be modeled by a function of the form `f(x)=e^x` ?

If I say x less than 7, do I mean `x<7` or `7-x` ?

(2) Considerations of domain/range (i.e. possible inputs/outputs):

Consider an engine warming up or population growth; over some period of time the growth can be modeled by an exponential growth function, but this cannot be a good model overall because the growth must slow. The engine cannot heat up indefinitely, nor can populations increase forever at an increasing rate.

If I solve a quadratic function for a time t and get 2 answers, one of which is negative, the negative solution may not make sense in the context of the problem.

(3) Complexity -- often models catch the general idea, but cannot be applied specifically. Human population models, already constrained by problems of getting truly accurate input data, cannot be relied upon to give precise answers for a given time, as births do not happen regularly.

(4) Using continuous models for discrete problems -- there are never 6.132billion + 1/2 people on earth -- or any other fractional number of people.

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