Mathematical modeling is the process of translating real-world scenarios and open-ended problems into mathematical relations. Real-world problems often have incomplete information and require sensible assumptions, relevant variables, appropriate functions, and careful verification and refinement.
A cyclical and iterative flow of the six stages of mathematical modeling.
A mathematical model is a mathematical relation that describes a real-world situation. It is a representation of a system or scenario used to gain qualitative and/or quantitative understanding of real-world problems and to predict future behaviour.
Mathematical modeling is the process of constructing mathematical models. A real-world problem is translated into a mathematical problem, solved, interpreted, refined, and communicated in terms of the real-world context.
Represented by a straight-line graph. Example: petrol consumption varies directly with distance travelled.
From initial data, \(k = \dfrac{45}{405} = \dfrac{1}{9}\), giving \(y = \dfrac{1}{9}x\).
Represented by a parabolic curve that rises and then falls, or falls and then rises.
Example: river depth \(y\) against distance \(x\) from the bank can be modelled as \(y = -0.0136x^2 + 0.408x\).
Curved graph showing rapid increase or decrease at a sharp rate.
where \(a > 0\), \(a \neq 1\), and \(C\) is the initial value. The natural base \(e \approx 2.7182818284\) may also be used.
Annual compound interest where savings grow exponentially:
Refined for compounding \(n\) times per year:
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What is the difference between a typical problem-solving question and an open-ended mathematical modeling question?
What form of function is suitable for modeling data that increases or decreases sharply at a non-constant rate, such as compound interest?
In compound interest modeling, how is the model \(A(t) = P(1 + r)^t\) refined if the interest is compounded \(n\) times per year?
A mathematical model always gives an exact, perfect answer for any real-world situation.
Real-world models rely on simplifying assumptions; hence, results are approximations and subject to limitations.
All real-world questions have only one unique correct model.
Different individuals or groups may choose different defensible assumptions and variables, leading to different valid models.
“Refining the model” means discarding all previous work.
Refining means re-evaluating assumptions and adapting the mathematical relation, for example moving from annual compounding to \(n\) compounding periods, to improve realism.
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1 Which of the following is the correct sequence of the first three components in the mathematical modeling process?
2 If a river cross-section is deepest in the middle and decreases to zero at both banks, which type of function is most appropriate to model its depth?
3 A variable is defined as a quantity with an value.
4 If compound interest is calculated \(n\) times a year, the yearly interest rate \(r\) is divided by \(n\) and the exponent \(t\) is multiplied by .
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