Define mathematical model .

A mathematical model simplifies a real-world situation so that it can be described using mathematics and used to make predictions. The description is deliberately imperfect: the aim is something simple enough to work with, not something that captures everything.

Why does only one branch of a reciprocal model usually matter?

Because the quantity being modelled is normally one that cannot be negative, such as a time, a length or a number of objects. The other branch is mathematically correct but describes values the situation cannot take, so it is discarded.

How can you find a weakness in a function used as a model?

Test the extremes : what the model predicts when the input is close to zero, and what it predicts when the input becomes very large. A model that behaves sensibly in the middle of its range can still give nonsense at either end.

ASMathsCambridge (CIE)Pure 1FlashcardsAlgebra & FunctionsModelling with Functions

Modelling with Functions (Cambridge (CIE) AS Maths: Pure 1): Flashcards

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Define mathematical model.

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Define mathematical model.
A mathematical model simplifies a real-world situation so that it can be described using mathematics and used to make predictions.
The description is deliberately imperfect: the aim is something simple enough to work with, not something that captures everything.
A model is built by making $______$ that simplify the situation, and it can later be $______$ if better information becomes available.
A model is built by making assumptions that simplify the situation, and it can later be refined if better information becomes available.
Both belong to the same cycle: you assume something in order to get started, then improve the model once you can see how well it fits.
The number of leaves on a tree after $d$ days is modelled by $f (d) = \frac{L}{d + 1}$ . What does $L$ represent?
$L$ is the number of leaves at the very start, when $d = 0$ .
Substituting the value that makes the rest of the expression disappear is the general way to work out what a constant in a model stands for.
Why does only one branch of a reciprocal model usually matter?
Because the quantity being modelled is normally one that cannot be negative, such as a time, a length or a number of objects.
The other branch is mathematically correct but describes values the situation cannot take, so it is discarded.
True or False?
A model of the form $f (d) = \frac{L}{d + 1}$ predicts that the quantity never reaches zero.
True.
No value of $d$ makes $\frac{L}{d + 1}$ equal to zero, so the curve gets ever closer to the horizontal axis without ever touching it.
If the quantity really does reach zero in practice, that is a limitation of the model rather than a fact about the situation.
How can you find a weakness in a function used as a model?
Test the extremes: what the model predicts when the input is close to zero, and what it predicts when the input becomes very large.
A model that behaves sensibly in the middle of its range can still give nonsense at either end.
A gas has pressure $P = \frac{1750000}{V}$ pascals, where $V$ is the volume in $m^{3}$ . What is the smallest whole volume keeping $P$ below $120000$ ?
Solving $\frac{1750000}{V} < 120000$ gives $V > 14.583 \dots$ , so the smallest whole volume is $15 m^{3}$ .
Rounding down to $14 m^{3}$ would break the condition, so a value produced by a model always has to be checked against what the situation actually allows.

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