A LevelMathsCambridge (CIE)Pure 3FlashcardsAlgebra & FunctionsFurther Modelling with Functions

Further Modelling with Functions (Cambridge (CIE) A Level Maths: Pure 3): Flashcards

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FrontFurther Modelling with Functions
Complete the three jobs done in mathematical modelling:
An assumption simplifies a real-life situation to make the mathematics $______$ .
A criticism points out a way in which the model is not $______$ .
A refinement changes the model to make its predictions more $______$ .

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Cards in this collection (6)

Complete the three jobs done in mathematical modelling:
An assumption simplifies a real-life situation to make the mathematics $______$ .
A criticism points out a way in which the model is not $______$ .
A refinement changes the model to make its predictions more $______$ .
The completed sentences are:
An assumption simplifies a real-life situation to make the mathematics easier.
A criticism points out a way in which the model is not suitable.
A refinement changes the model to make its predictions more realistic.
A criticism only identifies the flaw, whereas a refinement does something about it.
In the model $N = 64 + 4 m$ , $m$ is the number of months after 1 January. Why does the number sold during January correspond to $m = 1$ ?
Because January is not over until one whole month has passed, so the end of January is $m = 1$ rather than $m = 0$ .
Putting $m = 0$ would give the position at the very start of January, before any of that month's sales had happened.
True or False?
A quadratic function modelling a thrown stone describes only part of the curve.
True.
The stone is only in the air between the throw and the landing, so the model applies over a limited interval such as $0 \leq t \leq 10$ and the rest of the parabola describes nothing real.
Outside that interval the curve is still a perfectly good parabola, it simply stops corresponding to anything the stone does.
When does a modelling question need you to solve an equation rather than substitute?
Substitute when you are given the input and asked for the output, and solve when you are given the output and asked for the input.
Asking for the height of a stone after $4$ seconds is a substitution, whereas asking how long it takes to reach the sea sets the height to $0$ and solves for the time.
Why write a quadratic model in completed square form?
Because it hands you the model's maximum or minimum directly, with no differentiation needed.
Writing $h = 32 + 4.8 t - 0.8 t^{2}$ as $h = - 0.8 {(t - 3)}^{2} + 39.2$ shows at once that the greatest height is $39.2 m$ , reached at $t = 3 s$ .
True or False?
Every solution of an equation coming from a model is a valid answer to the modelling question.
False.
A model carries restrictions from the situation it describes, and any solution that breaks one of them has to be discarded.
Solving $32 + 4.8 t - 0.8 t^{2} = 0$ for the time a stone hits the sea gives $t = 10$ and $t = - 4$ , and only $t = 10 s$ makes sense for a time measured from the throw.

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