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  • Dashboard
Topics
Exponents & LogarithmsRounding & ErrorSequences & SeriesFinancial MathematicsMatricesComplex Numbers
Cartesian plane & linesFunction TheoryModellingTransformations & asymptotes
2D & 3D GeometryVoronoi DiagramsTrig equations & identitiesVectorsGraph Theory
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DifferentiationIntegrationDifferential Equations
Paper 3
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Perplex

Modelling (Lesson 2/10)

Cubic Models

1 / 2

Discussion

Amanda fixes magnets at positions ​x=−5cm​ and ​x=5cm. Then, she measures the force ​F​ applied to a third magnet at different positions. She plots her data on the graph below (negative positions and forces point toward Magnet ​1, positive positions and forces point toward Magnet ​2​).

Describe a curve that could be used to model force as a function of position.

Solution:

A good choice is a smooth, monotonically increasing S-shaped curve that goes through the points and has these visual features:

• At the left (around ​x≈–3​ cm) it sits down near ​F≈–8​ units and is quite steep.

• As you move toward the centre, the slope decreases, so by ​x≈–1​ cm it’s already flattening off.

• Between ​x≈–1​ and ​x≈1​ cm the curve is almost flat, passing through ​(0,0)​ with a gentle incline.

• Beyond ​x≈1​ cm the slope picks up again, becoming quite steep by ​x≈3​ cm where ​F≈8​ units.

• The result is one continuous “S” shape, negative and getting steeper to the left, inflecting near ​x≈–1, nearly straight through the middle, then again near ​x≈1​ to become steeper up to the right.

1 free

Modelling (Lesson 2/10)

Cubic Models

1 / 2

Discussion

Amanda fixes magnets at positions ​x=−5cm​ and ​x=5cm. Then, she measures the force ​F​ applied to a third magnet at different positions. She plots her data on the graph below (negative positions and forces point toward Magnet ​1, positive positions and forces point toward Magnet ​2​).

Describe a curve that could be used to model force as a function of position.

Solution:

A good choice is a smooth, monotonically increasing S-shaped curve that goes through the points and has these visual features:

• At the left (around ​x≈–3​ cm) it sits down near ​F≈–8​ units and is quite steep.

• As you move toward the centre, the slope decreases, so by ​x≈–1​ cm it’s already flattening off.

• Between ​x≈–1​ and ​x≈1​ cm the curve is almost flat, passing through ​(0,0)​ with a gentle incline.

• Beyond ​x≈1​ cm the slope picks up again, becoming quite steep by ​x≈3​ cm where ​F≈8​ units.

• The result is one continuous “S” shape, negative and getting steeper to the left, inflecting near ​x≈–1, nearly straight through the middle, then again near ​x≈1​ to become steeper up to the right.

1 free