Modelling with Functions (DP IB Analysis & Approaches (AA)): Revision Note

Modelling with functions

What is a mathematical model?

• A mathematical model simplifies a real-world situation so it can be described using mathematics

  • The model can then be used to make predictions

• Assumptions about the situation are made in order to simplify the mathematics

• Models can be refined (improved) if further information is available or if the model is compared to real-world data

How do I set up the model?

• The question could:

  • give you the equation of the model

  • tell you about the relationship

    • It might say the relationship is linear, quadratic, etc

  • ask you to suggest a suitable model

    • Use your knowledge of each model

    • E.g. if it is compound interest then an exponential model is the most appropriate

• You may have to determine a reasonable domain

  • Consider real-life context

    • E.g. if dealing with hours in a day then

    • E.g. if dealing with physical quantities (such as length) then

  • Consider the possible ranges

    • If the outcome cannot be negative then you want to choose a domain which corresponds to a range with no negative values

    • Sketching the graph is helpful to determine a suitable domain

Which models might I need to use?

• You could be given any model and be expected to use it

• Common models and examples include:

  • Linear

    • Arithmetic sequences

    • Linear regression

  • Quadratic

    • Projectile motion

    • The height of a cable supporting a bridge

    • Profit

  • Exponential

    • Geometric sequences

    • Exponential growth and decay

    • Compound interest

  • Logarithmic

    • Richter scale for the magnitude of earthquakes

  • Rational

    • Temperature of a cup of coffee

  • Trigonometric

    • The depth of a tide

How do I use a model?

• You can use a model by substituting in values for the variable to estimate outputs

  • For example: Let be the height of a football seconds after being kicked

    • will be an estimate for the height of the ball 3 seconds after being kicked

• Given an output you can form an equation with the model to estimate the input

  • For example: Let be the profit made by selling items

    • Solving will give you an estimate for the number of items needing to be sold to make a profit of 100

• If your variable is time then substituting will give you the initial value according to the model

• Fully understand the units for the variables

  • If the units of  are measured in thousand dollars then  represents $3000

• Look out for key words such as:

  • Initially

    • This means

  • Minimum/maximum

  • Limiting value

    • This means gets large

What do I do if some of the parameters are unknown?

• A general method is to form equations by substituting in given values

  • You can form multiple equations and solve them simultaneously using your GDC

  • This method works for all models

• For example, suppose is the mass in kg after hours

  • If you are told that the mass is 7 kg after 2 hours

    • Use to find the value of

    • 

    • 

• The initial value is the value of the function when the variable is 0

  • This is normally one of the parameters in the equation of the model