IB DP Maths: AA HL復習筆記2.4.3 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
- Consider real-life context
Which models might I need to use?
- You could be given any model and be expected to use it
- Common models 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
- Linear
How do I use a model?
- You can use a model by substituting in values for the variable to?estimate outputs
- For example: Let?h(t) be the height of a football?t?seconds after being kicked
- h(3) will be an estimate for the height of the ball 3 seconds after being kicked
- For example: Let?h(t) be the height of a football?t?seconds after being kicked
- Given an?output?you can?form an equation?with the model to?estimate the input
- For example: Let?P(n) be the profit made by selling?n?items
- Solving?P(n) = 100 will give you an estimate for the number of items needing to be sold to make a profit of 100
- For example: Let?P(n) be the profit made by selling?n?items
- If your variable is?time?then substituting?t?= 0 will give you the?initial value?according to the model
- Fully understand the?units for the variables
- If the units of?P?are measured in?thousand dollars?then?P?= 3?represents $3000
- Look out for?key words?such as:
- Initially
- Minimum/maximum
- Limiting value
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
- 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
Worked Example
c)? ? ? Find the time taken for the temperature of the coffee to reach 30°C.
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