IB DP Maths: AA HL复习笔记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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