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IB DP Maths: AA HL复习笔记2.4.3 Modelling with Functions
Category:
IB课程
,
教材笔记
,
福利干货
Date: 2022年7月12日 上午10:21
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 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
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
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
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 G
DC
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.
转载自savemyexams
Previous post: IB DP Maths: AA HL复习笔记2.4.2 Solving Equations
Next post: Edexcel IGCSE Biology 复习笔记 4.1.3 Biodiversity
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