Modelling with Differentiation inc. Optimisation
How can I use differentiation to solve modelling questions?
- Derivatives can be calculated for any variables – not just y and x
- In every case the derivative is a formula giving the rate of change of one variable with respect to the other variable
- For example if then
- is the rate of change of with respect to
- Differentiation can be used to find maximum and minimum points of a function (see Stationary Points & Turning Points)
- Therefore it can be used to solve maximisation and minimisation problems in modelling questions
- For example you may want to
- Maximise the volume of a container
- Minimise the amount of fuel used
- For example you may want to
Exam Tip
- Exam questions on this topic will often be divided into two parts:
- First a 'Show that...' part where you derive a given formula from the information in the question
- And then a 'Find...' part where you use differentiation to answer a question about the formula
- Even if you can't answer the first part you can still use the formula to answer the second part
Worked example
A cuboid has length cm, width cm, and height cm.
Show that the volume, cm3 is given by .
The volume of a cuboid is ""
Expand and simplify
Find the maximum volume of the cuboid.
Differentiate V with respect to x
At the maximum volume,
Solve for x
So the value of x, at the maximum volume is 0.3
Find the maximum volume by substituting x = 0.3 in to the formula for V
The maximum volume of the cuboid is 1.8 cm3
Prove that your answer is a maximum value.
Using the second derivative is usually the easiest way to find the nature of a stationary point
< 0
The value of the second derivative (at ) is negative, therefore V = 1.8 cm3 is a maximum volume