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Modelling with Differentiation (Cambridge O Level Additional Maths)

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Modelling with Differentiation

How is differentiation used in modelling questions?

  • Derivatives can be calculated for any variables – not just y and x
  • The derivative is a formula giving the rate of change of one variable with respect to the other variable
    • For example if A equals 4 pi r squared then fraction numerator straight d A over denominator straight d r end fraction equals italic 8 pi r
    • fraction numerator straight d A over denominator straight d r end fraction is the rate of change of A with respect to r
  • The phrase 'increasing at a rate of' means the rate of change of one variable with respect to time
    • fraction numerator straight d space space over denominator straight d t end fraction
  • Differentiation can be used to find maximum and minimum points of a function 
    • In modelling, this is called optimisation
    • Second derivative tests help to determine is the point is a maximum or minimum

Examiner Tip

  • Read the question carefully to determine which variables you will need to use
    • The question may give you a formula to help you 

Worked example

The volume, V, of a sphere of radius r is given by V space equals space 4 over 3 straight pi r cubed italic.

Find the rate of change of the volume with respect to the radius.
 



Differentiate the formula given for the volume of a sphere. 

fraction numerator straight d V over denominator straight d r end fraction equals 3 cross times 4 over 3 straight pi cross times r squared

fraction numerator bold d bold italic V over denominator bold d bold italic r end fraction bold equals bold 4 bold pi bold italic r to the power of bold 2

Optimisation

What is optimisation?

  • In general, optimisation is finding the best way to do something
  • In mathematics, optimisation is finding the maximum or minimum output of a function
    • For example, finding the maximum possible profit or minimum costs
  • Differentiation can be used to solve optimisation problems in modelling questions
    • For example you may want to
      • Maximise the volume of a container
      • Minimise the amount of fuel used

Example of using differentiation to maximise a function

Examiner 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 4 x cm, width x cm, and height open parentheses 3 over x minus 5 close parentheses cm.

(a)table row blank row blank row blank end table

Show that the volume, V cm3 is given by V equals 12 x minus 20 x squared.
 
The volume of a cuboid is "V equals l e n g t h cross times w i d t h cross times h e i g h t"

therefore V equals 4 x cross times x cross times open parentheses 3 over x minus 5 close parentheses

Expand and simplify

table row V equals cell 4 x squared open parentheses 3 over x minus 5 close parentheses end cell row V equals cell fraction numerator 12 x squared over denominator x end fraction minus 20 x squared end cell end table

bold therefore bold italic V bold equals bold 12 bold italic x bold minus bold 20 bold italic x to the power of bold 2
 

(b)

Find the maximum volume of the cuboid.
 
Differentiate V with respect to x

fraction numerator straight d V over denominator straight d x end fraction equals 12 minus 40 x

At the maximum volume, fraction numerator straight d V over denominator straight d x end fraction equals 0

therefore 12 minus 40 x equals 0

Solve for x

table attributes columnalign right center left columnspacing 0px end attributes row cell 40 x end cell equals 12 row x equals cell 12 over 40 equals 0.3 end cell end table

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

V equals 12 open parentheses 0.3 close parentheses minus 20 open parentheses 0.3 close parentheses squared equals 1.8

The maximum volume of the cuboid is 1.8 cm3
 

(c)

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 

fraction numerator straight d squared y over denominator straight d x squared end fraction equals negative 40 space space open parentheses less than 0 close parentheses

The value of the second derivative (at bold italic x bold equals bold 0 bold. bold 3) is negative

Therefore V = 1.8 cm3 is a maximum volume

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    Amber

    Author: Amber

    Expertise: Maths

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.