Maclaurin Series of Composites & Products (DP IB Analysis & Approaches (AA)): Revision Note

Maclaurin series of composites & products

How can I find the Maclaurin series for a composite function?

• A composite function is a ‘function of a function’ or a ‘function within a function’

  • For example sin(2x) is a composite function, with 2x as the ‘inside function’ which has been put into the simpler ‘outside function’ sin x

  • Similarly  is a composite function, with  as the ‘inside function’ and  as the ‘outside function’

• To find the Maclaurin series for a composite function:
   

• STEP 1
  Start with the Maclaurin series for the basic ‘outside function’

  • Usually this will be one of the ‘standard functions’ whose Maclaurin series are given in the exam formula booklet

• STEP 2
  Substitute the ‘inside function’ every place that x appears in the Maclaurin series for the ‘outside function’

  • E.g. for sin(2x) you would substitute 2x everywhere that x appears in the Maclaurin series for sin x

• STEP 3
  Expand the brackets and simplify the coefficients for the powers of x in the resultant Maclaurin series

How can I find the Maclaurin series for a product of two functions?

• To find the Maclaurin series for a product of two functions:
   

• STEP 1
  Start with the Maclaurin series of the individual functions

  • For each of these Maclaurin series you should only use terms up to an appropriately chosen power of x

    • See the Worked Example for how this is done!

• STEP 2
  Put each of the series into brackets and multiply them together

  • Only keep terms in powers of x up to the power you are interested in

• STEP 3
  Collect terms and simplify coefficients for the powers of x in the resultant Maclaurin series