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’

    • So for sin(2x), for example, 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

• This method can theoretically be used for quite complicated ‘inside’ and ‘outside’ functions

  • On your exam, however, the ‘inside function’ will usually not be more complicated than something like kx (for some constant k) or xⁿ (for some constant power n)

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 below to see 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