Differentiating & Integrating Maclaurin Series (DP IB Analysis & Approaches (AA)): Revision Note

Differentiating & Integrating Maclaurin Series

How can I use differentiation to find Maclaurin Series?

• If you differentiate the Maclaurin series for a function f(x) term by term, you get the Maclaurin series for the function’s derivative f’(x)

• You can use this to find new Maclaurin series from existing ones

  • For example, the derivative of sin x is cos x

  • So if you differentiate the Maclaurin series for sin x term by term you will get the Maclaurin series for cos x

How can I use integration to find Maclaurin series?

• If you integrate the Maclaurin series for a derivative f’(x), you get the Maclaurin series for the function f(x)

  • Be careful however, as you will have a constant of integration to deal with

  • The value of the constant of integration will have to be chosen so that the series produces the correct value for f(0)

• You can use this to find new Maclaurin series from existing ones

  • For example, the derivative of sin x is cos x

  • So if you integrate the Maclaurin series for cos x (and correctly deal with the constant of integration) you will get the Maclaurin series for sin x