Cyclic Functions under Differentiation

f(0)

1

f'(0)

1

f''(0)

1

f'''(0)

1

This Demonstration explores functions (such as sine and cosine) that are cyclic under differentiation with period four. Such a function is uniquely determined by its value and first three derivatives at a point. The thick curve is the cyclic function itself, and the dotted curves are the first four truncations of its Maclaurin series.