Taylor Series and Finite Differences

How can we use the concept of Taylor series to derive finite-difference operators? This video by Heiner Igel, LMU Munich, is part of the course "Computers, Waves, Simulations: A Practical Introduction to Numerical Methods using Python" on the COURSERA platform http://www.coursera.org

https://www.coursera.org/learn/comput...

Executable Jupyter notebooks here: https://github.com/heinerigel/coursera
and at https://www.seismo-live.org