added type hints and doctests to arithmetic_analysis/newton_method.py (#2259)

* added type hints and doctests to arithmetic_analysis/newton_method.py

Continuing #2128
Also changed some variable names, made them more descriptive.

* Added type hints and doctests to arithmetic_analysis/newton_method.py

added a type alias for Callable[[float], float] and cleaned up the exception handling

* added type hints and doctests to arithmetic_analysis/newton_method.py

improved exception handling

* Update newton_method.py

Co-authored-by: Christian Clauss <cclauss@me.com>