CCD.NA.Newt. Newton's Method: An early application of the first derivative, Newton's method for estimating roots of functions is often visualized with graphs, but an alternative that presents a different connection to using motion to understand the derivative is visualized with mapping diagrams.

Considering the derivative focus point for $x_0$ allows one to consider the question of when a linear function has value $0$ and find the corresponding number $x_1$ in the domain by connecting $0$ on the target to the focus point. This process is repeated using $x_1$ to find the next estimate $x_2$ and so on...

The first step of Newton's method for estimating roots visualized with a mapping diagram
using the derivative focus point to find $x_1$
$x_{n+1} =x_n - f(x_n)/f'(x_n)$ $f(x)=x^2- 2$ $f'(x)=2x$
$f(x)/f'(x) =(x^2 - 2) / (2x)$
3.00000000000000 7.00000000000000 6.00000000000000 1.16666666666667
1.83333333333334 1.36111111111111 3.66666666666667 0.371212121212122
1.46212121212121 0.137798438934803 2.92424242424243 0.0471227822264093
1.41499842989480 0.00222055660475729 2.82999685978961 0.000784649847605263
1.41421378004720 6.15675383563997E-7 2.82842756009440 2.17674085859724E-7
1.41421356237311 4.75175454539568E-14 2.82842712474623 1.67999893079162E-14
1.41421356237310 -4.44089209850063E-16 2.82842712474619 -1.57009245868378E-16
1.41421356237310 4.44089209850063E-16 2.82842712474619 1.57009245868378E-16
1.41421356237310 -4.44089209850063E-16 2.82842712474619 -1.57009245868378E-16