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