9.1.2        The Intermediate Value Theorem
Continuity can be understood by connecting it to the Intermediate Value Theorem (IVT) and solving equations of the form $f(x) = 0$.
IVT: If $f$ is a continuous function on the interval $[a,b]$ and $f(a) \cdot f(b) \gt 0$ then there is a number $c \in (a,b)$ where $f(c) = 0$.
Mapping diagrams provide an alternative visualization for the IVT. They can also be used to visualize a proof of the result using the "bisection method."

Bisection and IVT vizualized with GEOGEBRA.
This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

9.3.2         Newton’s Method
An early application
of the first derivative, Newton's method for estimating roots of functions is visualized with mapping diagrams.
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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

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com