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         Newtonâ€™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