DynamicNewton:=Manipulate[
Module[{func},
Clear[x];
func[x_,formula_]=Function[x,formula];
Module[{f=func[x,expr],g,list},
g[x_]=x-f[x]/f'[x];
list=NestList[g,x0,n];Column[{TraditionalForm@Text@Style[Row[{"Iterations ",Range[0,n],": ",N[ list]}],14],Plot[f[x],{x,a,b},PlotRange-> All,
Epilog->{
{Red,Arrowheads[Medium],Arrow[Most[Flatten[Map[{{#,0},{#,f[#]}} &,list],1]]]},
{PointSize[Medium],Point[{x0,0}]}},ImageSize->{600,325}]},Center]]],
{{expr,x^3-5x^2+1,"f(x)="},ControlType -> InputField},{{a,-5,"a"},ControlType -> InputField },{{b,5,"b"},ControlType -> InputField } ,
{{n,1,"n"},0,6,1,Appearance->"Labeled"},
{{x0,3,Subscript["x","0"]},a,b,Appearance->"Labeled" }]

DynamicNewton

Experiment 1

f(x) = sin(e^x) has 3 roots in the interval [0,2.5].  Use The Newton’s method app to approximate each one to 4 decimal places.  

a. How far below the first root can you start and still find the first root?  What happens if you start at a lower number?  
b. How far above the first root can you start and still find the first root?  What happens if you start at a higher number?

Experiment 2

f(x) = x*E^(-x) has a root at zero.  Look at this function on the interval [-1,7].  

a. What happens if you start with (Subscript[x, 0])=2?
b. What happens if you drag the slider to the left past 1?  How close to 0 do you have to be before you get a good approximation after 6 iterations? What if you only use 3 iterations?

Experiment 3

Let f(x) = (x^3)-x-3 on the range [-4,2].  

a. What happens if we start at a point near -3?
b. How can we fix this?