F[x_] = (x^3)-5 (x^2)+3 x + 5;

RootsF = NSolve[F[x]==0,x]

RootsPlot = Graphics[{PointSize[0.025],Red,Point[{x,0}/.RootsF]}];

PlotF = Plot[F[x],{x,-3,7},PlotStyle->{Blue, Thickness[0.007]}];

Show[PlotF,RootsPlot,PlotRange->{{-2,5},{-5,6}}, AspectRatio->1]

FindRoot[F[x]==0,{x,4},WorkingPrecision->20]

F[5]

DF[x_] = D[F[x],x]

DF[5]

ArrowPlots = Graphics[{Arrowheads[.05],Thickness[.010],Yellow, Arrow[{{5,-10},{5,-4.5}}], Black, Arrow[{{30/7,-10},{30/7,-4.5}}], Red, Arrow[{{3.9,10},{3.9,2}}]}];

TangentF = Plot[28 x - 120,{x,-3,7},PlotStyle->{Black, Thickness[0.007]}];

Show[ArrowPlots,PlotF,TangentF,RootsPlot,PlotRange->{{3.5,5.5},{-20,30}}, AspectRatio->1, Axes->True]

(Subscript[x, 0]) = SetPrecision[5.00,10];

(Subscript[x, 1]) =  ((Subscript[x, 0])-F[(Subscript[x, 0])]/DF[(Subscript[x, 0])])

F[(Subscript[x, 1])]

DF[(Subscript[x, 1])]

(F[(Subscript[x, 1])]-(Subscript[x, 1]) DF[(Subscript[x, 1])])

NextIntercept = Solve[F[(Subscript[x, 1])]+DF[(Subscript[x, 1])]  (x-(Subscript[x, 1]))==0,x]

ArrowPlots2 = Graphics[{Arrowheads[.05],Thickness[.010],Blue, Arrow[{{3.97495,-10},{3.97495,-4.5}}]}];

TangentF2 = Plot[15.2449 x - 60.5977,{x,-3,7},PlotStyle->{Red, Thickness[0.007]}];

Show[ArrowPlots,ArrowPlots2,PlotF,TangentF,TangentF2,RootsPlot,PlotRange->{{3.5,5.5},{-20,30}}, AspectRatio->1, Axes->True]

(Subscript[x, 0]) = N[5 ,30];

For[k=1,k≤8,k++, (Subscript[x, k]) = ((Subscript[x, k-1])  - F[(Subscript[x, k-1])]/DF[(Subscript[x, k-1])]); ]

Table[{(Subscript["x", k]),"=",N[(Subscript[x, k]),12]},{k,1,8}]//MatrixForm

(Subscript[x, 0]) = N[{0, 1, 5},30];

Newt[x_] = x-F[x]/DF[x]

N[NestList[Newt[#]&,(Subscript[x, 0]),10],12]//MatrixForm

NSolve[F[x]==0,x, WorkingPrecision->15]

Manipulate[TangentLine = F[a]+DF[a] (x-a);
AxisIntercept = Solve[F[a]+DF[a] (x-a)==0,x];
PlotIntercept = Graphics[{PointSize[0.025],Black,Point[{x,0}/.AxisIntercept[[1]]]}];TangentPlot = Plot[TangentLine,{x,-3,7},PlotStyle->{Red, Thickness[0.007]},PlotRange->{{-2,5},{-5,6}}, AspectRatio->1];Show[TangentPlot,PlotF,PlotIntercept,PlotRange->{{-2,5},{-5,6}}, AspectRatio->1],{{a,-1,"a"},-1,4.5,0.09}]

H[x_] = 8(x^5)-3(x^4)+2(x^3)+9x-5;

Plot[H[x],{x,-10,10},PlotStyle->{Blue, Thickness[0.007]}]

Plot[H[x],{x,-3,3},PlotStyle->{Blue, Thickness[0.007]}]

DH[x_] = D[H[x],x]

(Subscript[x, 0]) = N[{1,1+I,-1-I},40];

Newt[x_] = x-H[x]/DH[x]

(NewtMat = N[NestList[Newt[#]&,(Subscript[x, 0]),12],12])//MatrixForm

RootsH = {NewtMat[[12,1]],NewtMat[[12,2]],Conjugate[NewtMat[[12,2]]],NewtMat[[12,3]],Conjugate[NewtMat[[12,3]]]}

NSolve[H[x],x]

P = Coefficient[H[x],x,5] Product[x-RootsH[[k]],{k,1,5}]

Chop[Expand[P], (10^(-9))]

H[x]

Clear[H]

H[x_] = (E^x); K[x_] = Cos[2 x] + 5;

Plot[{H[x],K[x]},{x,0,3},PlotStyle->{{Red,Thickness[0.007]},{Blue,Thickness[0.007]}},PlotRange->{{0,3},{0,15}}]

F[x_] = H[x]-K[x]

DF[x_] = D[F[x],x]

(Subscript[x, 0]) = N[2,40];

Newt[x_] = x-F[x]/DF[x]

(NewtMat = N[NestList[Newt[#]&,(Subscript[x, 0]),5],12])//MatrixForm

F[x_] = SetPrecision[Tan[x]-1.26195000000000000,20]

DF[x_] = D[F[x],x]

Newt[x_] = x-F[x]/DF[x]

(Subscript[x, 0]) = SetPrecision[1,20]

(NewtMat = SetPrecision[NestList[Newt[#]&,(Subscript[x, 0]),5],20])//MatrixForm

{ArcTan[1.26195],ArcTan[SetPrecision[1.26195,20]]}

(5^13)

(6^13)

F[x_] = (x^13) - 8319407225;

Newt[x_] = x-F[x]/D[F[x],x]

(Subscript[x, 0]) = SetPrecision[6,20];

(NewtMat = N[NestList[Newt[#]&,(Subscript[x, 0]),5],12])//MatrixForm

N[(8319407225^(1/13)),12]

F[x_] = Sin[x];

Plot[F[x],{x,0,2π},PlotStyle->{Blue, Thickness[0.007]}]

(Subscript[x, 0])= SetPrecision[1.97603146838000000,20]

Newt[x_] =  SetPrecision[x-F[x]/D[F[x],x],20]

(NewtMat =  SetPrecision[NestList[Newt[#]&,(Subscript[x, 0]),30],20])//MatrixForm

N[-3π]

(Subscript[TLx, 0]) = Expand[F[(Subscript[x, 0])]+(D[F[x],x]/.x->(Subscript[x, 0]))  (x-(Subscript[x, 0]))]

(Subscript[TLx, 1]) = Expand[F[NewtMat[[2]]]+(D[F[x],x]/.x->NewtMat[[2]])  (x-NewtMat[[2]])]

Plot[{F[x],(Subscript[TLx, 0]), (Subscript[TLx, 1])},{x,0,2π}, PlotStyle->{{Blue, Thickness[0.008]},{Red, Thickness[0.006]},{Black, Thickness[0.006]}}]