The MVT and Approximations (ie, Taylor Series Light)

Example

f[x_] := Sin[x];
g[x_] := Cos[x];
h[x_] := -Sin[x];
L[x_] := x;
Error[x_] := f[x] - L[x];
x1Error[x_] := ArcCos[(Error[x]/x) + g[0]];
x2Error[x_] := - ArcSin[ (g[x1Error[x]] -g[0]) / x1Error[x]];

Print["Comparison of Sin[x] and L[x] = x. Let Error[x] = Sin[x] - L[x]. The three points below (in decreasing order) are x, the x from the MVT applied to Error (where we evaluate Error'[x]), and the x from the MVT applied to Error' (where we evaluate Error''[x]."];
Manipulate[Plot[{f[x],x}, {x,0,Pi/2},Epilog -> {PointSize[.025], Point[{x0,0}],  Point[{x1Error[x0],0}],  Point[{x2Error[x0],0}]}], {x0,0.000001,Pi/2}]