Paradox forgástest

f[x_]:=(1/x)

(Sqrt[1+((f'[x])^2)])

(∫_1^∞)((f[x])^2)(Sqrt[1+((f'[x])^2)])\[DifferentialD]x

Kocsik a kereszteződésben

T[t_]:=(Sqrt[(((x1-v1 t))^2)+(((x2-v2 t))^2)])

D[T[t], t]

Simplify[D[T[t], t], Assumptions->{x1-v1 t==0, x2-v2 t∈Reals}]

Taylor Polinom

Manipulate[
Show[
Plot[
{(∑_(i=0)^n)(((Derivative[i][f][x0])/(i!))(((x-x0))^i)), f[x]}
, {x, -3, 3}, PlotStyle->{Directive[Blue, Thick], Red}, AxesOrigin->{0, 0}, AxesLabel->{x, y}, PlotRange->3],
Graphics[{
Point[{x0, f[x0]}]
}]
]
,{{x0, 0}, -3, 3}, {n, 0, 20,1},{{f, (E^(#))&}, {Sin->"sin",Cos->"cos",(E^(#))&->"(E^x)", Cosh->"ch", (E^(-((#)^2)))&->"(E^(-(x^2)))", (1/(#-1))&->"1/x",If[#<0, -#, #]&->"|x|", If[#≠0, (E^(-(1/((#)^2)))), 0]&->"?"}}]

Fixpont Iteráció

Manipulate[
Show[
Plot[
{f[x], x},
{x, 0, 1}, AxesOrigin->{0, 0}, PlotRange->{0, 1}],
Graphics[{
Point[{x0, f[x0]}],
Line@Flatten[
NestList[
{f/@#[[1]], f/@#[[2]]}&, {{x0, x0}, {x0, f[x0]}},n
]
,1]
}]
]
,{n, 0, 20, 1},{{x0, 0.5}, 0, 1}, {{f, Cos}, { Cos->"cos", 0.1Sin[2Pi#]+#&->"0.1sin(x)+x", -0.1Sin[2Pi#]+#&->"-0.1sin(x)+x", 4#(1-#)&->"logistic"}}]

Newton gyökkeresés

Manipulate[
Show[
Plot[
f[x],
{x, -2Pi, 2Pi}, AxesOrigin->{0, 0}, PlotRange->{-2, 2}],
Graphics[{
Point[{x0, f[x0]}],
Line@Flatten[
NestList[
{{#[[2,1]], f[#[[2,1]]]},{#[[2, 1]]-((f[#[[2,1]]])/(f'[#[[2,1]]])), 0}}&, {{x0, 0}, {x0, f[x0]}},n
]
,1]
}]
]
,{n, 0, 20, 1},{{x0, 0.5}, -2, 2}, {{f, Cos}, { Cos->"cos", (E^(#))-1&->"E", Cosh[#]-1&->"ch", #Sin[1/#]&->"sin(1/x)"}}]

Numerikus Deriválás

f[x_]:=(E^x)
Manipulate[
Show[
Plot[
{f[x], If[a==b, f[a]+f'[a](x-a), f[a]+((f[b]-f[a])/(b-a))(x-a)]},
{x, -2, 2}, AxesOrigin->{0, 0}, PlotRange->{0, 6}],
Graphics[{
Point[{{a, f[a]}, {b, f[b]}}]
}]
]
,{{a, 0}, -2, 2},{{b, 1}, -2, 2}]

Numerikus Integrálás

f[x_]:=(E^(Sin[x]))
Manipulate[
Show[
Plot[
f[x],
{x, -2, 2}, AxesOrigin->{0, 0}, PlotRange->{0, 4}],
Graphics[{
Blue,Opacity[0.5],EdgeForm[Directive[Black]],
Table[
Rectangle[{a+((b-a)/n)i, 0},{a+((b-a)/n)(i+1), If[t, f[a+((b-a)/n)i], f[a+((b-a)/n)(i+1)]]}]
,{i, 0, n-1}]
}]
]
,{n, 1, 50, 1}, {{a, 0}, -2, 2}, {{b, 1}, -2, 2},{t,{True,False}}]

Lehajló Tartó

Clear[Sol]
Clear[q, T, M, κ,ϕ, e]
EI=1;

q[x_]:=1(*Piecewise[{
{0, x<-0.1},
{100(x+0.1), -0.1<x<0},
{100(-x+0.1), 0<x<0.1},
{0, 0.1<x}
}]*)
T=Interpolation[Table[{x, (∫_(-1)^x)q[ξ]\[DifferentialD]ξ}, {x, -1, 1, 0.01}]];
M=Interpolation[Table[{x, (∫_(-1)^x)T[ξ]\[DifferentialD]ξ}, {x, -1, 1,0.01}]];
κ[x_]:=((-M[x])/EI)
ϕ=Interpolation[Table[{x, (∫_(-1)^x)κ[ξ]\[DifferentialD]ξ}, {x, -1, 1, 0.01}]];
e=Interpolation[Table[{x, (∫_(-1)^x)ϕ[ξ]\[DifferentialD]ξ}, {x, -1, 1, 0.01}]];

Sol=Solve[{
(*(T[x]+a/.x->1)==0,*)
(M[x]+a x+b/.x->-1)==0,
(M[x]+a x+b/.x->1)==0,
(*(ϕ[x]+((a ((x^2)/2)+b x)/(-EI))+c/.x->-1)==0,*)
(*(ϕ[x]+((a ((x^2)/2)+b x)/(-EI))+c/.x->-1)==0,*)
(e[x]+((a ((x^3)/6)+b ((x^2)/2))/(-EI))+c x+d/.x->-1)==0
,(e[x]+((a ((x^3)/6)+b ((x^2)/2))/(-EI))+c x+d/.x->1)==0
}, {a, b, c, d}][[1]]

Show[
Plot[{q[x], T[x]+a/.Sol, M[x]+a x+b/.Sol, κ[x]+((a x+b)/(-EI))/.Sol, ϕ[x]+(((a ((x^2)/2)+b x)/(-EI))+c)/.Sol, e[x]+(((a ((x^3)/6)+b ((x^2)/2))/(-EI))+c x+d)/.Sol}, {x, -1, 1}, PlotRange->All,PlotStyle->{Red, Gray, Gray, Gray, Gray, Directive[Blue, Thick]}],
PlotRange->{-1, 1}
]

Simuló kör

f[x_]:=(x^2)
Manipulate[
Show[
Plot[
f[x]
, {x, -2, 2}, PlotRange->{0, 4} ,AspectRatio->1],
Graphics[{
Point[{{x0, f[x0]}, {x0, f[x0]}+{-f'[x0], 1}((1+((f'[x0])^2))/(f''[x0]))}],
Circle[{x0, f[x0]}+{-f'[x0], 1}((1+((f'[x0])^2))/(f''[x0])),((((1+((f'[x0])^2)))^(3/2))/(f''[x0]))],
Line[{{x0, f[x0]}, {x0, f[x0]}+{-f'[x0], 1}((1+((f'[x0])^2))/(f''[x0]))}],
Red,
Arrow[{{x0, f[x0]},{x0, f[x0]}+{-f'[x0], 1}((f''[x0])/(((1+((f'[x0])^2)))^2))}]
}]
]
,{{x0,0}, -2, 2}]

f[x_]:=((x^2)/2)
g[x_]:={x, f[x], 0}
Show[
ParametricPlot3D[
g[u]+Normalize[Cross[g'[u], -{-f'[u], 1, 0}((f''[u])/(((1+((f'[u])^2)))^2))+{0, 0, -1}]]v
, {u, -2, 2}, {v, -0.2, 0.2}, AxesLabel->{x, y, z}, Boxed->False, Mesh->{5,5}],
ParametricPlot3D[g[t], {t, -2, 2}, PlotStyle->Directive[Black, Thick]]
]