Zadanie 1

x/.Solve[1/(1+Exp[-x])==10^-64,x,Reals][[1]]//Simplify

-%/.Log->List//First//IntegerDigits//Length

x/.Solve[1/(1+Exp[-x])==10^-p,x,Reals,Assumptions->p>=64][[1]]//PowerExpand

N[%/.p->64]

Zadanie 2

wykresZad2=RegionPlot[x^2-y^3<=x y +1, {x,-2,2},{y,-2,2}]

NMinimize[{y,x^2-y^3<=x y +1},{x,y}]

Minimize[{y,x^2-y^3<=x y +1},{x,y}]

{X,Y}={x,y}/.%[[-1]]

ToRadicals[%]

a=865-24 (Sqrt[1299]);

-1/24*(1+1/CubeRoot[a]+CubeRoot[a])//N[#,32]&

%/X

Show[wykresZad2,Epilog->Point[{X,Y}],GridLines->{{X},{Y}}]

Zadanie 3

Integrate[Cos[x]^4,x]/.n->4

Zadanie 4

wykresZad4=ContourPlot[{Sin[x]^2+Cos[y]^2==1,x^4+y^4==y,y==x},{x,-2,2},{y,-2,2}]

FindInstance[y^4+y^4==y,y,Reals,3]

{Sin[x]^2+Cos[y]^2==1,x^4+y^4==y}/.y->(1/(2^(1/3)))/.x->((-1)/(2^(1/3)))

2^(-1/3)//N

Show[wykresZad4,Epilog->Point[{{0,0},{(1/(2^(1/3))),(1/(2^(1/3)))},{-(1/(2^(1/3))),(1/(2^(1/3)))}}]]

Zadanie 5

RegionPlot[1/x^2+1/y^2>1&&Max[Abs[x],Abs[y]]<8,{x,-12,12},{y,-12,12}]

Integrate[Boole[1/x^2+1/y^2>1&&Max[Abs[x],Abs[y]]<8],{x,-12,12},{y,-12,12}]

N[%]

Zadanie 6

$Assumptions=a∈Vectors[3]&&b∈Vectors[3]&&c∈Vectors[3]

(b-2a).c+2 a.(b+c)-b.(a+c)+(a+b).Cross[a,b]//TensorReduce

Zadanie 7

NestList[2-2/#&,x,12]//Simplify//DeleteDuplicates

Zadanie 8

r={x[t],y[t],z[t]}

g={0,-10,0}

Ω={0,0,1}

sys=m D[r,t,t]-m g-Cross[Ω,D[r,t]]

m=1;v0=10;α=1;

rozw=NDSolve[{sys[[1]]==0,sys[[2]]==0,sys[[3]]==0,x'[0]==v0 Sin[α],y'[0]==v0 Cos[α],z'[0]==0,x[0]==0,y[0]==0,z[0]==0},r,{t,0,6 v0 Sin[α]/10}][[1]]

{X,Y,Z}=r/.rozw;

ParametricPlot3D[{X,Y,Z},{t,0,4 v0 Sin[α]/10},PlotRange->{{0,16},{0,5},All},ViewPoint->{0,0,10},ViewVertical->{0, 1, 0}]

ParametricPlot[{X,Y},{t,0,4v0 Sin[α]/10}]

X/.FindRoot[Y==0,{t,4v0 Sin[α]/10}] (* zasięg *)

NMaximize[{Y,0<t<4v0 Sin[α]/10},t] (* wysokość *)

Zadanie 9

A={{1-Pi,1-Pi},{Pi,Pi}}

J=MatrixExp[{{0, I Pi},{I Pi,0}}]

MatrixPower[A,7]+MatrixExp[A]+J-E A//Simplify

Zadanie 10

Import["https://raw.githubusercontent.com/VA00/SymbolicRegressionPackage/master/SymbolicRegression.m"];

f=ZadanieNOF[]