1+1

2+2

a=1
b=3

(Sqrt[5])
(Sqrt[5.])
N[(Sqrt[5])]
N[(Sqrt[5]),(10^2)]
(Sqrt[5])//N

Clear[a,b]

(a+2(b^2))*((a/2)+(b^2))

3x

opice12=5
12opic

zak je krvelacny ucitel

h*f*b*v*a*b

Sin

a=5 (*komentar timto zpusobem*)

((a+2(b^2))*((a/2)+(b^2)))/10

LaplaceTransform[(E^t)+(t^2)Cos[2t],t,s]

ClearAll[a,b]
a=5;
b=7;
c=a+b

ArcCos[0.4];
Cos[%];

ArcCos[0.4]

Cos[%]

Sin[Pi]
Pi//Sin
Sin@Pi

Plot[(t^4)Cos[5t+0.4],{t,0,4Pi}]

y[1]==1

ClearAll[y]
DSolve[{(((2+x))^2)*y''[x]+2x*y'[x]+y[x]==0,y[1]==1,y'[1]==1},y[x],x];

Plot[y[x]/.%,{x,0,10}]

(*Zde zacina druhe cviceni*)

x=3

Plot[{((Sin'[x])^2),(1/((x^3)+1))},{x,0,10},PlotStyle->{{Red,Thick},{Black,Thick,Dashed}},GridLines->Automatic,AxesLabel->{"x","y"}]

Plot[{((Sin'[x])^2),(1/((x^3)+1))},{x,0,10}]

Plot[{D[((Sin[x])^2),x],(1/((x^3)+1))},{x,0,10}]

((D[Sin[x],x])^2)
((Sin'[x])^2)

mujlist={1,2,{luk,sip,3},{auto,{a,b,kuba}}};

mujlist//TreeForm

mujlist[[1]]
mujlist[[3]]
mujlist[[3,2]]
mujlist[[3]][[2]]
mujlist[[0]]
mujlist//Length
mujlist[[5]]
mujlist[[-1]]

ClearAll[x]

xylist={x,y,y,y,x,x,x};
xylist/.x->3

prirazeni={a->1,b->2,x->3,z->4};
xylist/.prirazeni
xylist/.{x->z,z->5}
(*xylist//.{x->z,z->5}*)(*zajimavost*)

ClearAll[x,k]
rce=(x^4)+((x^2)/(k^2))+k*x+(2/3)==0;
rce;
reseni=Solve[rce,x];(*schovane reseni rovnice*)
%//Length;
reseni//Length;
x1=x/.reseni[[1]];
x2=x/.reseni[[2]];
x3=x/.reseni[[3]];
x4=x/.reseni[[4]];
soucet=Abs[x1]+Abs[x2]+Abs[x3]-Abs[x4];
Plot[soucet,{k,-5,-(2/3)},PlotStyle-> {Blue,Thick,Dashed},AxesLabel-> {"k","f(k)"}(*,PlotRange-> All*)]

reseni2=x/.reseni;
x1=reseni2[[1]];
x2=reseni2[[2]];
x3=reseni2[[3]];
x4=reseni2[[4]];
soucet=Abs[x1]+Abs[x2]+Abs[x3]-Abs[x4];
Plot[soucet,{k,-5,-(2/3)},PlotStyle-> {Blue,Thick,Dashed},AxesLabel-> {"k","f(k)"}(*,PlotRange-> All*)]

reseni3=Abs[x/.reseni];
x1=reseni3[[1]];
x2=reseni3[[2]];
x3=reseni3[[3]];
x4=reseni3[[4]];
soucet=x1+x2+x3-x4;
Plot[soucet,{k,-5,-(2/3)},PlotStyle-> {Blue,Thick,Dashed},AxesLabel-> {"k","f(k)"}(*,PlotRange-> All*)]

res=Solve[{(x^2)+(y^2)==k,y-3(x^2)+(5/2)==k},{x,y}];
xka=x/.res;
Plot[xka,{k,0,5},AxesLabel->{"k","f(k)"},PlotStyle-> {{Blue,Thick},{Red,Thick},{Black,Thick,Dashed},{Brown,Thick,Dashed}},GridLines-> Automatic]

(*3.cviceni*)

ClearAll["Global`*"]

ClearAll[f,g]
f[x_]:=(x/3)*Tanh[2*x+1]
g[psi_]:=((Sin[psi]+psi)/(Cos[psi]-(psi^2)))

f[1];
f[1.];

Plot[g[6*f[(((Sqrt[t])+1)/t)]],{t,8,10},AxesLabel->{"t","f(t)"},PlotStyle-> {Blue,Thick},GridLines-> Automatic,PlotRange-> {Automatic,{-20000,20000}},PlotLabel-> "g[6*f[(((Sqrt[t])+1)/t)]"]

res=Solve[(((7/5)*t+(t^2))/x)+(x^3)+((x+3)/((t^4)-1))==((Sin[t])^3),x];
listRes=x/.res;
x1=listRes[[1]];
x2=listRes[[2]];
x3=listRes[[3]];
x4=listRes[[4]];
funkce[t_]=((Abs[x1]-Abs[x3])/(Abs[x2]+Abs[x4]));
Plot[funkce[2*Tan[2*x+(1/2)(x^2)]],{x,0,2},AxesLabel->{"x","f(x)"},PlotStyle-> {Brown,Thick},GridLines-> Automatic]

Solve[Sin[x]==x,x]

Solve[Sin[x]==0,x]

FindRoot[Sin[x]==x,{x,2}]

FindRoot[Sin[x]==0,{x,2}]

FindRoot[Sin[x]==0,{x,(Pi/2)}]

FindRoot[Sin[x]==1,{x,(Pi/2)}]

ClearAll[g,f]
g[fce_,x0_]:=fce'[x0]*(x-x0)+fce[x0];
f[t_]:=(t^2)+Cos[t];
Plot[{f[x],g[f,2]},{x,-1,3},AxesLabel->{"x","f(x)"},PlotStyle-> {{Brown},{Orange,Dashed}},GridLines-> Automatic]

Manipulate[g[fce_,x0_]:=fce'[x0]*(x-x0)+fce[x0];
f[t_]:=(t^2)+Cos[t];
Plot[{f[x],g[f,man]},{x,-1,3},AxesLabel->{"x","f(x)"},PlotStyle-> {{Brown},{Orange,Dashed}},GridLines-> Automatic,PlotRange-> {Automatic,{-1,9}}],{man,-1,3}]

Manipulate[g[fce_,x0_]:=fce'[x0]*(x-x0)+fce[x0];
f[t_]:=(t^2)+Cos[t];
Show[Plot[{f[x],g[f,man]},{x,-1,3},AxesLabel->{"x","f(x)"},PlotStyle-> {{Brown},{Orange,Dashed}},GridLines-> Automatic,PlotRange-> {Automatic,{-1,9}}],Graphics[{PointSize[Large],Pink,Point[{man,f[man]}]}]],{man,-1,3}]

Manipulate[g[fce_,x0_]:=fce'[x0]*(x-x0)+fce[x0];
f[t_]:=(t^2)+Cos[t];
graf=Plot[{f[x],g[f,man]},{x,-1,3},AxesLabel->{"x","f(x)"},PlotStyle-> {{Brown},{Orange,Dashed}},GridLines-> Automatic,PlotRange-> {Automatic,{-1,9}}];
bod=Graphics[{PointSize[Large],Green,Point[{man,f[man]}]}];
Show[graf,bod]
,{{man,0},-1,3}]