Egri chiziqlarning maxsus nuqtalarini kompyuter algebrasi tizimi yordamida tekshirish
![Ta`rif 1. Chiziq deb quyidagi tenglamalarni qanoatlantiruvchi
nuqtalar to`plamiga aymatmiz:
x = φ( t) , y = Ψ ( t) ,( a ˂ t ˂ b )
(1)
bunda
ϕ(t) va ψ(t) -uzluksiz funksiyalar bo`lib, uzluksiz hosilalarga egadir.
(1) tenglamalar chiziqning parametrik tenglamalari deb, t esa parameter
deb ataladi.
Teorema . Agar (1) chiziqning t=t
0 ga mos nuqtasida
x0= ϕ(t0) va
y0= ψ(t0)
bo`lib, bu nuqtadagi ϕ(t0) va ψ(t0) hosilalarning aqalli bittasi
noldan farqli bo`lsa, u holda bu nuqtaning yetarlicha kichik atrofi
regulyar yoydan, t
0 nuqtaning o`zi esa oddiy nuqtadan iborat bo`ladi.
Masalan,
ϕ(t0)≠ 0 deylik. Ushbu tenglamani X = φ(t) = ˃ X – φ (t) = 0
X – φ(t) = F (x , t ) deb olamiz va
F(x,t)= x−ϕ(t)= 0, (2)
bundan:
∂F(x,t)
∂t = ∂
∂t[x−ϕ(t)]=−ϕ(t)
Bu hosila, shartga ko`ra,
x= x0,t= t0 nuqtada noldan farqli. Mavjudlik
teoremasiga asosan,
x= x0,t= t0 ga yaqin x va t uchun faqat bitta
(uzluksiz monoton)
t= t(x)
funksiya mavjudki *
,
y(x0,t0) nuqta atrofida (2) tenglamani qanoatlantiradi
va
x= x0 qiymatda t= t0 qiymatni qabul qiladi. Endi t=t(x) ni (1)
tenglamalarning ikkinchiga qo`yamiz:
y = Ψ
[ t( x )] = f ( x ) . ❑
(3)
Bu funksiya
(x0y0) nuqta atrofida uzluksiz](https://docx.uz/documents/397aa3a1-b3cd-475b-9acb-3f5e3bbf44ff/page_10.png?v=1)
![14 – chizma
IV – MAPLE DASTURI YORDAMIDA MISOLLAR
TECHISH .
> restart;" 1-MISOL";" 1-MISOL"
>
> "CHIZIQNING MAXSUS NUQTALARI(oshkormas kurnishda berilgan
bo'lsa)";
"CHIZIQNING MAXSUS NUQTALARI(oshkormas kurnishda berilgan bo'lsa)"
> F:=y^2+x^2-x^3;
:= F y2 x2 x3
> f[x]:=diff(F,x);
:= fx 2x 3x2
> f[y]:=diff(F,y);
:= fy 2y
> d:=diff(f[x],y)^2-diff(f[x],x)*diff(f[y],y);
:= d 4 12 x
> for i from 1 to 2 do s:=[solve({f[x],f[y]},
{x,y})];x[i]:=subs(s[i],x);y[i]:=subs(s[i],y);
print([x[i],y[i]]) od;
:= s
, { } , x 0 y 0 { } , x 2
3 y 0
:= x1 0
:= y1 0
[ ] ,0 0](https://docx.uz/documents/397aa3a1-b3cd-475b-9acb-3f5e3bbf44ff/page_16.png?v=1)
![:= s
, { } , x 0 y 0 { } , x 2
3 y 0 := x
2 2
3
:= y2 0
,2
3 0
> for i from 1 to 2 do if y[i]^2+x[i]^4-x[i]^2=0 then
print([x[i],y[i]]," maxsus nuqta");delta:=-12*x[i]-4 else
print([x[i],y[i]],"maxsus nuqta emas") fi od;
, [ ] ,0 0 " maxsus nuqta"
,
,2
3 0 "maxsus nuqta emas"
> for i from 1 to 2 do if y[i]^2+x[i]^4-x[i]^2=0 then
x:=x[i];y:=y[i] fi od; delta:=d;
:= -4
> if delta>0 then print("Tugun maxsus nuqta") else if delta<0 then
print("Yakkalangan maxsus nuqta") else print("agar bu nuqtadan
o'tuvchi regulyar yoy narmaldan bir tomonda urunmadan ikki
tomonda bo'lsa bu nuqta 1- tip qaytish nuqtasi de-
di");print("agar bu nuqtadan o'tuvchi regulyar yoy narmal va
urunmada bir tomonda bo'lsa 2-tip qaytish nuqtasi de-di");
print(" agar bu nuqtadan o'tuvchi regulyar yoy narmal va
urunmadan ikki tomonda bo'lsa o'z-o'ziga qaytish maxsus nuqta
de-di");print(" bu tasdiqlarni grafikka qarab aniqlash mumkin")
fi;fi;
"Yakkalangan maxsus nuqta"
> f:=proc(x)if x>-1 then (x^3-x^2)^(1/2) else -(-x^2-x^3)^(1/2) fi
end; g:=proc(x)if x>-1 then -(x^3-x^2)^(1/2) else (-x^2-
x^3)^(1/2) fi end;
> plot([f,g],-10..10,color=green,thickness=5);
:= f proc () end proc x if then else end if -1 x ^ ( ) ^x 3 ^x 2 ( ) /12 ^ ( ) ^x 2 ^x 3 ( ) /12
:= g proc () end proc x if then else end if -1 x ^ ( ) ^x 3 ^x 2 ( ) /12 ^ ( ) ^x 2 ^x 3 ( ) /12](https://docx.uz/documents/397aa3a1-b3cd-475b-9acb-3f5e3bbf44ff/page_17.png?v=1)
![>
> restart; "2- MISOL";
> "CHIZIQNING MAXSUS NUQTALARI(oshkormas kurnishda berilgan
bo'lsa)";"3- MISOL"
"CHIZIQNING MAXSUS NUQTALARI(oshkormas kurnishda berilgan bo'lsa)"
> F:=y^2+x^4-x^2;
:= F y2 x4 x2
> f[x]:=diff(F,x);
:= fx 4x3 2x
> f[y]:=diff(F,y);
:= fy 2y
> d:=diff(f[x],x)*diff(f[y],y)-diff(f[x],y);
:= d 24 x2 4
> for i from 1 to 3 do
s:=[solve(f[x],x)];h:=[solve(f[y],y)];x[i]:=subs(s[i]);y[i]:=sub
s(h[1]); print([x[i],y[i]]) od;
:= s
, , 0 2
2 2
2
:= h [ ]0
:= x1 0
:= y1 0
[ ] ,0 0
:= s
, , 0 2
2 2
2](https://docx.uz/documents/397aa3a1-b3cd-475b-9acb-3f5e3bbf44ff/page_18.png?v=1)
![:= h [ ]0
:= x2
2
2
:= y2 0
,2
2 0
:= s
, , 0 2
2 2
2
:= h [ ]0
:= x3 2
2
:= y3 0
, 2
2 0> for i from 1 to 3 do if y[i]^2+x[i]^4-x[i]^2=0 then
print([x[i],y[i]]," maxsus nuqta");delta:=-12*x[i]-4 else
print([x[i],y[i]],"maxsus nuqta emas") fi od;
, [ ] ,0 0 " maxsus nuqta"
,
,2
2 0 "maxsus nuqta emas"
,
, 2
2 0 "maxsus nuqta emas"
> for i from 1 to 3 do if y[i]^2+x[i]^4-x[i]^2=0 then
x:=x[i];y:=y[i] fi od; delta:=d;
:= -4
> if delta<0 then print("Tugun maxsus nuqta") else if delta>0 then
print("Yakkalangan maxsus nuqta") else print("agar bu nuqtadan
o'tuvchi regulyar yoy narmaldan bir tomonda urunmadan ikki
tomonda bo'lsa bu nuqta 1- tip qaytish nuqtasi de-
di");print("agar bu nuqtadan o'tuvchi regulyar yoy narmal va
urunmada bir tomonda bo'lsa 2-tip qaytish nuqtasi de-di");
print(" agar bu nuqtadan o'tuvchi regulyar yoy narmal va
urunmadan ikki tomonda bo'lsa o'z-o'ziga qaytish maxsus nuqta
de-di");print(" bu tasdiqlarni grafikka qarab aniqlash mumkin")
fi;fi;
"Tugun maxsus nuqta"](https://docx.uz/documents/397aa3a1-b3cd-475b-9acb-3f5e3bbf44ff/page_19.png?v=1)
![> f:=proc(x)if x>0 then (x^2-x^4)^(1/2) else -(x^2-x^4)^(1/2) fi
end; g:=proc(x)if x>0 then -(x^2-x^4)^(1/2) else (x^2-x^4)^(1/2)
fi end;
> plot([f,g],-1..1,color=green,thickness=5); := f proc () end proc x if then else end if 0 x ^ ( ) ^x 2 ^x 4 ( ) /12 ^ ( ) ^x 2 ^x 4 ( ) /12
:= g proc () end proc x if then else end if 0 x ^ ( ) ^x 2 ^x 4 ( ) /12 ^ ( ) ^x 2 ^x 4 ( ) /12
> restart;"3- MISOL";
>
"3- MISOL"
> "CHIZIQNING MAXSUS NUQTALARI(oshkormas kurnishda berilgan
bo'lsa)";
"CHIZIQNING MAXSUS NUQTALARI(oshkormas kurnishda berilgan bo'lsa)"
> F:=y^2-x^3-x^2;
:= F y2 x3 x2
> f[x]:=diff(F,x);
:= fx 3x2 2x
>
:= d 12 x 4
> f[y]:=diff(F,y);
:= fy 2y
> d:=diff(f[x],x)*diff(f[y],y)-diff(f[x],y)^2;
:= d 12 x 4
> for i from 1 to 2 do s:=[solve({f[x],f[y]},
{x,y})];x[i]:=subs(s[i],x);y[i]:=subs(s[i],y);
print([x[i],y[i]]) od;
:= s
, { } , x 0 y 0 { } , x -2
3 y 0
:= x1 0](https://docx.uz/documents/397aa3a1-b3cd-475b-9acb-3f5e3bbf44ff/page_20.png?v=1)
![:= y1 0
[ ] ,0 0
:= s
, { } , x 0 y 0 { } , x -2
3 y 0
:= x2
-2
3
:= y2 0
, -2
3 0> for i from 1 to 2 do if y[i]^2-x[i]^3-x[i]^2=0 then
print([x[i],y[i]]," maxsus nuqta") else
print([x[i],y[i]],"maxsus nuqta emas") fi od;
, [ ] ,0 0 " maxsus nuqta"
,
, -2
3 0 "maxsus nuqta emas"
> for i from 1 to 2 do if y[i]^2-x[i]^3-x[i]^2=0 then
x:=x[i];y:=y[i] fi od; delta:=d;
:= -4
> if delta<0 then print("Tugun maxsus nuqta") else if delta>0 then
print("Yakkalangan maxsus nuqta") else print("Grafigiga qarang")
fi;fi;
"Tugun maxsus nuqta"
> f:=proc(x)if x>0 then (x^3+x^2)^(1/2) else -(x^2+x^3)^(1/2) fi
end; g:=proc(x)if x>0 then -(x^3+x^2)^(1/2) else (x^3+x^2)^(1/2)
fi end;
> plot([f,g],-1..1,color=blue,thickness=5);
:= f proc () end proc x if then else end if 0 x ^ ( ) ^x 3 ^x 2 ( ) /12 ^ ( ) ^x 3 ^x 2 ( ) /12
:= g proc () end proc x if then else end if 0 x ^ ( ) ^x 3 ^x 2 ( ) /12 ^ ( ) ^x 3 ^x 2 ( ) /12](https://docx.uz/documents/397aa3a1-b3cd-475b-9acb-3f5e3bbf44ff/page_21.png?v=1)
![>
> restart; "4- MISOL";
> "CHIZIQNING MAXSUS NUQTALARI(oshkormas kurnishda berilgan
bo'lsa)";"4- MISOL"
"CHIZIQNING MAXSUS NUQTALARI(oshkormas kurnishda berilgan bo'lsa)"
> F:=x^2*y^2-x^2-y^2;
:= F x2y2 x2 y2
> f[x]:=diff(F,x);
:= fx 2xy2 2x
> f[y]:=diff(F,y);
:= fy 2x2y 2y
> d:=diff(f[x],x)*diff(f[y],y)-diff(f[x],y)^2;
:= d ( ) 2y2 2 ( ) 2x2 2 16 x2y2
> for i from 1 to 5 do s:=[solve({f[x],f[y]},
{x,y})];x[i]:=subs(s[i],x);y[i]:=subs(s[i],y);
print([x[i],y[i]]) od;
:= s [ ] , , , , { } , x 0 y 0 { } , x 1 y 1 { } , x 1 y -1 { } , x -1 y 1 { } , x -1 y -1
:= x1 0](https://docx.uz/documents/397aa3a1-b3cd-475b-9acb-3f5e3bbf44ff/page_22.png?v=1)
![:= y1 0
[ ] ,0 0
:= s [ ] , , , , { } , x 0 y 0 { } , x 1 y 1 { } , x 1 y -1 { } , x -1 y 1 { } , x -1 y -1
:= x2 1
:= y2 1
[ ] ,1 1
:= s [ ] , , , , { } , x 0 y 0 { } , x 1 y 1 { } , x 1 y -1 { } , x -1 y 1 { } , x -1 y -1
:= x3 1
:= y3 -1[ ],1 -1
:= s [ ] , , , , { } , x 0 y 0 { } , x 1 y 1 { } , x 1 y -1 { } , x -1 y 1 { } , x -1 y -1
:= x4 -1
:= y4 1
[ ],-1 1
:= s [ ] , , , , { } , x 0 y 0 { } , x 1 y 1 { } , x 1 y -1 { } , x -1 y 1 { } , x -1 y -1
:= x5 -1
:= y5 -1
[ ] , -1 -1
> for i from 1 to 5 do if x[i]^2*y[i]^2-x[i]^2-y[i]^2=0 then
print([x[i],y[i]]," maxsus nuqta");delta:=-12*x[i]-4 else
print([x[i],y[i]],"maxsus nuqta emas") fi od;
, [ ] ,0 0 " maxsus nuqta"
, [ ] ,1 1 "maxsus nuqta emas"
, [ ] ,1 -1 "maxsus nuqta emas"
, [ ] , -1 1 "maxsus nuqta emas"
, [ ] , -1 -1 "maxsus nuqta emas"
> for i from 1 to 5 do if y[i]^2-x[i]^3-x[i]^2=0 then
x:=x[i];y:=y[i] fi od; delta:=d;
:= 4
> ifdelta <0 thenprint (" Tugunmaxsusnuqta ") elseifdelta >0
thenprint (" Yakkalanganmaxsusnuqta ")
elseprint (" agarbunuqtadano ' tuvchiregulyaryoynarmaldanbirtomondau
runmadanikkitomondabo ' lsabunuqta 1- tipqaytishnuqtaside -
di "); print (" agarbunuqtadano ' tuvchiregulyaryoynarmalvaurunmadabir
tomondabo ' lsa 2- tipqaytishnuqtaside - di "); print ("
agarbunuqtadano ' tuvchiregulyaryoynarmalvaurunmadanikkitomondabo '
lsao ' z - o ' zigaqaytishmaxsusnuqtade - di "); print ("
butasdiqlarnigrafikkaqarabaniqlashmumkin ") fi ; fi ;](https://docx.uz/documents/397aa3a1-b3cd-475b-9acb-3f5e3bbf44ff/page_23.png?v=1)
!["Yakkalangan maxsus nuqta"> f:=proc(x)if x>0 then (x^2/(x^2-1))^(1/2) else -(x^2/(x^2-
1))^(1/2) fi end; g:=proc(x)if x>0 then -(x^2/(x^2-1))^(1/2)
else (x^2/(x^2-1))^(1/2) fi end;
> plot([f,g],-10..10,color=yellow,thickness=5);
f xproc ( ) :=
if then else end if0 x ^( )/ ( )^x 2 ^x 2 1 ( )/1 2 ^( )/ ( )^x 2 ^x 2 1 ( )/1 2
end proc
g xproc ( ) :=
if then else end if0 x ^( )/ ( )^x 2 ^x 2 1 ( )/1 2 ^( )/ ( )^x 2 ^x 2 1 ( )/1 2
end proc
> restart; "5- MISOL";
> "CHIZIQNING MAXSUS NUQTALARI(oshkormas kurnishda berilgan
bo'lsa)";
"5- MISOL"
"CHIZIQNING MAXSUS NUQTALARI(oshkormas kurnishda berilgan bo'lsa)"
> F:=4*y^2-x^5-5*x^4;
:= F 4y2 x5 5x4
> f[x]:=diff(F,x);
:= fx 5x4 20 x3](https://docx.uz/documents/397aa3a1-b3cd-475b-9acb-3f5e3bbf44ff/page_24.png?v=1)
![> f[y]:=diff(F,y); := fy 8y
> d:=diff(f[x],x)*diff(f[y],y)-diff(f[x],y);
>
:= d 160 x3 480 x2
:= d 12 x 4
> for i from 1 to 2 do s:=[solve({f[x],f[y]},
{x,y})];x[i]:=subs(s[i],x);y[i]:=subs(s[i],y);
print([x[i],y[i]]) od;
:= s [ ] , { } , x 0 y 0 { } , x -4 y 0
:= x1 0
:= y1 0
[ ] ,0 0
:= s [ ] , { } , x 0 y 0 { } , x -4 y 0
:= x2 -4
:= y2 0
[ ],-4 0
> for i from 1 to 2 do if 4*y[i]^2-x[i]^5-5*x[i]^4=0 then
print([x[i],y[i]]," maxsus nuqta");delta:=-12*x[i]-4 else
print([x[i],y[i]],"maxsus nuqta emas") fi od;
, [ ] ,0 0 " maxsus nuqta"
, [ ] , -4 0 "maxsus nuqta emas"
> for i from 1 to 2 do if 4*y[i]^2-x[i]^5-5*x[i]^4=0 then
x:=x[i];y:=y[i] fi od; delta:=d;
:= 0
> ifdelta <0 thenprint (" Tugunmaxsusnuqta ") elseifdelta >0
thenprint (" Yakkalanganmaxsusnuqta ")
elseprint (" agarbunuqtadano ' tuvchiregulyaryoynarmaldanbirtomondau
runmadanikkitomondabo ' lsabunuqta 1- tipqaytishnuqtaside -
di "); print (" agarbunuqtadano ' tuvchiregulyaryoynarmalvaurunmadabir
tomondabo ' lsa 2- tipqaytishnuqtaside - di "); print ("
agarbunuqtadano ' tuvchiregulyaryoynarmalvaurunmadanikkitomondabo '
lsao ' z - o ' zigaqaytishmaxsusnuqtade - di "); print ("
butasdiqlarnigrafikkaqarabaniqlashmumkin ") fi ; fi ;
"agar bu nuqtadan o'tuvchi regulyar yoy narmaldan bir tomonda urunmadan ikki tomond \
a bo'lsa bu nuqta 1- tip qaytish nuqtasi de-di"
"agar bu nuqtadan o'tuvchi regulyar yoy narmal va urunmada bir tomonda bo'lsa 2-tip qa \
ytish nuqtasi de-di"
" agar bu nuqtadan o'tuvchi regulyar yoy narmal va urunmadan ikki tomonda bo'lsa o'z- \
o'ziga qaytish maxsus nuqta de-di"
" bu tasdiqlarni grafikka qarab aniqlash mumkin"](https://docx.uz/documents/397aa3a1-b3cd-475b-9acb-3f5e3bbf44ff/page_25.png?v=1)
![> f:=proc(x)if x>0 then ((x^5+5*x^4)/4)^(1/2) else
-((x^5+5*x^4)/4)^(1/2) fi end; g:=proc(x)if x>0 then -
((x^5+5*x^4)/4)^(1/2) else ((x^5+5*x^4)/4)^(1/2) fi end;
> plot([f,g],-1..1,color=red,thickness=3);
f xproc ( ) :=
0 x ^( )/1 4 ^x 5 /5 4 ^x 4 ( )/1 2if then
^( )/1 4 ^x 5 /5 4 ^x 4 ( )/1 2else
end if
end proc
g xproc ( ) :=
0 x ^( )/1 4 ^x 5 /5 4 ^x 4 ( )/1 2if then
^( )/1 4 ^x 5 /5 4 ^x 4 ( )/1 2else
end if
end proc
> restart;"6- MISOL";"6- MISOL"
> "CHIZIQNING MAXSUS NUQTALARI(oshkormas kurnishda berilgan
bo'lsa)";
"CHIZIQNING MAXSUS NUQTALARI(oshkormas kurnishda berilgan bo'lsa)"
> F:=(x^2+y^2)*(y-a)^2-l^2*y^2;
:= F ( ) x2 y2 ( ) y a 2 l2y2
> f[x]:=diff(F,x);
:= fx 2x( ) y a 2
> f[y]:=diff(F,y);
:= fy 2y( ) y a 2 2( ) x2 y2 ( ) y a 2l2y](https://docx.uz/documents/397aa3a1-b3cd-475b-9acb-3f5e3bbf44ff/page_26.png?v=1)
![> d:=diff(f[x],x)*diff(f[y],y)-diff(f[x],y); := d 2( ) y a 2( ) 2( ) y a 2 8y( ) y a 2x2 2y2 2l2 4x( ) y a
> solve({f[x],f[y]},{x,y});"yechimlarni tekshish uchun pastki
tengliklarni tugirlang";
, { } , x 0 y 0 { } , x 0 y ( ) RootOf 2_Z 2 3_Z a a2 l2
"yechimlarni tekshish uchun pastki tengliklarni tugirlang"
> for i from 1 to 2 do s:=[solve({f[x],f[y]},
{x,y})];x[i]:=subs(s[i],x);y[i]:=subs(s[i],y);
print([x[i],y[i]]) od;
:= s [ ] , { } , x 0 y 0 { } , x 0 y ( ) RootOf 2_Z 2 3_Z a a2 l2
:= x1 0
:= y1 0
[ ] ,0 0
:= s [ ] , { } , x 0 y 0 { } , x 0 y ( ) RootOf 2_Z 2 3_Z a a2 l2
:= x2 0
:= y2 ( ) RootOf 2_Z 2 3_Z a a2 l2
[ ] ,0 ( ) RootOf 2_Z 2 3_Z a a2 l2
> for i from 1 to 2 do if 4*y[i]^2-x[i]^5-5*x[i]^4=0 then
print([x[i],y[i]]," maxsus nuqta");delta:=-12*x[i]-4 else
print([x[i],y[i]],"maxsus nuqta emas") fi od;
, [ ] ,0 0 " maxsus nuqta"
, [ ] ,0 ( ) RootOf 2_Z 2 3_Z a a2 l2 "maxsus nuqta emas"
> l:=0;a:=1;for i from 1 to 2 do if 4*y[i]^2-x[i]^5-5*x[i]^4=0
then x:=x[i];y:=y[i] fi od; delta:=d;
:= l 0
:= a 1
:= 4
> ifdelta <0 thenprint (" Tugunmaxsusnuqta ") elseifdelta >0
thenprint (" Yakkalanganmaxsusnuqta ")
elseprint (" agarbunuqtadano ' tuvchiregulyaryoynarmaldanbirtomondau
runmadanikkitomondabo ' lsabunuqta 1- tipqaytishnuqtaside -
di "); print (" agarbunuqtadano ' tuvchiregulyaryoynarmalvaurunmadabir
tomondabo ' lsa 2- tipqaytishnuqtaside - di "); print ("
agarbunuqtadano ' tuvchiregulyaryoynarmalvaurunmadanikkitomondabo '
lsao ' z - o ' zigaqaytishmaxsusnuqtade - di "); print ("
butasdiqlarnigrafikkaqarabaniqlashmumkin ") fi ; fi ;
"Yakkalangan maxsus nuqta"
>](https://docx.uz/documents/397aa3a1-b3cd-475b-9acb-3f5e3bbf44ff/page_27.png?v=1)
Egri chiziqlarning maxsus nuqtalarini kompyuter algebrasi tizimi yordamida tekshirish