SOʻZBOSHI
Komil inson gʻoyasi azal-azaldan xalqimizning ezgu orzusi, uning
ma’naviyatining uzviy bir qismi boʻlib kelgan. U sharq falsafasidan oziqlanib,
yanada kengroq ma’no-mazmun kasb etib kelmoqda.
Erkin fuqarolik jamiyatini barkamol, ezgu gʻoyalari, hayotiy e’tiqodi
mustahkam boʻlgan insonlargina bunyod eta oladi. Shuning uchun yangilanayotgan
jamiyatimizda
sogʻlom
avlodni
tarbiyalash,
erkin
fuqaro
ma’naviyatini
shakllantirish, ma’naviy-ma’rifiy ishlarni yuksak darajaga koʻtarish orqali barkamol
insonlarni voyaga yetkazishga muhim e’tibor berilmoqda. Mamlakatimizda
“Kadrlar tayyorlash Milliy dasturi”asosida ta’lim-tarbiya tizimining tubdan isloh
etilayotgani ham ana shu ulugʻvor maqsadni amalga oshirish yoʻlidagi muhim
qadamlardir.
Hozirgi davr yoshlari ruhiyatida chuqur va mustahkam bilimlarni
shakllantirish, milliy istiqlol gʻoyalariga sadoqatni, ona-Vatanga mehr-muhabbatni,
bu yoʻldagi fidoyilikni tarbiyalashni davom ettirish oliy ta’limning asosiy
vazifalaridandir.
“Ta’lim toʻgʻrisida”gi Qonun va “Kadrlar tayyorlash Milliy dasturi”
vazifalarini amalga oshirishda va yuqori malakali mutaxassislar tayyorlashda aniq
fanlarga ehtiyoj kuchayib bormoqda, chunki asosiy muhandislik maxsus fanlari ana
shu fanlar asosida qurilgan boʻladi.
Oliy matematika fanidan har tomonlama chuqur bilim olish uchun faqat asosiy
nazariy mavzularni oʻzlashtirish kifoya qilmasdan, maxsus tanlangan misol va
masalalarni yetarlicha mustaqil yechish qobiliyatiga ham ega boʻlish zarur. Shu
sababli talabalarga auditoriyadan tashqari mustaqil bajarish uchun topshiriqlar
to‘plami va ularni bajarish bo‘yicha metodik yordam beradigan o‘quv
qo‘llanmalarni yaratish katta ahamiyatga ega.
3
Maskur oʻquv qoʻllanma oliy malakali ta’lim boʻyicha yangi davlat ta’lim
standartlarining irrigatsiya, qishloq xoʻjaligi va texnik yoʻnalishlar
uchun
matematik ta’limga qoʻyilgan talablarga mos keladi.
Oʻquv qoʻllanmadagi misol va masalalar chiziqli algebra, analitik geometriya,
matematik analiz va differensial tenglamalar boʻyicha oliy qishloq xoʻjalik
muassasalaridagi matematika (I, II, III ) kursining toʻliq hajmiga mos tanlangan.
Toʻplamga qishloq xoʻjalik oliy ta’lim
muassasalarida oʻqitalidigan oliy
matematika kursining barcha boʻlimlarini qamrab olgan va boblar boʻyicha
sistematik joylashtirilgan 1000 ga yaqin masalalar kiritilgan. Topshiriqlarning
turliligi va etarlicha koʻp soni talabalarga matematika fani boʻyicha bilimlarini
mustahkamlashga va oʻqituvchilarga fanning barcha boʻlimlari boʻyicha amaliy
mashgʻulotlar, nazorat ishlari va mustaqil ish boʻyicha topshiriqlar berish uchun
qulaydir. Masalalar toʻplami qishloq xoʻjaligi va texnik oliy ta’lim muassasalarining
talabalar uchun va fanni mustaqil oʻrganuvchilarga moʻljallangan. Ilovada asosiy
formulallar keltirilgan. Oʻquv qoʻllanma oxirida har bir topshiriqga mos
masalalarning namunaviy yechimlari va oʻz bilimini tekshirish uchun nazorat
savollari keltirilgan.
Toʻplam muallifning Toshkent irrigatsiya va meioratsiya institutida koʻp
yillik dars berish jarayonlarida toʻplangan ma’lumotlari va orttirilgan tajribasi
asosida yozildi.
Oʻquv qoʻllanma
irrigatsiya, qishloq xoʻjaligi va texnik yoʻnalishlar
talabalari hamda «Маtематiка I, II, III» fani oʻqituvchilari, shuningdek oʻzining
matematik bilimini oshirish uchun mustaqil oʻrganuvchilar uchun foydali boʻladi,
deb hisoblayman.
4
1.CHIZIQLI ALGEBRA
1-topshiriq. Determinantni hisoblang.
1.1.
1.3.
1.5.
1.7.
1.9.
1.11.
2
1
2
0
3
4
1
2
2
1
0
1
5
1
2
3
2
4
1
0
4
1
1
3
0
6
4
2
1
3
2
2
1
4
0
1
2
2
3
1
2
1
1
3
4
3
3
5
3
2
4
1
2
0
2
4
1
0
2
1
2
3
1
2
2
1
3
0
1
1
5
1
2
4
2
1
2
3
1
1
2
0
1
1
2
3
3
6
2
5
1
2
2
3
1
0
6
4
2
3
1
0
2
3
5
1
2
3
2
0
2
7
2
1
2
3
4
1
1
1
1
0
4
2
3
2
3
4
0
2
3
0
2
1
0
5
1
3
3
1
4
3
4
3
2
1
4
1
1
5
2
1
4
3
0
2
2
3
0
4
1
2
3
4
1
2
5
0
1
1
4
1
1
2
3
1
2
3
4
1
2
4
1
1
1
1
4
1
2
1
2
2
1.2.
1.4.
1.6.
1.8.
1.10.
1.12.
5
1.13.
1.15.
1.17.
1.19.
1.21.
1.23.
1.25.
5
0
4
2
1
1
2
1
4
1
2
0
1
1
1
1
1.14.
6
2
10
4
5
7
4
1
2
4
2
6
3
0
5
4
1.16.
6
0
1
1
2
2
0
1
1
1
3
3
4
1
1
2
2
0
1
3
6
3
9
0
0
2
1
3
4
2
0
6
0
2
1
7
4
8
2
3
10
1
5
4
2
2
0
3
3
2
1
1
1
1
2
1
3
4
4
0
8
3
2
1
1
2
3
4
4
5
1
5
2
1
4
3
3
2
8
2
3
4
1
2
5
3
1
3
4
3
2
1
2
4
6
8
1
1
0
3
3
5
1
2
3
2
1
1
0
1
1
2
1
2
1
3
3
1
3
0
4
0
1
2
1
2
1
2
3
2
0
2
3
2
0
5
1
1
2
3
4
3
5
0
4
5
1
0
1
0
2
3
1
2
3
3
0
1
3
4
3
1
2
0
4
1
2
0
5
0
6
1
1
2
1
1
2
2
1
3
3
1
2
1
1
3
2
1
5
0
4
2
1.18.
1.20.
1.22.
1.24.
1.26.
6
1.27.
1.29.
5
3
7
1
1
2
3
4
3
2
0
2
2
0
1
1
2
1
4
6
3
3
1
0
3
2
3
4
4
2
1
2
1
2
0
4
1
8
2
3
2
3
1
1
3
2
0
4
3
1
2
4
5
3
7
1
2
0
1
3
3
2
0
2
1.28.
1.30.
2-topshiriq. A va B matritsalar berilgan. Quyidagilar topilsin: 1) 2 A  3B ;
2) A  B ; 3) B  A ; 4) A1 ;
4
6 

3 ,
1
1
1
7
3 

6  ,
4
2
2
2
1

4 ,
5
3
1
4 

6 ,
 3

2.1. A   2

 3
5
 2

2.2. A   8

 3
 1

2.3. A   1

 3
 4

2.4. A   2

 1
2

2.5. A   3

4
 1

2.6. A   2

 4
5) A  A1 .
4
2

3



1 

6,

4
1
4
3
5
3 
1
5
2 

4 .
2
1
 7

B 5

 1
5
3
1

1  .
2
3
 0

B 2

 1
1
5
1

0 .
1
2
 1

B 2

 1
1
1

3 .
2
1 
 1

B 2

 4
0
5
4 

3  .
3
2
 2

B 3

 1
2 5
3
1
5 

0 .
 2

B   3

 4
1 

1 ,

1
7
8
3






2

2.7. A   1

0
1

2 ,
6
3

1
1
 3

2.8. A   3

 4
1
7

5
 2

2.9. A   4

 2
1
4 

3 ,
0

1 ,
5
9

1 
7
 1

2.10. A   4

 0
7
 6

2.11. A   1

 10
9
9
3
1
1
 5

2.12. A   7

 4
8
 1

2.13. A   2

 3
0
 3

2.14. A   4

 2
4
1

2.15. A   3

2
0
1
3
7
3

3 ,

2
4

1 ,

7
4 

5  ,
2

2 ,

1
5
3
3

0 3
1
1
3
 1

B 1

 3
0
 0

B 5

 7
8
2
5
1

B  3

0

3 
8
2

5 .
0
2
0
6
4 

4 .
4
1


9
2

2 .
5
2
1
4
1

3 .
5
2


5
2
1
 3

B   3

 1
0
 1

B 0

 2
 3

B   3

 5

7,

8
2

5 .
0
6

B  1

4
 1

B 1

 2

0
0

1 ,
 4

B   4

 3
5

1 .

3 
1
1

7 .
3
2
7
2
1 

6 .
1
1
5
4

1 .
0
6



4 
 3

2.16. A   1

 5
4
2
0
 3

2.17. A   1

 1

1
2
1
5
1 

1  ,
3
2
7
2

3 ,
3
1
4

4  ,

2
2
 5

2.21. A   1

 8
1
3
4
2 

1  ,

1 
2

4 ,
4
2
5
2

3 ,
1
2
3
3
2

1  ,
1
3
4
 0

B 2

 3


9
4
0

1 .
1
2
1
1
2

1 .
7
1


2 5

2 1.

0 2
4
3 

1 .
1
5 
5
3

B  0

1
3
3

B  7

1

6
1

2 .
9
2
5
2
5

1 .
6
0


7
5 

1 .
2
2
 1

B 1

 4
4
3
4

2 .
1
2
 3

B 3

 5
2
1
1 

2 .
3
0
 5

B 3

 1

5
0
2
 0

B 2

 2

3
2
 2

B 5

 1
3

B 3

1

8
 2

2.20. A   3

 1
 2

2.24. A   1

 4
2

2 ,
0
 3

2.19. A   1

 4
 3

2.23. A   1

 0

1 
1
 8

2.18. A   5

 10
5

2.22. A   1

3
3 

3 ,
4



 3

2.25. A   4

 2
1
 1

2.26. A   3

 4
2
3

4
3
5
1 

3 ,

0
1
 6

2.28. A   9

 0
1
 2

2.29. A   2

 1
1
11 

5 ,
2

7
3
1 

1 ,
1
1
2
3

0
6
1
1
1

3 .
3


1 
 4

B 3

 0
7
2
6 

1  .
1
2
 3

B 0

 1
0
2
1

7 .
3
2
6
4
0

6  .
2
3
0
1
5

2 
3
7
 2

B 4

 4

1
3
0

1 .
7
2
 3

B 2

 1

1
0
7
 1

B 2

 1
5

6 ,
0
5
 2

B 5

 1

7 
2
 8

2.27. A   1

 1
6

2.30. A   3

2
0

2 ,




3-topshiriq. Berilgan algebraik tenglamalar sistemasini 1) Kramer formulasi,
2) Gauss мetodi, 3) teskari matritsa yordamida yeching.
 x  2 y  z  5,

3.1.  x  2 y  2 z  2,
3x  y  4 z  2.

2 x  2 y  3z  4,

3.2.  x  2 y  z  5,
3x  z  1.

 x  3 y  z  2,

3.3. 2 x  2 y  z  1,
2 x  3 y  3z  4.

2 x  y  z  4,

3.4.  x  3 y  z  7,
3x  y  4 z  12.

10
3x  4 y  5,

3.5.  x  y  z  1,
 x  3 y  z  3.

 x  3 y  3z  11,

3.6.  x  2 y  3z  1,
3x  3 y  z  1.

3x  2 z  11,

3.7. 2 x  2 y  3z  3,
 x  y  4 z  1.

2 x  y  3z  3,

3.8. 4 x  2 y  5 z  5,
3x  4 y  7 z  2.

 x  3 y  2 z  4,

3.9. 2 x  6 y  z  2,
4 x  8 y  z  2.

2 x  4 y  z  4,

3.10. 3x  6 y  2 z  4,
4 x  y  3z  1.

 x  2 y  3z  6,

3.11. 4 x  y  4 z  9,
3x  5 y  2 z  10.

3x  3 y  2 z  2,

3.12. 4 x  5 y  2 z  1,
5 x  6 y  4 z  3.

3x  2 y  4 z  8,

3.13. 2 x  4 y  5 z  11,
4 x  3 y  2 z  1.

2 x  y  z  2,

3.14. 3x  2 y  2 z  2,
 x  2 y  z  1.

 x  2 y  3z  5,

3.15. 2 x  y  z  1,
 x  3 y  4 z  6.

2 x  3 y  z  2,

3.16.  x  5 y  4 z  5,
4 x  y  3z  4.

2 x  4 y  3z  1,

3.17.  x  2 y  4 z  3,
3x  y  5 z  2.

 x  2 y  3z  7,

3.18.  x  3 y  2 z  5,
 x  y  z  3.

 x  y  2 z  3,

3.19. 5 x  2 y  7 z  22,
2 x  5 y  4 z  4.

 x  2 y  3z  0,

3.20. 2 x  y  4 z  5,
3x  y  z  2.

3x  2 y  z  5

3.21. 2 x  3 y  z  1
2 x  y  3z  11

 x  2 y  3z  6

3.22. 2 x  3 y  4 z  20
3x  2 y  5 z  6

11
2 x  y  3z  7

3.23.  x  2 y  z  4
3x  3 y  2 z  1

x  2 y  z  1

3.24. 2 x  3 y  z  4
x  y  2z  1

2 x  2 y  3 z  0

3.25.  x  2 y  z  6
2 x  y  2 z  2

3x  2 y  2 z  1

3.26. 2 x  3 y  z  3
 x  y  3z  2

4 x  3 y  2 z  9

3.27. 2 x  5 y  3z  4
5 x  6 y  2 z  18

 x  4 y  2 z  3

3.28. 3x  y  z  5
3x  5 y  6 z  9

7 x  5 y  31

3.29. 4 x  11z  43
2 x  3 y  4 z  20

2 x  y  2 z  3

3.30.  x  2 y  4
2 y  z  2

4-topshiriq. Bir jinsli chiziqli algebraik tenglamalar sistemasi yechilsin.
 x  2 y  4z  0
4.1. а)  2 x  y  3z  0
x  3y  z  0

5 x  3 y  2 z  0
б)  2 x  4 y  3z  0
3x  7 y  5 z  0

x  2y  z  0
4.2. а)  2 x  3 y  2 z  0
3x  2 y  5 z  0

5 x  y  2 z  0
б) 3x  2 y  3z  0
2 x  y  z  0

 x  3 y  2z  0
4.3. а)  2 x  y  3z  0
3x  5 y  4 z  0

 x  2 y  5z  0
б)  2 x  4 y  z  0
3x  2 y  4 z  0

8 x  y  3z  0
4.4. а)  x  5 y  z  0
4 x  7 y  2 z  0

2 x  y  4 z  0
б) 7 x  5 y  3z  0
5 x  4 y  z  0

 2 x  y  3z  0
4.5. а) 3x  y  2 z  0
 x  3 y  4z  0

3x  2 y  z  0
б)  2 x  3 y  5z  0
5 x  y  4 z  0

12
6 x  5 y  4 z  0
4.6. а)  x  y  z  0
3x  4 y  3z  0

5 x  y  6 z  0
б)  4 x  3 y  7 z  0
x  2y  z  0

 x  y  2z  0
4.7. а)  2 x  y  3z  0
3 x  2 z  0

x  2y  z  0
б) 3x  3 y  5 z  0
4 x  y  6 z  0

 x  4 y  3z  0
4.8. а)  2 x  5 y  z  0
 x  7 y  2z  0

2 x  y  2 z  0
б) 3x  2 y  3z  0
5 x  y  z  0

 7 x  y  3z  0
4.9. а) 3x  2 y  3z  0
 x  y  2z  0

7 x  6 y  z  0
б) 3x  3 y  4 z  0
4 x  3 y  5z  0

x  2y  z  0
4.10. а) 3x  y  2 z  0
2 x  3 y  5z  0

 2 x  y  3z  0
б)  x  3 y  2 z  0
x  2y  z  0

3 x  y  3 z  0
4.11. а)  2 x  3 y  z  0
 x  y  3z  0

3x  y  z  0
б)  2 x  3 y  4 z  0
5 x  2 y  3z  0

2 x  y  5z  0
4.12. а)  x  2 y  3z  0
5 x  y  4 z  0

 2 x  y  3z  0
б)  x  2 y  4 z  0
x  y  z  0

 4 x  y  10 z  0
4.13. а)  x  2 y  z  0
2 x  3 y  4 z  0

x  y  z  0
б)  2 x  3 y  4 z  0
3x  2 y  5 z  0

 x  3 y  4z  0
4.14. а) 5 x  8 y  2 z  0
2 x  y  z  0

x  5y  z  0
б)  2 x  3 y  7 z  0
3x  2 y  6 z  0

7 x  6 y  z  0
4.15. а)  4 x  5 y  0
 x  2 y  3z  0

x  8y  7z  0
б) 3x  5 y  4 z  0
 4 x  3 y  3z  0

13
3x  2 y  z  0
4.16. а)  2 x  3 y  2 z  0
4 x  y  4 z  0

 2 x  4 y  3z  0
б)  x  3 y  2 z  0
3x  y  z  0

3x  2 y  0
4.17. а)  x  y  2 z  0
4 x  2 y  5z  0

5 x  y  2 z  0
б) 3x  y  z  0
 2 x  2 y  3z  0

x  3y  z  0
4.18. а)  2 x  5 y  2 z  0
 x  y  5z  0

4 x  y  4 z  0
б) 3x  2 y  z  0
7 x  y  3z  0

5 x  5 y  4 z  0
4.19. а) 3x  y  3z  0
x  7 y  z  0

2 x  y  2 z  0
б)  4 x  y  5z  0
 2 x  2 y  3z  0

2 x  y  z  0
4.20. а) 3x  2 y  4 z  0
 x  5 y  3z  0

 x  2 y  5z  0
б)  x  2 y  4 z  0
2 x  9 z  0

3x  2 y  z  0
4.21. а)  2 x  y  3z  0
4 x  3 y  4 z  0

4 x  y  5z  0
б)  2 x  3 y  2 z  0
 2 x  2 y  3z  0

5 x  4 y  2 z  0
4.22. а) 3 y  2 z  0
 4 x  y  3z  0

5 x  8 y  5 z  0
б) 7 x  5 y  z  0
2 x  3 y  4 z  0

3 x  y  2 z  0
4.23. а)  x  y  z  0
 x  3 y  3z  0

5 x  3 y  4 z  0
б) 3x  2 y  z  0
8 x  y  3z  0

 2 x  y  3z  0
4.24. а)  x  2 y  5z  0
3 x  y  z  0

3x  2 y  3z  0
б)  2 x  3 y  z  0
5 x  y  2 z  0

 x  2 y  3z  0
4.25. а)  2 x  y  z  0
3x  3 y  2 z  0

 x  3 y  2z  0
б) 3x  y  4 z  0
2 x  2 y  z  0

14
 4 x  y  3z  0
4.26. а) 8 x  y  7 z  0
2 x  4 y  5z  0

 x  3 y  5z  0
б)  x  2 y  3z  0
2 x  y  2 z  0

2 x  5 y  z  0
4.27. а)  4 x  6 y  3z  0
 x  y  2z  0

 x  2 y  4z  0
б) 5 x  y  2 z  0
4 x  y  2 z  0

x  y  z  0
4.28. а)  2 x  3 y  4 z  0
 4 x 11y  10 z  0

5 x  6 y  4 z  0
б) 3x  3 y  z  0
 2 x  3 y  3z  0

3x  2 y  z  0
4.29. а)  2 x  3 y  2 z  0
4 x  y  4 z  0

 2 x  4 y  3z  0
б)  x  3 y  2 z  0
3x  y  z  0

x  2y  z  0
4.30. а) 3x  y  2 z  0
2 x  3 y  5z  0

 2 x  y  3z  0
б)  x  3 y  2 z  0
x  2y  z  0

2. TEKISLIKDAGI ANALITIK GEOMETRIYA
5-topshiriq. ABC uchburchaklarning koordinatalari berilgan . Quyidagilarni
topish kerak : 1) AB tomon uzunligini; 2) AB va BC tomon tenglamalarini va
ularning burchak koeffitsiyentlarini; 3) A burchakni; 4) CD balandlik tenglamasi
va uning uzunligini 5) AE mediana tenglamasi va bu mediananing CD balandlik
bilan kesishish nuqtasi K ning koordinatalarini; 6) K nuqtadan o‘tib, AB tomonga
parallel bo‘lgan to‘g‘ri chiziq tenglamasini; 7) CD to‘g‘ri chiziqqa nisbatan A
nuqtaga simmetrik joylachgan N nuqta koordinatalarini;
8) ABC uchburchak
yuzasini.
5.1. А(1; -1), B (4; 3), C(5; 1).
5.2. А (0; -1), B (3; 3), C(4; 1).
5.3. А (1; -2), B(4; 2), C(5; 0).
5.4. А (2; -2), B(5; 2), C(6; 0).
5.5. А (0; 0), B(3; 4), C(4; 2).
5.6. А (0; 1), B(3; 5), C(4; 3).
15
5.7. А (3; -2), B(6; 2), C(7; 0).
5.8. А (3; -3), B(6; 1), C(7; -1).
5.9. А (-1; 1), B(2; 5), C(3; 3).
5.10. А (4; 0),
5.11. А(2; 2 ), B (5; 6), C(6; 4).
5.12. А (4; -2), B (7; 2), C(8; 0).
5.13. А (0; 2),
B(3; 6), C(4; 4).
5.14. А (4; 1),
5.15. А (3; 2),
B(6; 6), C(7; 4).
5.16. А (-2; 1), B(1; 5), C(2; 3).
B(7; 4), C(8; 2).
B(7; 5), C(8; 3).
5.17. А (4; -3), B(7; 1), C(8; -1).
5.18. А (-2; 2), B(1; 6), C(2; 4).
5.19. А (5; 0), B(8; 4), C(9; 2).
5.20. А (2; 3), B(5; 7), C(6; 5).
5.21. A  4, 2  , B  6, 4  , C  4,10  .
5.22. A  4,1 , B  3, 1 , C  7, 3 .
5.23. A  5, 2  , B  0, 4  , C  5, 7  .
5.24. A 1, 7  , B  3, 1 , C 11, 3 .
5.25. A  2, 6  , B  3,5 , C  4, 0  .
5.26. A  1, 4 , B  9,6  , C  5, 4  .
5.27. A  4, 4  , B 8, 2  , C  3,8 .
5.28. A  3, 3 , B  5, 7  , C  7, 7  .
5.29. A  5,1 , B 8, 2  , C 1, 4  .
5.30. A  6, 9  , B 10, 1 , C  4,1 .
3. VEKTORLAR ALGEBRASI
6-topshiriq.
АВСD piramida uchlarining koordinatalari berilgan boʻlsa,
quyidagilar talab qilinadi: 1) AB ,
AС ,
AD
vektorlarni tuzish va ularning
uzunliklarini topish; 2) AB va AC vektorlar orasidagi burchaklarni topish; 3)
AD vektorning AB vektordagi proeksiyasini topish; 4) ABC yoqning yuzasini
topish; 5) ABCD piramidaning hajmini topish.
6.1. А(1; 2; 1), B (-1; 5; 1), C(-1; 2; 7), D(1; 5; 9).
6.2. А (2; 3; 2), B (0; 6; 2),
C(0; 3; 8), D(2; 6; 10).
6.3. А (0; 3; 2), B(-2; 6; 2), C(-2; 3; 8), D(0; 6; 10).
6.4. А (2; 1; 2), B(0; 4; 2),
C(0; 1; 8), D(2; 4; 10).
16
6.5. А (2; 3; 0), B(0; 6; 0),
C(0; 3; 6), D(2; 6; 8).
6.6. А (2; 2; 1), B(0; 5; 1),
C(0; 2; 7), D(2; 5; 9).
6.7. А (1; 3; 1), B(-1; 6; 1), C(-1; 3; 7), D(1; 6; 9).
6.8. А (1; 2; 2), B(-1; 5; 2), C(-1; 2; 8), D(1; 5; 10).
6.9. А (2; 3; 1), B(0; 6; 1),
6.10. А (2; 2; 2), B(0; 5; 2),
6.11. А(1; 3; 2),
C(0; 3; 7), D(2; 6; 9).
C(0; 2; 8), D(2; 5; 10).
B (-1; 6; 2), C(-1; 3; 8), D(1; 6; 10).
6.12. А (0; 1; 2), B (-2; 4; 2), C(-2; 1; 8), D(0; 4; 10).
6.13. А (0; 3; 0), B(-2; 6; 0),
C(-2; 3; 6), D(0; 6; 8).
6.14. А (2; 1; 0), B(0; 4; 0),
C(0; 1; 6), D(2; 4; 8).
6.15. А (0; 2; 1), B(-2; 5; 1),
C(-2; 2; 7), D( 0; 5; 9).
6.16. А (1; 1; 1), B(-1; 4; 1),
C(-1; 1; 7), D(1; 4; 9).
6.17. А (1; 2; 0), B(-1; 5; 0),
C(-1; 2; 6), D(1; 5; 8).
6.18. А (0; 1; 0), B(-2; 4; 0), C(-2; 1; 6), D(0; 4; 8).
6.19. А (0; 1; 1), B(-2; 4; 1), C(-2; 1; 7), D(0; 4; 9).
6.20. А (0; 2; 0),
B(-2; 5; 0), C(-2; 2; 6), D(0; 5; 8).
6.21. А (0; 5; 6), B(1; 12; -4), C(3; 8; -7), D(3; 0; -4).
6.22. А(4; -1; 6),
B (-2; 5; 2), C(0; 3; 0), D(2; 3; 8).
6.23. А (0; -3; 0), B (2; 4; -2), C(-3; 0; 2), D(0; -6; 7).
6.24. А (9; 0; 8), B(2; -6; 2),
C(1; -4; 0), D(12; -2; 10).
6.25. А (2; 5; -1), B(0; 2; 4),
C(3; -2; -3), D(3; -2; 7).
6.26. А (5; 0; -2), B(3; 7; -10), C(1; -2; 1), D( 1; -4; 0).
6.27. А (0; 0; 0), B(3; 4; 0),
C(0; -6; 1), D(0; 4; 5).
6.28. А (2; -1; 4), B(0; 0; 8), C(-6; 0; 0), D(0; 0; 0).
17
6.29. А (7; 10; -4), B(5; -1; -4), C(5; 4; -2), D(9; 3; -6).
6.30. А (0; 0; 4), B(0; 3; 1), C(2; 10; 5), D(-3; 3; 8).
7-topshiriq.
a, b va c vektorlar berilgan. Quyidagilar boʻyicha: а) uch
vektorning aralash koʻpaytmasini hisoblang; b) vektor koʻpaytmaning modulini
toping; c) ikki vektorning skalyar koʻpaytmasini hisoblang; d) ikki vektorni
kollinear yoki optogonal shartlari boʻyicha tekshirib koʻring; e) uch vektorni
komplanarlikka tekshiring.
7.1. a  4i  2 j  3k , b  3 j  5k , c  6i  6 j  4k
а) 5a, b,3c ;
b) 7a, 4c ;
c) 3a,9b ;
d) a, c ; e) 3a, 9b, 4c .
7.2. a  3i  j  5k , b  2i  4 j  8k , c  3i  7 j  k ;
а) 2a, b,3c ;
b) 9a, 4c ; c) 5b, 6c ;
d) b, c ; e) 2a,5b, 6c .
7.3. a  9i  4 j  5k , b  i  2 j  4k , c  5i  10 j  20k ;
а) 2a, 7b,5c ;
b) 6b, 7c ;
c) 7a, 4c ;
d) b, c ;
e) 2a, 7b, 4c .
d) a, c ;
e) 3a, 4b,8c .
d) a, c ;
e) 2a,3b, 4c .
d) a, c ;
e) 7a, b, c .
7.4. a  4i  5 j  4k , b  5i  j , c  2i  4 j  3k ;
а) a, 7b, 2c ;
b) 5a, 4b ; c) 7c, 3a ;
7.5. a  4i  2 j  3k , b  2i  k , c  12i  6 j  9k ;
а) 2a,3b, c ;
b) 4a,3b ;
c) b, 4c ;
7.6. a  2i  4 j  2k , b  9i  2k , c  3i  5 j  7k ;
а) 7a,5b, c ;
b) 5a, 4b ; c) 3b, 8c ;
7.7. a  7i  4 j  5k , b  i  11 j  3k , c  5i  5 j  3k ;
а) 3a, 7b, 2c ;
c) 4a, 5c ; d) a, c ;
b) 2b, 6c ;
7.8. a  3i  4 j  k , b  i  2 j  7k , c  3i  6 j  21k ;
18
e) 4a, 2b,6c .
а) 5a, 2b, c ;
b) 4b, 2c ;
c) a, c ;
d) b, c ;
e) 2a, 3b, c .
d) a, b ;
e) a, 6b,3c .
d) b, c ;
e) 2a, 7b, 4c .
7.9. a  4i  2 j  k , b  3i  5 j  2k , c  j  5k ;
а) a, 6b,3c ;
c) a, 4c ;
b) 2b, a ;
7.10. a  9i  3 j  k , b  3i  15 j  21k , c  i  5 j  7k ;
а) 2a, 7b,3c ;
b) 6a, 4c ; c) 5b, 7a ;
7.11. a  3i  2 j  k , b  2 j  3k , c  3i  2 j  k ;
а) a, 3b, 2c ;
c) 2a, 4b ; d) a, c ;
b) 5a,3c ;
e) 5a, 4b,3c .
7.12. a  2i  3 j  k , b  j  4k , c  5i  2 j  3k ;
а) a,3b, c ;
b) 3a,2c ;
c) b, 4c ;
d) a, c ;
e) a,2b,3c .
c) 9a, 7c ; d) a, b ;
e) a, 6b,5c .
7.13. a  2i  4 j  3k , b  5i  j  2k , c  7i  4 j  k ;
а) a, 6b, 2c ;
b) 8b,5c ;
7.14. a  2i  7 j  5k , b  i  2 j  6k , c  3i  2 j  4k ;
а) a, 6b, c ;
c) 7a, 4b ; d) b, c ;
b) 5b,3c ;
e) 7a, 4b,3c .
7.15. a  4i  j  3k , b  2i  3 j  5k , c  7i  2 j  4k ;
а) 7a, 4b, 2c ;
b) 3a,5c ;
c) 2b, 4c ;
d) b, c ;
e) 7a, 2b,5c .
7.16. a  5i  6 j  4k , b  4i  8 j  7k , c  3 j  4k ;
а) 5a,3b, 4c ;
c) 7a, 2c ; d) a, b ;
b) 4b, a ;
e) 5a, 4b, 2c .
7.17. a  4i  3 j  7k , b  4i  6 j  2k , c  6i  9 j  3k ;
а) 2a, b, 2c ;
c) 5a, 3b ;
b) 4b, c ;
d) b, c ;
e) 2a, 4b,7c .
7.18. a  i  5k , b  3i  2 j  2k , c  2i  4 j  k ;
а) 3a, 4b, 2c ;
b) 7a, 3c ;
c) 2b,3a ;
7.19. a  5i  2 j  2k , b  7i  5k , c  2i  3 j  2k ;
19
d) b, c ;
e) 7a, 2b, 3c .
а) 2a, 4b, 5c ;
b) 3b,11c ; c) 8a, 6c ;
d) a, c ;
e) 8a, 3b,11c .
d) a, c ;
e) 3a, 2b,3c .
7.20. a  2i  4 j  2k , b  7i  3 j , c  3i  5 j  7k ;
а) a, 2b,3c ;
b) 3a, 7b
c) c, 2a ;
7.21. a  4i  6 j  2k , b  2i  3 j  k , c  i  5 j  3k ;
а) 5a, 7b, 2c ;
d) a, b ;
e) 3a, 7b, 2c .
b) 7b, 6a ; c) 5a, 4c ; d) a, b ;
e) 5a,3b, 4c .
b) 4b,11a ; c) 3a, 7c ;
7.22. a  4i  6 j  2k , b  2i  3 j  k , c  3i  5 j  7k ;
а) 6a,3b,8c ;
7.23. a  6i  4 j  6k , b  9i  6 j  9k , c  i  8k ;
а) 2a, 4b,3c ;
b) 3b, 9c ;
c) 3a, 5c ;
d) a, b ;
e) 3a, 4b, 9c .
b) 6a, 4c ; c) 2a,5b ; d) a, c ;
e) 6a, 7b, 2c .
7.24. a  3i  j  2k , b  i  5 j  4k , c  6i  2 j  4k ;
а) 4a, 7b, 2c ;
7.25. a  9i  4k , b  2i  4 j  6k , c  3i  6 j  9k ;
а) 3a, 5b, 4c ;
c) 2a,8c ;
b) 6b, 2c ;
d) b, c ;
e) 3a, 6b, 4c .
d) b, c ;
e) 4a, 6b,9c .
7.26. a  3i  8 j , b  2i  3 j  2k , c  8i  18 j  8k ;
а) 4a, 6b,5c ;
b) 7a,9c ; c) 3b, 8c ;
7.27. a  3i  2 j  7k , b  i  5k , c  6i  4 j  k ;
а) 2a, b, 7c ;
b) 5a, 2c ;
d) a, c ;
e) 2a,3b, 7c .
d) b, c ;
e) a, 2b, 6c .
c) 2a, 7c ; d) b, c ;
e) 2a, 4b,3c .
c) 3b, c ;
7.28. a  5i  3 j  4k , b  2i  4 j  2k , c  3i  5 j  7k ;
а) a, 4b, 2c ;
b) 2b, 4c ;
c) 3a, 6c ;
7.29. a  7i  2k , b  2i  6 j  4k , c  i  3 j  2k ;
а) a, 2b, 2c ;
b) 4a,3c ;
7.30. a  3i  j  5k , b  2i  4 j  6k , c  i  2 j  3k ;
20
а) 3a, 4b, 5c ;
b) 6b,3c ;
c) a, 4c ;
d) b, c ;
e) 3a, 4b, 5c .
4. IKKINCHI TARTIBLI EGRI CHIZIQLAR
8-topshiriq. Toʻgʻri chiziq va ikkinchi tartibl egri chiziqlar berilgan .
Quyidagilar topilsin: 1) ikkinchi tartibli egri chiziqning barcha elementlari;
2) ikkinchi tartibli egri chiziq bilan toʻgʻri chiziqning kesishgan nuqtalari;
3) egri va toʻgʻri chiziqlarni grafikda tasvirlang.
8.1. а) x 2  3 y 2  36,
y  2x  9  0 ;
b) y  x 2  1,
y  x  1.
8.2. а) y 2  x 2  12 , y  2 x ;
b) x 2  y 2  4 y  12  0 ,
y  x  2.
8.3. а) x 2  2 y 2  8 , x  2 y  0 ;
b) x 2  2 x  y 2  7  0 ,
y  x 1  0.
8.4. а) 9 x 2  16 y 2  144 , y  2 x  3  0 ;
b) y  x 2  x , y  x  4 .
8.5. а) x 2  2 y 2  28 , y  x  4  0 ;
b) x 2  y 2  4 y  0,
y  x  4  0.
8.6. а) 3x 2  16 y 2  48  0, у  3х  12 ;
b) x 2  y 2  2 у  3  0,
y  x  3.
8.7. а) ( x  2) 2  ( y  3) 2  9 , x  y  5  0 ; b) 7 x 2  9 y 2  63,
y  2 x  6  0.
21
8.8. а)
y 2  12 x,
y  x  1;
3x 2  4 y 2  12,
b)
y  x  2  0.
8.9. а) ( x 5)2  ( y 6)2  16, x  y  1  0 ;
b) 4 x2  9 y2  36,
y  x  3.
8.10. а) 12x2  13 y2  156,
y
1
x  1,
6
b) y  x 2  4 x,
y  x  2.
8.11. а) 3x2  y  63,
b) y  x 2  2,
y  2x,
2
y  5x  2 .
8.12. а) y  x 2 ,
b) 4 x 2  25 y 2  100,
y  3x  2,
3x  10 y  25  0.
8.13. а) x 2  y 2  2  0,
y  2 x  1,
2
b) 16 x  25 y  400,
2
3 y  4 x  16  0.
8.14. а) y  2  x , y  3x  2,
2
b)
x
2
2
 4 y  25,
x  2 y  7  0.
8.15. а) x 
y
2
 4 y,
2y  x  5  0,
b)
x
2
2
 4 y  20,
y  2 x  10.
b) 2 x 2  y 2  4  0,
8.16. а) y  x  2 x, y  3x  8,
2
y  x  2.
8.17. а)
x
2
 4x 
y
2
 12  0, y  x  2,
b)
y
2
 4 x  4,
2
y  5x .
22
8.18. а)
2
x
 2x 
y
2
 4 y  13  0,
y  x  3,
b) 2 x 2  y 2  4,
y  x  4  0.
8.19. а) y  x 2  2, y  5x  2  0,
b) 3x 2  y 2  63,
y  2 x  0.
b) 3 x  2 y  84  0,
8.20. а) x 2  2 y  0, 2 y  2 x  3,
2
2
y  3x.
8.21. а) 2 x 2  y 2  4  0, y  x  2,
b) y  x 2  2 x,
2 y  3x  8.
8.22. а) x 2  4 y 2  20, y  2 x  8,
b) y 2  4 y  x,
2 y  x  5.
8.23. а) x 2  4 y 2  25, 2 y  x  7  0,
b) y  2  x 2 ,
y  3x  2.
8.24. а) 16 x 2  25 y 2  400, 3 y  4 x  16,
b) x 2  y 2  2  0,
y  2 x  1.
8.25. а) 4 x 2  25 y 2  100, 3x  10 y  25  0,
b) y  x 2 ,
y  3x  2.
8.26. а) 3x 2  y 2  63, y  2x,
b) y  x 2  2,
y  5x  2.
1
6
8.27. а) 12 x 2  13 y 2  156, y  x  1,
b) y  x 2  4 x,
y  x  2.
8.28. а) 4 x 2  9 y 2  36, y  x  3,
b) y 2  12 x,
y  x  1.
23
8.29. а) 7 x 2  9 y 2  63, y  2 x  6,
b) x 2  y 2  2 y  3  0,
y  x  3.
8.30. а) 3x 2  4 y 2  12, y  x  2,
b) y 2  12 x  0,
y  x  1  0.
9-topshiriq. Ikkinchi tartibli egri chiziqlarning kanonik tenglamalarini tuzing:
а) ellips; b) giperbola; c) parabola ( A, B - egri chiziqda yotuvchi nuqtalar, F - fokus,
a - katta (haqiqiy) yarimoʻq, b - kichik (mavhum) yarimoʻq,  - ekssentpisitet;
y  kx - giperbola asimptotalari tenglamalari, D - egri chiziq direktrisasi, 2c -
fokuslar orasidagi masofa).
9.1.
7
а) b  2 2 ,   ;
2
, 2a  12 ;
2
b) k 
9
c) OY - simmtriya oʻqi va A  45,15 .
9.2.
 15 
,1 ;
2


а) A  0, 2  , B 
b) k 
2 10
11
, ;
9
9
b) k 
11
, 2c  12 ;
5
c) D : y  5 .
9.3.
10
а) 2a  22 ,   ;
11
c) OX - simmtriya oʻqi va A  7,5 .
9.4.
 20 
, 2  ;
3


b) A  8, 0  , B 
2
3
а)   , A  6,0  ;
c) D : y  1 .
9.5.
а) 2a  22 ,  
57
;
11
2
3
b) k  , 2c  10 13 ;
c) OX - simmtriya oʻqi va A  27,9  .
24
9.6.
b) a  5 ,   7 ;
а) b  4 , F  9, 0  ;
5
c) D : x  6 .
9.7.
а) b  15 ,  
10
;
25
3
b) k  , 2a  16 ;
4
c) OX - simmtriya oʻqi va A  4, 8 .
9.8.

 21 1 
17 
,  ;
 , B 
3 
 2 2
a) A  

1
b) k  ,  
2
5
;
2
c) D : y  1 .
9.9.

40 
;
3 
a) A  3, 0  , B 1,

b) k 
15
2
,
;
3
3
c) D : y  4 .
12
9.10. а) b  5 ,   ;
1
b) k  , 2a  6 ;
13
3
c) OY - simmtriya oʻqi va A  9,6  .
b) b  2 10 , F  11,0  ;
9.11. а) a  4 , F  3, 0  ;
c) D : x  2 .

7
8
9.12. а)   , A 8,0  ;
b) A  3, 

 13 
3
, 6  ;
 , B 
5
5


c) D : y  4 .
9.13. а)  
b) A  80,3 , B  4 6,3 2  ;
21
, A  5,0  ;
5
c) D : y  1 .
9.14. а) 2a  24 ,  
22
;
6
b) k 
c) OX - simmtriya oʻqi va B  7, 7  .
25
2
, 2c  10 ;
3
9.15. а) a  6 , F  4,0  ;
b) b  3 , F  7,0  ;
c) D : x  7 .
7
8
5
b) k  , 2a  12 ;
9.16. a) b  2 15 ,   ;
6
c) OY - simmtriya oʻqi va A  2,3 2  .
9.17. а) 2a  50 ,   3 ;
29
, 2c  30 ;
14
b) k 
5
c) OY - simmtriya oʻqi va A  4,1 .
14
b) a  13 ,   ;
9.18. а) b  15 , F  10,0  ;
13
c) D : x  4 .
 14 
,1 ;
3


9.19. а) A  0, 3  , B 
b) k 
21
11
, ;
10
10
c) D : y  4 .
9.20. а) b  7 , F 13,0  ;
b) b  4 , F  11,0  ;
c) D : x  13 .
9.21. а) b  2 , F  4 2,0  ;
b) a  7 ,  
c) D : x  5 .
12
b) a  11 ,   ;
9.22. а) b  7 , F  5, 0  ;
11
c) D : x  10 .

9.23. а) A  3, 0  , B  2,

5
;
3 
3
4
5
4
b) k  ,   ;
c) D : y  2 .
26
85
;
7
9.24. а) b  2 ,  
5 29
;
29
b) k  12 , 2a  26 ;
13
c) OX - simmtriya oʻqi va A  5,15 .
b) A  6, 0  , B  2 2,1 ;
3
5
9.25. а)   , A  0,8 ;
c) D : y  0 .
9.26.
а) a  9 , F  7,0  ;
b) b  6 , F 12,0  ;
1
4
c) D : x   .
 32 
9.27. а)   , A  0,  11  ;
b) A 
,1 , B  8, 0  ;
 3 
5
6
c) D : y  3 .
4
b) a  9 ,   ;
9.28. а) b  5 , F  10,0  ;
3
c) D : x  12 .
9.29. а) a  13 , F  5,0  ;
b) b  44 , F  7,0  ;
3
8
c) D : x   .
17
9.30. а) 2a  30 ,   ;
b) k 
15
17
, 2c  18 ;
8
c) OY - simmtriya oʻqi va A  4, 10  .
5. FAZODAGI ANALITIK GEOMETRIYA
10-topshiriq. A, B, C va D nuqtalarning koordinatalari berilgan
boʻlsa,
quyidagilar topilsin: 1) AD toʻqri chiziqning kanonik tenglamasini; 2) A, B
va C
nuqtalardan oʻtuvchi Q tekislik
tenglamasini; 3) D nuqtadan oʻtib, Q
tekislikka perpendikulyar boʻlgan toʻgʻri chiziqning kanonik tenglamasini; 4)
27
D nuqtadan Q tekislikgacha boʻlgan masofani; 5) AD toʻqri chiziq bilan Q
tekislik orasidagi burchakni.
10.1. А(3, -2, 5),
B (-2, 4, 3), C(1, -1, 6), D(2, 0, -1).
10.2. А (1, 2, 4),
B (3, 0, 1), C(0, -1, 1), D(2, 1, -1).
10.3. А (3, 0, 4),
B(6, 3, 0), C(0, -9, 1), D(1, 2, 10).
10.4. А (2, 7, 3),
B(4, 5, 6), C(2, -3, 0), D(5, 1, 12).
10.5. А (-3, -4, -2), B(9, 11, 0), C(0, 3, 10), D(-3, 8, 1).
10.6. А (4, -2, 10), B(0, 3, 9), C(-5, 3, 3), D(2, 1, 0).
10.7. А (2, -5, 0),
B(-4, 3, 2), C(-1, 1, 5), D(3, 5, -4).
10.8. А (3, -6, -3), B(2, 6, 0), C(2, -4, 2), D(0, 2, 7).
10.9. А (3, -5, 0),
B(4, 3, 2), C(-1, 1, 4), D(4, 5, -1).
10.10. А (10, 4, 0),
B(0, 3, 3), C(1, 6, -2), D(-2, -4, 5).
10.11. А(0, 2, 0),
B (-2, 5, 0), C(1, -1, 6), D(2, 0, -1).
10.12. А (1, 2, 4),
B (3, 0, 1), C(-2, 2, 6),
D(0, 5, 8).
10.13. А (2, 1, 2),
B(0, 4, 2), C(0, 1, 8),
D(2, 4, 10).
10.14. А (1, -4, 0), B(2, -6, 2), C(12, -2, 10), D(9, 0, 8).
10.15. А (2, 1, 0),
B(-2, 6, 0), C(-2, 3, 5),
D( 0, 6, 8).
10.16. А (-3,-6, 2), B(1,-2, 0), C(-1, 5, -8), D(-3, -4, 3).
10.17. А (1, 3, 2),
B(-1, 6, 2), C(-1, 3, 8),
10.18. А (1, 2, 0),
B(-1, 5, 0), C(-1, 2, 6), D(1, 5, 8).
D(1, 6, 10).
10.19. А (-2, 0, -2), B(2, 4, -4), C(0, 11, -12), D(-2, 2, -1).
10.20. А (1, 2, 1), B(-1, 5, 1), C(-1, 2, 7), D(1, 5, 9).
10.21. A  6,1,1 ,
B  4, 6, 6  ,
C  4, 2, 0  ,
D 1, 2,6  .
10.22. A  4, 2,5 ,
B  0,7,1 ,
C  0, 2, 7  ,
D 1,5,0  .
28
10.23. A  6, 6,5 ,
B  4,9,5  ,
C  4,6,11 , D  6,9,3 .
10.24. A 1,8, 2  ,
B  5, 2,6  ,
C  5, 7, 4  ,
D  4,10,9  .
10.25. A  4, 4,10  , B  7,10, 2  ,
C  2,8, 4  ,
D  9, 6,9  .
10.26. A  4, 6,5 ,
B  6,9, 4  ,
C  2,10,10  , D  7,5,9  .
10.27. A  3,1, 4  ,
B  1,6,1 ,
C  1,1,6  ,
D  0, 4, 1 .
10.28. A 10,9,6  , B  2,8, 2  ,
C  9,8,9  ,
D  7,10,3 .
10.29. A  3,5, 4  ,
B 8,7, 4  ,
C  5,10, 4  ,
D  4,7,8  .
10.30. A  7, 2, 2  ,
B  5,7, 7  , C  5, 3,1 ,
D  2,3,7  .
6. FUNKSIYA VA UNING LIMITI
11-topshiriq. Berilgan limitlar (Lopital qoidasidan foydalanmagan holda)
hisoblansin.
11.1. а) lim
x 2
с)
3x 2  5 x  2
;
2x2  x  6
b) lim
x 
x 1
 2x  3 
d) lim

 .
x 
 2x  5 
arctg 2 x
lim
;
x0
4x
2 x 2  15 x  25
;
11.2. а) lim
x 5
5  4x  x 2
с)
 2x  1
b) lim 5x
;

x

3
2x
3
2 x
lim
3x  2 
d) lim 

x   3 x  4 
lim
4 x  7x  3
;
2
2 x  x 1
b)
lim
arcsin 2 x
;
4x
 4x  3 
d) lim 

x   4 x  1 
2
x 1
с)
2
x 
1  cos 4 x
;
2 arcsin 2 2 x
x 0
11.3. а)
2 x 2  3x  1
;
3x 2  x  4
x0
lim
x 
29
.
3  2x  x
x
2
2
 4x  1
;
2 x 3
.
3x 2  5 x  4
;
b) lim 3
x 
x  x 1
2x  9x  9 ;
lim  5x  6
x
2
11.4. а)
2
x 3
с)
lim
x0
11.5. а) lim
x 4
 2x  5 
d) lim


x 
 2x  1 
sin 3x
;
tg 5 x
5x  x 2  4
;
x2  2x  8
b) lim
x 
.
2x2  x  4
;
3  x  4x2
 5x  1 
d) lim


x 
 5x  4 
tg 2 x
c) lim
;
x 0
x sin 2 x
3 x
2 x 1
.
x2  7x  1
b) lim
;
x 
3x 2  x  3
x2  2x  8
11.6. а) lim
;
x2
2 x 2  5x  2
 3x  1 
d) lim

 .
x 
 3x  4 
tgx  sin x
c) lim
;
x 0
x3
2x
3x 2  2 x  1
11.7. а) lim
;
x 1
x2  4x  3
3x 2  5 x  4
b) lim
;
x 
2x2  x  1
 2x  7 
d) lim


x 
 2x  3 
sin 2 3x
c) lim
;
x 0
tg 2 2 x
6  x  x2
11.8. а) lim
;
x  3
3x 2  8 x  3
4 x 1
.
2x3  2x  1
b) lim
;
x 
3x 2  4 x  2
 4x  1 
d) lim


x 
 2x  3 
3x
c) lim
;
x 0
arcsin 6 x
x3 1
11.9. а) lim
;
x 1
5x 2  4 x  1
1 2 x
.
5  2 x  3x 2
b) lim
;
x 
x2  x  3
 5x  2 
d) lim


x 
 5x  3 
1  cos 6 x
c) lim
;
x 0
x sin 3x
x 2  2x  8
11.10. а) lim
;
x 2
8  x3
3 2 x
.
x 2  3x  4
b) lim
;
x 
2 x 3  5x  1
30
 x 2
d) lim


x 
 x 3
c) lim sin 3xctg 5x ;
x 0
11.11. а) lim
x 3
x2  x  6
;
x 2  6x  9
b) lim
x  2
x2  4
;
1  4x  3
1 2 x
x 2  4x  4
;
x2  4
b) lim
x 3
.
2x  3  3
;
x2  9
2 

1

d) lim


x 
 4x  3 
arctg 3x
c) lim
;
x 0
6x
x3  8
11.13. а) lim
;
x  2
x2  x  2
b) lim
x 4
x 2  7 x  10
11.14. а) lim
;
x 5
x 2  10 x  25
b) lim
x 1
( x  5) 2
;
x 2  3x  10
b) lim
x 1
1 3 x
2 x 3
.
x32
;
x 1
6 x 1
2 

1

d) lim


x 
 3x  2 
x 2  6x  9
11.16. а) lim
;
x 3
x 3  27
x2  x  2
b) lim
;
x 2
4x  1  3
1  cos x
;
x2
4  3x  x1 .
d) lim
x 1
x
31
.
x 1
;
3x  7  2
sin 4 x
c) lim
;
x 0
tgx
c) lim
x 0
.
x 2
;
x  6x  8
4 

1

d) lim


x 
 4x  1
c) lim xctg 4 x ;
x 0
4 x 1
2
2 

1

d) lim


x 
 2x  5 
tg 2 x
c) lim
;
x0
5x 2
11.15. а) lim
x 5
.
3 

1

d) lim


x 
 x  4
xtgx
c) lim
;
x 0
1  cos 4 x
11.12. а) lim
x 2
4 x
.
2x 2  x  3
11.17. а) lim 2
;
x 1
x  2x  1
c) lim
x 0
b) lim ( 9 x 2  4 x  3x) ;
x 
1  cos 6 x
;
tg 3x
d) lim 5  2 x  x 2 .
x
x 2
x 2  25
b) lim
;
x 5
2x  1  3
2 x 2  5x  3
11.18. а) lim 2
;
x  3
3x  11x  6
tg 3x
c) lim
;
x 0
sin 5 x
11.19. а) lim
x  2
c) lim
x 0
d) lim 7  2 x 
2 x 2  3x  2
;
( x  2) 2
( 2 x  4 x 2  3 x) ;
b) lim
x 
2 x  3 x1 ;
d) lim
x  1
4x
;
arctg 2 x
1
b) lim
x  3
sin 2 2 x
c) lim
;
x0
x2
а)
lim
x 2
c)
11.22.
а)
c)
3
2x 2  x  6
b)
;
d)
sin 7 x
;
x0 tg 2 x
lim
x  2x  8
2
lim
x 
lim
x 4
1 x  2
;
4  1  5x
2 x  5 x2 .
d) lim
x 2
3x 2  5 x  2
5x  x 2  4
.
x 3
( x  1) 2
11.20. а) lim
;
x 1
4x 2  x  5
11.21.
2
x 3
b)
;
d)
32
4  2x  9x 2
;
3x
 2x  3 
lim 
 .
x  2 x  1 
lim
x 
tg 2 x
lim
;
x0 sin 2 x
5 x 2  3x  1
4x 3  x 2  1
3x  5
2
2x
 4x  1 
lim 
 .
x  4 x  3 
;
11.23.
а)
2 x 3
;
x7
lim
x 7
c)
11.24.
а)
lim
11.25.
а)
x 0
x2
lim
2x  1  5
;
x3
11.26.
 11x  6
b)
;
1  cos 6 x
;
x 0 1  cos 2 x
d)
а)
x2  x  6
b)
lim
lim
а)
x2
lim
2x  2
lim
lim
;
x 1
3x 2
3x 2  x  2
;
33
;
7 x 2  3x  1
5x 2  6
;
7x
lim
5x 3  2 x  3
1  4x 3
;
6x
 x  4
lim 
 .
x   x  1 
lim
3x 3  5 x  1
7x3  6
d)
 5x  2 
lim 
 .
x  5 x  1 
b)
3x 2  x  1
d)
;
 3x  4 
lim 
 .
x  3 x  5 
;
2x
lim
x 
cos 4 x  cos3 4 x
2 x 2  5x  7
lim
x 
sin 7 x  sin 3x
;
x 0
x  sin x
x 0
а)
;
lim
x 2
c)
 x  21
4x 3  1
2x
x 
c)
c)
11.28.
d)
x 3  2x  3
 6x  5 
lim 
 .
x  6 x  1 
x 
2 x 2  5x  3
lim
d)
b)
1  cos 4 x
;
x 0 2 x  tg 2 x
x 3 2 x 2
11.27.
;
lim
x 3 3x 2
lim
x 
cos 2 x  cos3 2 x
x 3
c)
b)
x 5
2
;
4x
 7x  1 
lim 
 .
x  7 x  5 
b)
lim
x 
4x 2  x  1
;
5x  3
1  cos 3x
lim
;
x 0 x  sin 2 x
c)
11.29.
а)
lim
2 x 2  7 x  15
;
lim
6x  1  5
x 2
x 4
c)
lim
x 0
lim
4x 4  2x  1
5x 3  6
x 
d)
lim
а)
x
b)
1  cos 2 x
;
x 0 x  tg 3 x
c)
lim 2 x  3 x  2 .
x 2
x 2  2 x  15
x 5
11.30.
d)
lim 3  2 x  x 1 .
2x
x 1
b)
;
x  sin 3x
cos x  cos3 x
lim
4x 2  x  5
x  3 x 3
d)
;
 x2  1
lim 7  6 x  3 x 3 .
x
x 1
7. FUNKSIYANING HOSILASI
12-topshiriq. Berilgan funksiyalarning
12.1. а) y 
3x  x
c) y  arctg
dy
hosilasini toping.
dx
sin 2 3x
b) y 
;
3 cos 6 x
;
x2  2
1 x
;
1 x
d) y  2
x
 x 3tgx ;
e) y  (2 x  3) tgx .
12.2.
а) y 
c) y  e
5x  4
x  5x  2
2
arctg x 2 1
;
b) y  (2 arcsin x  1  x 2 ) 5 ;
;
3x 2  2
d) y  ln
;
3x 2  2
;
3
e) y  (ctg 4 x) sin 4 x .
34
;
12.3. а) y  3
3x  1
x  9x  1
3
b) y  (3arctg2 x  ln(1  4 x 2 ))4 ;
;
3x 2  4
d) y  ln
;
3x 2  4
1
c) y  ln arccos
;
2x
e) y  (sin 2 x) tg 2 x .
12.4. а) y 
2x  3
x  8x  4
3
3
b) y  (4 tg 2 x  tg 2 x) 5 ;
;
x4  3
d) y  ln 4
;
x 3
1
c) y  ln arctg ;
x
4
1
x
e) y  ( x  1) .
4
12.5. а) y  3
2x  1
x  6x  1
c) y  e arccos
1 x 2
b) y  (5tgx  cos2 x) 4 ;
;
3
3x  1
;
3x  1
d) y  ln 3
;
e) y  (cos2 x) tg 2 x .
12.6. а) y  3
4x  3
x  4x  1
3
c) y  ln tge
2 x
b) y  (2 arccos
;
x
 1 x)4 ;
2x 2  3
d) y  ln
;
2x 2  3
;
4
e) y  (ctgx) sin x .
2
12.7.
а) y 
5x  6
3
x3  6x  2
b) y  (3ctg x  ln sin x) 3 ;
2
;
3
2
x
1
d) y  ln 3
;
3
2x  1
ln( 2 x  1)
c) y 
;
2x  1
x
1 

e) y   x 
.
x 

35
12.8.
x 3  10
а) y 
x  8x
4
b) y  (6 arcctg3 x  arcctg 3x) 4 ;
;
1
;
x
c) y  ln tg
d) y  ln 3
10  3x 2
;
x 3  10 x
1
x
e) y  ( x  ln x) .
12.9.
а) y 
3x  2
3
x 2  3x  1
c) y  e arcctg
4 x 1
b) y  (2tg 3 x  sin 2 3x) 6 ;
;
2x  3
;
x 2  4x  3
d) y  ln 4
;
x2
 1
e) y  1   .
 x
12.10. а) y 
5x  2
x  5x  1
2
c) y  e arcctg
x 2 1
b) y  (3cos 2 x  cos2 x) 4 ;
;
5  4x
;
x  8 x  10
d) y  ln
;
2
e) y  (arcsin x ) 2 x .
12.11. а) y 
2x  7
x 2  8 x  14
c) y  ln arccos
b) y  (5 ctg 2 x 
;
1
;
x
d) y  ln 8
1 3
) ;
sin 2 x
4x 2  1
;
4x 2  1
e) y  (tg 2 x) cos 2 x .
12.12. а) y 
3x  4
x  9x  6
2
b) y  (5sin x  cos 2 x) 3 ;
2
;
c) y  ln cos ;
d) y  ln
4 x
e) y  (1  x 2 ) arcsin x .
36
3
x3  2
;
x3  2
12.13. а) y 
3x  4
b) y  (3sin 2 x  cos2 2 x) 3 ;
;
x 3  3x  2
d) y  ln
c) y  ln arcsin 1  x ;
2
3
2  x2
;
x3  6x
e) y  (2 x  3) tgx .
12.14. а) y 
x3
b) y  (2 arcctgx  ln(1  x 2 ))4 ;
;
x  6x  9
3
3x 2  2
d) y  ln
;
x3  2x
c) y  ln tgx ;
3
4
e) y  (1  cos x) x .
2
12.15. а) y 
2x
x  5x  3
3
2
b) y  (3cos 3 x  sin 2 3x) 3 ;
;
2x  1
c) y  arctg
;
2x  1
d) y  ln
x3  3
;
x3  9x
e) y  ( x 3  2) sin x .
12.16. а) y 
3x
x  4x  1
3
2
b) y  (2 arcsin x  arccosx) 4 ;
;
2x 2  2
d) y  ln
;
x 3  3x
c) y  ln arctg x  1 ;
3
e) y  ( x 2  1) arctgx .
12.17. а) y 
c) y  e
4x
x  5x  2
3
2
arctg2 2 x 1
b) y  (5tg 2 x  x 2 ) 3 ;
;
d) y  ln
;
e) y  (arcsin x)
1 x 2
.
37
4
x2  4
;
x 3  12 x
4x  1
12.18. а) y 
b) y  (4 tg
;
x  16 x  2
2
d) y  ln
c) y  arcsin 1  4 x ;
2
x
 x )3 ;
3  x2
;
x3  9x
3
e) y  ( x  sin x) x .
2
2x  3
12.19. а) y 
x 2  4x  3
b) y  (4 tg
;
x
 x )3 ;
4  3x 2
d) y  ln
;
x3  4x
c) y  ln sin( 2 ) ;
x
5
e) y  (tg 2 x) tg 2 x .
3x  8
12.20. а) y 
x 2  3x  4
c) y  e arcsin
1 x
b) y  (2cos x  sin 2 x) 3 ;
2
;
d) y  ln 4
;
5  x2
;
x 3  15 x
e) y  ( x  1) arctg x .
2
b) y  3x  arctg 7 x ;
6
12.21. a) y  6 x3  2 x  3 ;
c) y   sin x tgx ;

d) y 
1  x3  5 x 2
;
ln x

x
2
y

x

3
e)
.
12.22. a) y 
4
 2 x  35
b) y  3cos x  tg 2 3x ;
;
arctg 7 x3
d) y 
;
2
5x  1
2
c) y   tgx  x ;

38

8 x  4 x2  3
e) y 
.
sin 7 x
5
12.23. a) y 
c) y 
1  2 x 3
b) y  3sin x  arctgx2 ;
;
2 x 4  3x  1
;
2x  5
d) y 
ln x
x 2  3x  6
;
e) y   cos x tgx .
12.24. a) y 
1
5 2x  1
b) y  5cos x  arctg x ;
;
7 x3  2 x  e  x
c) y 
;
tg 9 x
cos e x  2 x6
d) y 
;
ln 3x
x
e) y   sin x  .
12.25. a) y 
1
 2 x  14
b) y  2cos x  tg x ;
;
d) y   5 x  1ln x ;
c) y  ln x  esin 2 x ;
2 x3  4 x  1
e) y 
.
sin 5 x
12.26. a) y  5 ln 1  x  ;
tgx
b) y  e  arcsin9 x3 ;
x
d) y   cos 2 x  ;
4
c) y  cos2 6 x  e x ;
e) y 
7 x 4  e2 x  3
.
sin 5 x
b) y  3sin x  ln 2 x3 ;
12.27. a) y  4 2 x  cos x ;
39
c) y 
6 x2  5x  1
ln 1  2 x 
d) y 
;

cos3 1  x 2
;
x2  4 x  1
e) y   5 x  3sin x .
1
12.28. a) y 
;
1  sin x
2 x3  4 x  3
b) y 
;
arcsin 2 x
tgx
d) y  sin 2 9 x  e ;
c) y  6 x  tg 3 x ;
e) y   cos7 x  x .
cos 4 x  2 x7
12.29. a) y 
;
tg 2 x
b) y 
2
c) y  sin x  e x ;
ln 1  3x 
x
;
2
d) y  arctg 2 x  5 x ;
e) y   tgx  x  1 .
7
12.30. a) y  2  x  x3 ;
b) y  tg x  arcsin x2 ;
2
ex  x  1
c) y 
;
ln 6 x
d) y 
1
2 x3  3 x  1
tgx
e) y   cos x  .
13-topshiriq. Lopital qoidasi asosida limitlarni hisoblang:
3
13.1. lim
x a
x 3 a
.
x a
ln cos x
.
x0
x
13.2. lim
40
;
x  arctg x
.
3
x0
x
e ax  cos ax
13.3. lim bx
.
x0 e  cos bx
13.4. lim
e x 1
13.5. lim
.
x0 cos x  1
e x  ex
13.6. lim
.
x0 sin x  cos x
e tg x  e x
13.7. lim
.
x0 tg x  x
xm  am
13.8. lim n
.
x0 x  a n
2
x4
13.10. lim x .
x e
ln x
.
x0 ln sin x
13.9. lim
13.11. lim x
x 0
2
1
ex .


13.12. lim a 2  x 2 tg
xa
2
х
lim tg x 2 x .

2
ax  l x
13.19. lim
.
x 0
tgx
13.21.
.
1
1 

13.15. lim 1  2  .
x
x 
x
2a
 1 
13.14. lim x   e x  1 .


x


1 

13.13. lim  tg x 
.

cos x 
x 
13.17.
x
13.16.
lim x
13.18.

x0
lim
x
13.20. lim
x 1
e x 1
lim
.
x0 sin x
 .
ln e x 1

1
x x
1 e .
ln x
.
1  x3
1 
 x

.
x1 x  1 ln x 
13.22. lim 
41

13.23.
ea x 1
lim
.
x 0 sin bx
13.25.
lim
e  e  2х
.
x0
x  sin x
13.26.
13.27.
ln sin 2 x
lim
.
x 0 ln sin x
ex 1
13.28. lim
.
x  0 sin 2 x
13.29.
a

lim  x  sin  .
x
x
x
13.24.
x
13.30.
lim e  x
x0
x
x

lim  2  
x a 
a
lim x
1
1 x
x 1

tg
1
x
.
x
2a
.
.
8. HOSILA YORDAMIDA FUNKSIYANI TEKSHIRISH
14-topshiriq. Differensial
hisob
usullaridan
foydalanib
y  f x 
funksiyani toʻla tekshiring . Tekshirish natijalari asosida funksiyaning grafigini
quring. Funksiyani quyidagi
sxema asosida tekshirish tavsiya etiladi:
1)
funksiyaning aniqlanish sohasini topish; 2) funksiyani uzluksizlikka tekshirish; 3)
funksiyaning juft va toqligini aniqlash;
4)
funksiyaning oʻsish va kamayish
oraliqlarini va ekstremum nuqtalarini topish; 5) funksiya grafigining qavariqlik va
botiqlik oraliqlarini hamda egilish nuqtasini topish;
6)
funksiya grafigining
asimptotalarini topish.
14.1. y  2 x  33 x 2 .
14.2. y  x  ln x  2 .
4
14.3. y  x  2 
.
x2
4x3
14.4. y 
.
3x 2  1
42
e x 1
14.5. y 
.
x
14.6. y 
x3
14.7. y 
.
2 x  1
14.8. y  ln( x 2  2 x  2) .
14.9. y  2 x ln x .
14.10. y 
x3
.
2( x  1) 2
x3
14.11. y  2
.
3( x  3)
14.12. y 
2( x  1) 2
.
x2
x 2  6x  9
14.13. y 
.
( x  1) 2
2( x  1) 2
14.14. y 
.
x2
14.15.
y  4 xe
x2
2
4x3
14.16. y 
.
9(3  x 2 )
.
14.17. y  4 xe  x .
14.19. y 
14.21. y 
x 2  2x  7
.
x 2  2x  3

8x
.
x  22
4x  8
.
2
x 1

14.18. y 
x 2
 .
2 x2
14.20. y 
4
.
x 2  2x  3
14.22. y 
x2  8
.
2
x2


x3
14.24. y 
.
2
x 9
3x 2  6 x
14.23. y 
.
x 1
14.25. y 
x3
.
2
x 1
14.27. y 
x2  5
.
x3
14.28. y 
4 x2  5
.
x
14.29. y 
x3  8
.
2
2x
14.30. y 
2x  1
.
2
x
14.26. y  x  e2 x  1 .
43
9. ANIQMAS VA ANIQ INTEGRALLAR
15-topshiriq.
Aniqmas
integrallar
topilsin
(15.1-15.30 misollarda
integrallash natijalarini hosila olib tekshirib koʻring).
15.1. а) 
5  5 х 3  3х
х
dx ;
b)

7  5x dx .
15.2. а) 
2 x 3  55 х 2  6
dx ;
3x
b)

15.3. а) 
x  5x 2  7
dx ;
3x 2
b)

b)

b)

b)

15.4. а) 
15.5. а) 
15.6. а) 
2 x  3х 2  7
3
x
dx ;
5  4 х 3  2х
dx ;
x2
4  x x  2х3
х
dx ;
3
1  5 х dx .
3
(1  x) 2 dx .
1
2 x
dx
3
3 x
dx .
dx .
dx
(2  x) 3
dx .
15.7. а) 
3  37 х 3  4 х 2
dx ;
x
b)  (1  4 x) 5 dx .
15.8. а) 
4x 3  x3  1
dx ;
х
b)  (2  5x) 3 dx .
3x 3  5 x  7
dx ;
15.9. а) 
x
b)  (1  7 x) 3 dx .
3  x  4х3
dx ;
15.10. а) 
x2
b)

1  5 х dx .
4  6 х5  7х
dx ;
x
b)

7  4 х dx .
15.11. а) 
44


3
15.12. а)   x x  3  5 dx ;
x




x
15.13. а)   x 3  4  7 dx ;
x


b)
4  3 х 2  3х 2
dx ;
x
b)
15.14. а) 
 x

dx

b)
3
3

b)

dx ;
b)

15.19. а) 
8  5 x  3х 3
dx ;
x2
b)

15.20. а) 
2  5 х3  7х 2
dx ;
x3
b)

b)

15.17. а)   6 x 3  5 x 3  dx ;
8
x

15.18. а) 
2x 4  3 x  9
х
15.21. а)   5 x 3  4 

3 
dx ;
x4 
15.22. а) 
7  x  5х 3
dx ;
x
b)

15.23. а) 
65 х 7
dx ;
x2
b)


4x 2

15.24. а)   7 x 3  3  3 dx ;
x


15.25. а)   5 x 

b)
3

 9 dx ;
7
x

b)
45

2  7x
3
.
(5  4 x) 2
dx
b)
2  3 х2  7х 4
dx ;
x
.
(1  3x) 2
dx


15.16. а) 
.
dx

b)
15.15. а)   3  2 x 4  6 dx ;
x

(2  x) 2
3
.
5
3  2 x dx .
4
2  5 x dx .
1  6 x dx .
3
dx
(3  x) 3
dx
7 x
3
dx .
dx .
7  4 х dx .
dx
3  7x
3
dx .
dx
(1  4 x) 4
3
dx
3

(1  3x) 2
dx
3
3  7x
dx .
dx .
dx .
2  7 х 6  4х 3
dx ;
x
b)

15.27. а)  
 4  3 dx ;
x
x


b)

 2x3 7

15.28. а)  
  9 dx ;
 x x

b)

b)

b)

15.26. а) 
3 x

6
 7 x3
8

15.29. а)   2  3  2 dx ;
x
 x

 7x3

15.30. а)   3  5 x 2  5 dx ;
 x

5
5  2 x dx .
4
1  5 x dx .
3
1  4 x dx .
dx
(3  x) 3
3
dx .
(2  3x) 2 dx .
16-topshiriq. Bevosita integrallang:

x 2 dx
16.3.

x 3 dx
5
16.5.

sin x dx
.
cos 6 x
16.7.

cos x dx
3
16.9.

dx
.
sin 2 x ctg x 1
16.1.
16.11.
2 3
 x x  5 dx .
.
16.4.
 sin
16.6.

tg 7 x
.
16.8.

dx
.
2
3
cos x tg x
16.10.

x4  2
e
5
8
x cos x dx .
dx
.
cos 2 x
x
 e x dx .
2
x
x

16.2.
x3  5
sin 2 x

.
dx .
16.12.
46

ln 6 x dx
.
x
arctg3 x dx
.
1 x2
16.14.

16.16.
e
arcsin x
16.13.

16.15.
 arcsin
16.17.

x dx
.
x 2 10

e x dx
.
ex  4
16.21.

2 x dx
.
16.22.

16.23.
 x  3 .
16.24.

sin 2 x
dx .
1  cos 2 x
16.25.

16.26.

ln 3 x
dx .
x
16.27.
e
cos x
sin x dx .
16.28.
e
16.29.
e
 x5
4
16.19.
dx
5
x 1 x
4 1
x
2
.
16.18.
16.20.
xdx
9
2
dx
.
x ln x
16.30.
x dx .
1 x2
x
cos e x dx .

x 2 dx
.
x3  5

e 3 x dx
.
e3x  a 2

dx .
x 3 dx
1 x
5 x
8
.
dx .
e x dx
1 e
2x
.
17-topshiriq. Aniqmas integrallarni toping.
17.1. а)  e sin x sin 2 xdx ;
b)  arctg xdx ;
2
c) 
dx
;
x3  8
d)
47
1
dx
.
3
x 1
17.2. а) 
c) 
xdx
;
(x  4) 6
2 x 2  3x  1
dx ;
x3 1
x 3 dx
17.3. а) 
c) 
17.5.
1 x
d)
dx
;
cos x(3tgx  1)
d)
cos 3xdx
;
4  sin 3x
3
cos2 x
d)
c) 
x2  1 x
 3 (1  x) dx .
cos xdx
 1  cos x ;
1
b)  x arcsin dx ;
x
;
( x  3)dx
c)  3
;
x  x 2  2x
17.7. а) 
x  3  3 ( x  3) 2
b)  x 2 e 3 x dx ;
x2
dx ;
x 3  5 x 2  8 x  40
sin xdx
dx

b)  x 4 2 x dx ;
2
dx
;
3
2
x  x  2x  2
17.6. а) 
dx
 sin x  tgx .
b)  x3 x dx ;
;
3x  7
dx ;
x  4 x 2  4 x  16
а) 
c) 
8
d)
3
17.4. а) 
c) 
b)  e x ln(1  3e x )dx ;
2
d)
(
( 4 x  1)dx
x  4) 4 x 2
.
( x  arctgx )dx
;
1 x2
b)  x ln( x 2  1)dx ;
( x 2  3)dx
;
x 4  5x 2  6
d)
17.8. а) 
arctg xdx
;
1 x
1
x5
dx .
3
x5
b)  x sin x cos xdx ;
48
;
x 2 dx
c)  4
;
x  81
d)
sin x
dx ;
3  2 cos x
17.9. а)  3
b)  x 2 sin 4 xdx ;
( x 2  x  1)dx
c)  4
;
x  2x3  3
17.10. а)

arcsin x
1 x
2
d)
( x 3  6)dx
c)  4
;
x  6x 2  8
17.11. а) 
c) 
d)
3
dx ;
x ln 2 x
x2
dx .
dx
 2 sin x  cos x  2 .

4
3
x 2 (3 x  1) 2
dx .
b)  5 arcsin 2 xdx ;
d)
cos xdx
;
2  2 sin 2 x

2dx
3
(2 x  1)  2 x  1
2
.
b)  3x 2 ln 3xdx ;
(5 x  1)dx
;
x 2  2 x  15
17.14. а) 
3
b)  x 2 ln 2 xdx ;
dx ;
( x  12)dx
;
x2  x  6
17.13. а) 
c) 
d)
( x  4)dx
;
x 2  2x  8
17.12. а) 
c) 
1 x
2

( x  1)(6 x  1)
b)  x ln 2 xdx ;
dx ;
(arccosx) 2
dx
 3 cos x  4 sin x .
d)
arctg 2 xdx
;
1  4x 2

2 xdx
3
( x  1)  x  1  1
2
3
b)  2 xarctgxdx ;
2 x 2  3x  12
dx ;
c)  3
x  x 2  6x
d)
49

6
6
x 1
x7  6 x5
dx .
.
e ctg 2 x dx
17.15. а)  2
;
sin 2 x
c) 
(5 x  2)dx
;
x 2  2x  8
17.16. а) 
c) 
b)
e 2 x dx
1 e
2x
d)  sin 3 x cos2 xdx .
;
b)
dx
;
x 3  2 x  3x
17.17. а) 
dx
x 1  ln x
2
c) 
;
b)
1  ln x
dx ;
x
1
x 1  ln x
2
d)
 3 3  2 cos x
sin 3 x
 cos2 x dx .
d)  sin 2 x cos3 xdx .
1
dx ;
( x  1) ln 2 ( x  1)
sin x dx
dx ;
b)  x 3 ln xdx ;
dx ;
b)  xarctg 2 xdx ;
2x 2  x  1
dx ;
c) 
x3  x
17.21. а)
2x
b)  x ln xdx ;
x4
dx ;
c)  4
x  5x 2  4
17.20. а) 
 2 x  1e
cos3 x
dx .
d) 
sin 2 x
x
dx ;
3
x  3x  2
17.19. а) 
 x  3cos 3xdx ;
d)  sin 7 xdx .
x2  3
dx ;
c)  3
x  2 x 2  3x
17.18. а) 
 2 x  1sin xdx ;
d)  sin 3 x cos3 xdx .
b)
,
50

ln 2 x
x2
dx,
c)
17.22. а)
c)
17.23. а)
c)
17.24. а)
c)
17.25. а)
c)
17.26. а)
x2
 x 2  2x  5
dx,
ln tg x 
 sin x  cos x dx,

3x  1
2 x 2  5x  1
3 arctg x
 1  x2
dx,
2x  3
 x 2  2x  7



dx
ex  4
3  2x 2
1  3x
1  4x 2
dx,
,
x2
x 1
dx,
dx,
 2x 2  x  1

dx,
dx,
2  ln x
dx,
x
51
d)
 1  sin x .
b)
2
 x e
d)
dx
 4 sin x  6 cos x .
b)
x
dx
2
x
2
dx,
 sin 4 x dx,
x3  1
d)
 x 3  x 2 dx.
b)
 x  sin 4 x dx,
3x 2  15
d)
 x  1  x 2  5x  6 dx.
b)
x
2
 5 x dx,
6x 2
d)
 x  1  x 2  3x  2dx.
b)
x
2
 e x dx,
c)
17.27. а)
c)
17.28. а)
c)
17.29. а)
c)
17.30. а)
c)


3x  4
x  6 x  13
2
arcsin 3 x
1 x
2
,
dx,
4x  1
 4 x 2  4 x  5 dx,
sin 2 x dx
 3 sin 2 x  4

,
x 8
3  2x  x 2
e 2 x dx
 5  e 2x  1
dx,
5x  1
 x 2  4 x  1 dx,


x  arctg x
x 1
2
 x 2  4 x  3  x  5dx.
b)

d)
dx
 4  5 cos x .
b)
x
d)
dx
 2 sin x  cos x  2 .
b)
,
dx,
x5
3x 2  6 x  1
dx,
2 x 2  26
d)

x  ln 2 x dx,
 cos 6 x dx,
2
arctg 4 x
x 1
2
d)
cos x
 1  cos x dx.
b)
e
d)
 3 cos x  4 sin x .
3x
 sin x dx,
dx
18-topshiriq. Aniq integrallar hisoblansin.
 2

 3 3x 2  x  dx ;
18.1. a)  
x 1
0 

3
1
b)  x 2 e  x dx .
0
52
dx,


2 

18.2. a)  sin 2 x  3 
dx ;
x  1
0 
6
4
b)  sin 5 x cos 3xdx .
0

2
0
18.3. а)  2 sin 2 x sin 7 xdx ;
b)

1
0
x3
dx .
1 x  2

5
5x
dx ;
18.4. а) 
0 1  3x
b)  4 x sin 2 xdx .
0

1
2
b)  3 ln( x  1)dx .
18.5. а)  cos xdx ;
3
0
0
4
3 

18.6. а)   2 x 
dx ;
x
1
9
18.7. а)
x

x 1
4
18.8. а)

4

b)  x sin xdx .
dx ;
0
1
b)  x 2 e x dx .
0
2
x
dx ;
x 1
b)  cos xdx .
x
4 (1  x 2 ) 3 dx ;
b)  x 2 cos xdx .

0
18.10. а)
0
x 1
dx ;
x 1
9
4
18.9. а)
1
b)  xe 3 x dx .
0

9
0

18.12. а)
5

1
5x  3
dx .
2
x
1
3
3 cos x
dx ;
18.11. а) 
3
0 sin x
6
b)
x
4
7x
dx ;
1  3x
b)
 2x
3
1
3x  7
dx .
 7x  6
2
x 2  3x  5
1 x 3  5x 2  6 x dx .
2
18.13. а)  2 x3 x dx ;
b)
0
8
3
4
18.14. а) 
dx ;
1 x
3 5
b)

0
53
5
dx .
3x  16  3x

18.15. а)
3
2
2x
dx ;
x 1 1

0
b)  2 cos5 x sin 2 xdx .
0
2
8
4
18.16. а) 
dx ;
x 1
3 5
18.17. а)
4
0
e
5x
dx ;
2x  1  1

0
18.18. а)
b)  x 3 e x dx .
b)  3 ln xdx .
1
2 x 1
1 3x dx ;
2
5
b)
x
0
3x  2
dx .
2
 3x  2

6
11
2
4
x
18.19. а) 
dx ;
6 3
0 (1  x )
b)  cos 5 x cos 3xdx .
0

18.20. а)
5

1
3
18.21. а)
6
4
dx ;
3x  1  3x

b)  cos3 x sin 2 xdx .
0
3
x 1  x dx ;
3
б)
2
2
0
12 3

18.22. а)
0
0
12 x 5
x 1
6
1
x 2 dx
18.23. а)  2
;
x

1
0
18.24. а)
б)
dx ;
2
dx
x  2x  4
2
2
2
б)  ( y  1) ln ydy ;
1
2

0
2
18.26. а)

2
0
18.25. а)
2 x
0
 sin x  cos xdx ;

 xe
б)
dx ;
1

dx
 2 x 2  3x  2 ;

0
2
cos x
dx ;
1  cos x
б)
5

xdx
4  x2
dx
 x 2  4x  21 ;
б)
;


54
3
6
sin 2 x
dx ;
cos3 x
;
1
18.27. а)

0

18.28. а)
0
dx
;
4  3x
б)

6
б)
0
18.29. а)
2
 ( x  3) sin xdx ;
2
2
б)  ln( 3x  2)dx ;
1
6
e
18.30. а)
;
0
 sin x  cos xdx ;

x 2  2x  4
2
 12 ctg 3xdx ;

dx


0
dx
x 1  ln x
2
1
б)
;
 xe
1
2 x
dx ;
2
19-topshiriq. 19.1-19.15 masalalarda egri chiziqlar bilan chegaralangan shakl
yuzasini hisoblang:
2
19.1. y  x  1 , x  4 , x  0 , y  0 .
x
19.2. y  0 , y  e , x  0 , x  1 .
19.3. y  0 , y  sin 2 x , x  0 , x 
2
.
3
2
19.4. y  0 , y   x  4 , x  3 .
2
19.5. y  x  2 x , y  x .
x
x
19.6. y  e , y  e , x  1 .
2
19.7. x  4 y , y 
19.8. y  e
x
2
e
x
2
8
.
x2  4
, x  0, y  0 , x  2 .
19.9. y  3  2 x , y   x , y  4 .
2
19.10. x  y  0 , y  x  2 .
55
19.11. Kardoidadan iborat shakl yuzasni toping:
r  21  cos   .
19.12. r  4 sin 2 .
19.13. Egri chiziqning bitta oʻramasi yuzasini toping: r  cos 2 .
19.14. Ellips yuzasini hisoblang: x  3 cos t , y  2 sin t .
19.15.
19.17.
y  x2  4 x,
y  x  4.
y  x2 , y  2  x2 .
19.18. y  ln x,
x  e,
19.19. y  2 x  4,
2
2
19.20. y  6 x  x ,
y  0.
x  0.
y  0.
19.21.
y  x3 ,
y  8, x  0.
19.22.
y 2  1  x, x  3.
19.23. y  3  2 x  x ,
2
y  0.
x2 y 2
19.24. 2  2  1.
a
b
19.25-19.30 masalalarda egri chiziq yoyi uzunligini toping:
2
3
19.25. Egri chiziq yoyi uzunligini toping: y  x , x  0 dan x  1 gacha
 y  0 .
19.26. Egri chiziq yoyi uzunligini toping: y  ln cos x , x  0 dan x 
gacha.
56

4
y
19.27. Egri chiziq yoyi uzunligini toping:
1 x

e  e  x  , x  1 dan
2
x  1 gacha.
19.28. Egri chiziq yoyi uzunligini toping: y  ln x , x  3 dan x  8
gacha.
1
3
2
3
19.29. Egri chiziq yoyi uzunligini toping: x  t  t , y  t  2 , t  0 dan
t  3 gacha.
3
3
19.30. Egri chiziq yoyi uzunligini toping: x  cos t , y  sin t ,
agar
0  t  2 .
20-topshiriq.  shaklning koʻrsatilgan oʻq atrofida aylanishidan hosil
boʻlgan jism hajmini (verguldan soʻng ikkita raqamgacha aniqlikda) toping.
20.1.  : y 2  4  x, x  0, OY
20.2.  :
20.3.  :
oʻq atrofida.
x  y  2 , x  0, y  0, OX oʻq atrofida.
x2 y 2

 1, OY oʻq atrofida.
9
4
20.4.  : y3  x2 , y  1, OX oʻq atrofida.
20.5.  : x2  8 y  0, y  0, x  3, OY oʻq atrofida.
20.6.  : y  x3 , x  0, y  8, OY oʻq atrofida.
20.7.  : y 2  ( x  4)3 , x  0, OY oʻq atrofida.
20.8.  : y   x2  8, y  x2 , OX oʻq atrofida.
20.9.  : y  2  x2  8, y  x2 , OX oʻq atrofida.
20.10.  : xy  4, 2x  y  6  0, OX oʻq atrofida.
57
20.11.  : x3  ( y  1)2 , x  0, y  0, OX oʻq atrofida.
20.12.  :
x2 y2

 1, OX oʻq atrofida.
16 1
x
20.13.  : y  e ,
20.14.  :
x  1, x  0, y  0, OX oʻq atrofida.
2 y  x  0, x  4, y  0,
ОХ oʻq atrofida.
20.15.  : y  2x  x2 , y  0, OX oʻq atrofida.
20.16.  : y 2 
4x
, x  3, OX oʻq atrofida.
3
20.17.  : y  e x , x  0, x  1, OX oʻq atrofida.
20.18.  : y  x2 , 8x  y 2 OY oʻq atrofida.
20.19.  : x  2 cos t, y  5sin t, OY oʻq atrofida.
20.20.  : y 2  4 x, x2  4 y, OY oʻq atrofida.
20.21.  : y  sin x, y  0, (0  x   ), OY oʻq atrofida.
20.22.  : x  1  y 2 , y 
3
x, y  0, OX oʻq atrofida.
2
20.23.  : y 2  ( x  1)3 , x  2, ось OX .
20.24.  : y 2  x, x 2  y, OX oʻq atrofida.
20.25.  : y  x  1 ,
x  5, y  0
OX oʻq atrofida.
20.26.  : y 2  ( x  4)3 , x  0, OX oʻq atrofida.
3
20.27.  : y  x , x  0, y  8,
OY oʻq atrofida.
2
20.28.  : y  x  x , y  0, OX oʻq atrofida.
2
3
20.29.  : y  ( x  4) , x  0, OX oʻq atrofida.
2
20.30.  : 2 y  x , 2 x  2 y  3  0, OX oʻq atrofida.
58
21-topshiriq. Aniq integrallarning integrallash
boʻlakka boʻlib, Simpson formulasi
oraligʻini
yordamida taqriban
hisoblang. Barcha
hisoblarni mingdan birlargacha yaxlitlab hisoblang.
8
21.1.

12
x 3  16dx .
21.2.
2
9

x 3  32dx .
21.4.

x 2  10dx .
1
3
1
21.5.
x 3  9dx .
2
7
21.3.


8
x 2  19dx .
21.6.
9

9
x 3  5dx .
21.8.

12

10
x 2  4dx .
21.10.
2
7
x 2  1dx .

12
x 2  8dx .
21.12.
3

21.14.

x 3  36dx .
3
8

6
x 3  8dx .

21.16.
2
x 3  64dx .
4
8

21.17.
x 2  2dx .
7
x 3  3dx .
1
21.15.

2
11
9
x 3  11dx .
21.18.
2

x 3  1dx .
1
11
21.19.

0
21.11.
21.13.
x 3  2dx .
1
0
21.9.
x 2  4dx .
2
10
21.7.


4
x  2dx .
3

21.20.
3
9 x  28dx .
4
1
2
3
21.22.  ln(3x  4)dx .
21.21.1.  ln(3x  1)dx .
1
0
2
6
21.23.  3 3x  1dx .
21.24.  ln(3x  8)dx .
0
3
59
10 ta teng
1
4
21.25.

21.26.  ln(3x  13)dx .
5 x  4dx .
4
1
4
6
21.27.  ln(3x  2)dx .
21.28.
1

0
10
21.29.

0,5 x  1dx .
3
3
9 x  26dx .

21.30.
9 x  37dx .
4
2
22-topshiriq.
Xosmas
integrallarni
hisoblang
uzoqlashuvchiligini koʻrsating.
22.1. а)

2
dx
2 x x  1 ;
b)
3

22.2. а)
22.3. а)
22.4. а)
b)
b)
3
b)
.
2
dx
 (x  3)
2
.
0
4
xdx
3 x 2  4 ;
b)

0
dx
3
(x  4)
3
ln xdx
2 x ;
b)

0
b)

2
3
(x  3) 2
dx
3
4
x
 xe dx ;
b)

x
2

dx
4 x x ;
e
b)
x
1
60
2
dx
10
3 x
 xe dx ;
0
22.9. а)
.
dx
 (x  1)
3
xdx
 (x 2  1) 2 ;
0
22.8. а)
3
.
1

22.7. а)
1 x
2
xdx
 x4  9 ;
3

22.6. а)

0

2
x 2 dx
1
dx
2 x ln x ;

22.5. а)
dx
 (x  3)
(x  2) 2
dx
.
4
2
dx
.
ln x
.
.
.
yoki
uning

dx
9 x ln 3 x ;
22.10. а)

2
7
2

b)
1
0
.
4  x2
0
x
 xe dx ;
22.11. а)
xdx

b)
3
dx
.
7x


dx
22.12. а)  2
;
1 x  x  1
dx
4
b)
 1  cos 2 x .
0


xdx
22.13. а) 
;
3
0 x  2
2
b)  tgxdx .
0


dx
 x 2  4 x  5 ;
22.14. а)
b)
dx
;

 1  cos x
2

22.15. а)
dx
 x 2  4 x  13 ;

1
b)
0
3
b)
1

2
dx
22.17. а) 
;
4
e x ln x 
b)

b)


0
xdx
3
x2 1
3x
2
 2dx
.
x
dx
0 x ln 3 x .
4
x
 xe 2 dx ;
b)

5 3
0
22.20. а)
x  1
e 2
xdx
22.18. а) 
;
4
0  x  3


.
3
0

22.19. а)
dx

0
.
1 x2
0
2 x
 x e dx ;
22.16. а)
dx

dx
 x  5
4
.
3x  2
dx .
3
x
0
2
;
b)


1
xdx
22.21. a) 
;
4
0 16x  1
b)
3
0

3
16 xdx
22.22. a) 
;
4
1 16 x  1
b)

1
61
dx
;
4  4x
dx
x 2  6x  9
;

x dx
22.23. a) 
16x  1
4
0

16x  1
4
1

22.25. а)
;
b)
( x  4) 3
;
b)
22.27. а)
1
 x2
x 
dx ;
22.28. а)   3
 2
x

1
x

1
 

22.29. а)  e
b)
 xdx ;
2
dx
x
(4  x )  arctg
2
x 2 dx

31( x  1)
3
x 2 dx
b)

0
64  x 6
1
x 4 dx
0
1  x5
3



dx
 (x  4)3 ;
2
0
22.30. а)
3
1
0
3x
;
ln( 3x  1)
dx ;
3x  1

5
b)
2

5
1
dx
;
9
1  2x

(3  x )
4
b)
1
x dx ;
dx
3
1
dx
 2x2  2x  1 ;
0
1
x2
0
1
22.26. а)
0

1
2
0
b)
;
xdx

e3 
1
xdx
22.24. a) 
3
1
3
;
b)
2
2

0
;
;
;
3 sin 3 x
dx ;
cos x
10. KOʻP ARGUMENTLI FUNKSIYALAR
23-topshiriq. Berilgan funksiyalarning birinсhi va ikkinchi tartibli hususiy
hosilalari topilsin.
23.1. z  2 cosx 2 y   5x 2 y  4 y .
23.2. z 
23.3. z  12e x
23.4. z  2 xy  y 2 .
2
y
 7 x 3 y  13 y  9 .
62
x2 y2
.

a2 b2
x y
.
x
23.5. z  x cosxy   8x 2 y 2  7 x .
23.6. z  arcsin
23.7. z  4 cosx 2  y 3   9 xy 3  5 .
23.8. z  arcsin
23.9. z  5e x 3 y  5x 2 y 2  45x .
23.10. z  4 sinxy   3 y  15xy 4 .
23.11. z  3 x 2  y 2  5 x 3 y  8x .
y
23.12. z  arccos .
x
23.13. z  2 x  y  5xy 4  8x  2 .
23.14. z  ln x  x 2  y 2 .
23.15. z  7 ln x 3  y 2   9 x 3 y  2 x .
23.16. z  arctg
2
23.17. z  arctg
x
1
3
2
 y2

3


x y
.
1  xy
23.18. z  ln x 2  y  .
y
x
 arctg .
x
y
23.19. z  6 ln x 2  y 2   4 x 3 y 2  8 y  5 .
23.21. z 
x y
.
x y
23.20. z  ln x 2  4 y .
23.22. z  arcsin xy .
.


23.23. z  sin 2 ax  by .
23.24. z  ln e x  e y .
23.25. z 
23.26. z  sin  x 2  y 2 .

x sin y .



x
.
y
2
2
23.27. z  ln x  3 y .
23.28. z  ln tg
x2  y2
23.29. z  2
.
x  y2
23.30. z  ln y 2  2 x  10 .
24-topshiriq. z  f x, y  funksiya, x0 y0 
nuqta va a


vektor
berilgan. Quyidagilarni topish kerak: 1) A nuqtadagi gradientni; 2) A nuqtada
a
vektor yoʻnalishi boʻyʻicha hosilani.
63
x y
,
x2  y2
A(1;-2),

 
a i 2j .
24.2. z  2 x 2  8x 2 y 3 ,
A(2;1),

 
a  i 3j .
24.3. z  x 4  5x 2 y 2  3 ,
A(2;-2),



a  2i  5 j .
24.4. z  2 x 2  3xy  4 y 2 ,
A(2;-2)

 
a  i 3j .
24.5. z  x 2  3xy  4 y 2  x ,
A(1;3),



a  8i  6 j .
24.6. z  3x 2  2 xy  y 2 ,
A(1;2),



a  4i  3 j .
24.7. z  x 2  3xy 2 ,
A(1;3),

 
a i 2j .
24.8. z  2 x 3 y  3x 2 y 2 ,
A(1;-2),



a  6i  8 j .
24.9. z  3x 2 y 2  5xy 2 ,
A(1;1),
 

a  2i  j .
24.10. z  2 x 2  3xy  y 2 ,
A(2;1),



a  3i  4 j .
24.11. z  x 2  xy  y 2 ,
A(1;1),
 

a  2i  j .
24.12. z  2 x 2  3xy  y 2 ,
A(2;1),



a  3i  4 j .
24.13. z  ln( 5x 2  3 y 2 ) ,
A(1;1),



a  3i  2 j .
24.14. z  ln( 5x 2  4 y 2 ) ,
A(1;1), a  2i  j .
24.15. z  5x 2  6 xy ,
A(2;1),

 
a i 2j .
24.16. z  arctg ( xy 2 ) ,
A(2;3),



a  4i  3 j .
2
24.17. z  arcsin x

A(1;2), a  5 j  12 j .
24.1. z 
 ,
y 






 

a  2i  j .
24.18. z  ln 3x 2  4 y 2  ,
A(1;3),
24.19. z  3x 4  2 x 2 y 3 ,
A(-1;2), a  4i  3 j .
24.20. z  3x 2 y 2  5 y 2 x ,
A(1;1),

64


 

a  2i  j .



a  3i  5 j .
24.21. z  x 3 y  4 x 2 y 2 ,
A(1;3),
24.22. z  x 2 y 2  xy 2 ,
A(1;-2),
  
a  i 3j .
24.23. z  5x 2  3xy  2 y 2 ,
A(3;-1),
 

a  4i  3 j .
24.24. z  3x 2  5xy  2 y 2 ,
A(-2;1),


a  3i  5 j .
24.25. z  5x 2  7 xy  2 y 2 ,
A(5;-3),



a  3i  4 j .
24.26. z  3x 2  7 x 2 y 3 ,
A(-3;1),
 

a  3i  1 j .
24.27. z  x 4  3x 2 y 2  2 ,
A(1;-3),



a  4i  3 j .
24.28. z  x 2  7 xy  2 y 2 ,
A(-2;5)
24.29. z  3x 2  xy  2 y 2  x ,
A(0;3),



a  3i  4 j .
24.30. z  4 x 2  xy  3 y 2 ,
A(-3;2),

 
a  i 5j .



a  2i  5 j .
11. DIFFERENSIYAL TENGLAMALAR
25-topshiriq.
Quyidagi
differensial
tenglamalarning umumiy
yechimi(umumiy integrali ) topilsin.
25.1. a) xy   4 2 x 2  y 2  y ,
b) y  cos2 x  y  tgx ,
с) y  e y y  0 .
y 2 10 y
 5,
25.2. a) 4 y   2 
x
x
b) (1  x 2 ) y   2 xy  (1  x 2 ) 2 ,
с) y  y  2 y .
25.3. a) xy   y  x 2  y 2 ,
b) xy  y  3  0 ,
с) y   y   ( y ) 2 .
65
25.4. a) y  
y
y
 tg ,
x
x
b) y  cos x  ( y  1) sin x ,
с) y   12 y 2  0 .
25.5. a) y  
x y
,
yx
b) x 2 y  2 xy  3 ,
с) 2 y   e 4 y .
25.6. a) xyy  3x 2  y 2 ,
b) xy   y  x  1  0 ,
с) ( y  2) y   2( y ) 2 .
25.7. a) xy   y  x 2  y 2  0 ,
b) (1  x 2 ) y   y  arctgx ,
с) 2 yy   3  ( y ) 2 .
25.8. a) xy   y ln
y
0 ,
x
b) y  1  x 2  y  arcsin x ,
с) y   3 y  1 .
25.9. a) xy   4 x 2  y 2  y ,
b) y   2 ytg 2 x  sin 4 x ,
с) ( y  1) 2 y   ( y ) 3 .
y2 8y
 8,
25.10. a) 2 y   2 
x
x
b) y  sin x  y cos x  1,
с) xy   y  4x 3 ,
25.11. a) y  
8x  5 y
,
5x  2 y
b) y   2 xy  3x 2 e  x ,
2
с) xy   y   x 2 cos x .
b) xy   y  x 2 cos x ,
25.12. a) 4 xyy   y 2  3x 2  0 ,
66
с) x 3 y  4 ln x .
25.13. a) y  
x y
,
x y
b) y   y  tgx  1 ,
cos x
с) xy   y   x 2  0 .
25.14. a) xy  3 2 x 2  y 2  y ,
b) y   2 y  e x  x ,
с) y   y ctgx  sin x .
b) y  xy  x 3 y 3 ,
25.15. a) xy   x 2  y 2  y ,
с) y  
x
(1  x 2 ) 3
25.16. a) xy   xe
y
x
.
b) x 2 y   y 2  2 xy  0 ,
 y  0,
с) xy   2 y   2 x 4 .
25.17. a) ( x 2  y 2 ) y   2 xy ,
b) xy   y  y 2 ,
с) xy   ln x  1.
2
2
25.18. a) x y  y  2 xy  0 ,
b) xy  y  y 2 x ,
с) ytgy  2( y) 2 .
25.19. a) 2 x 2 y  x 2  y 2  0 ,
b) y  y  xy 2 ,
с) 3 yy   ( y ) 2  0 .
b) y  y   y 2 e 2 x ,
25.20. a) xy   x 2  y 2  y ,
с) xy   2 y   x 3 .
b) y  
25.21. a) y  tg x  ctg y ,
67
n
a
y n,
x
x
y3
c) y   y ctg x 
.
sin x
25.22. a)


xy  x dy  ydx  0 ,
c) xy  y 
b) y  y  cos x ,
x2  y2 .


2
25.23. a) x  2 xy dx  4  x dy  0 ,
b) y  
n
y  ex xn ,
x
3
c) y  xy  xy .




2
2
25.24. a) xy  2 x dx  x y  y dy  0 , b)  xy   y arctg
c) y  
y
ex
y
 x,
x
y
.
x

2
2
25.25. a) 1  y dx  y 1  x dy  0 ,
b) 3 y   2 xy  xe  x y 4  0 ,
2
2 x
c) y e y   3 .


y2
b) y   2  2 ,
x
25.26. a) y x  3 y  x ,

2

2
2
c) y  3x dy  2 xydx  0 ,
1 y2
 0,
25.27. a) y  
1 x2
b) y  
3 2
c) 2 xy   3 y  y x .
68
2y
 x  13 ,
x 1
25.28. a) y  

1  2x
y 1  0 ,
2
x
b) 1  x
2
 y  2xy  1  x  ,
2 2
x 2
c) y   y  e y .
25.29. a)  y  x  dx   y  x  dy  0 ,
b). y  
x y
,
x y
c) y  ytgx  ctgx .
25.30. a) y sin x  y ln y ,
b) y  
x y
 ,
y x
c)
1 y2
y 
.
1 x2
26-topshiriq. Oʻzgarmas
koeffitsientli
ikkinchi
tartibli bir jinsli
boʻlmagan chiziqli differensial tenglamalar berilgan. Koʻrsatilgan boshlangʻich
shartlarni qanoatlantiruvchi hususiy yechimlari topilsin.
26.1. y   5 y   6 y  12 cos 2 x ,
y(0)=1,
y(0)  3 .
26.2. y   5 y   6 y  (12 x  7)e  x ,
y(0)=0,
y(0)  0 .
26.3. y   2 y   y  16e x ,
y(0)=1
26.4. y  6 y  9 y  10e3 x ,
y(0)=3,
y (0)  2 .
26.5. y  4 y  13 y  26 x  5 ,
y(0)=1,
y(0)  0 .
y(0)  2 .
4
,
3
y(0) 
1
.
27
26.6.
y  6 y  9 y  x 2  x  3 ,
y (0) 
26.7.
y  y  3cos x  sin x ,
y(0)=0,
y(0)  1 .
26.8.
y  y  6 y  6 x 2  4 x  3 ,
y(0)=3,
y(0)  5 .
69
26.9.
y  3 y  3e3 x ,
y(0)=2,
y(0)  4 .
26.10.
y  4 y  5 y  5x  4 ,
y(0)=0,
y(0)  3 .
26.11.
y  y  2 y  cos x  3sin x ,
y(0)=1,
y(0)  2 .
26.12.
y  2 y  y  2 sin x ,
y(0)=1,
y(0)  2 .
26.13.
y   6 y   9 y  2e 3 x ,
y(0)=1,
y(0)  3 .
26.14.
y  16 y  7 cos3x ,
y(0)=1,
y(0)  4 .
26.15.
y  4 y  3 y  8e5 x ,
y(0)=3,
y(0)  7 .
26.16.
y  2 y  6 x 2  6 x  2 ,
y(0)=1,
y(0)  1 .
26.17.
y  y  2 y  4e2 x  2 x  1,
y(0)=3,
y(0)  5 .
26.18.
y  5 y  10 x  3 ,
y(0)=2 ,
y(0)  4 .
26.19.
y  y  6 sin 2 x ,
y( )  1,
y( )  4 .
26.20.
y   4 y  (3x  1)e  x ,
y(0)  0,
y(0)  4 .
26.21.
y"6 y'9 y  x 2  x  3;
26.22.
4
y 0   ,
3
y ' 0  
y" y  9 x  e 2x ;
y0  0,
y' 0  5.
26.23.
y"2 y'5 y  5x 2  4 x  2;
y0  0,
y' 0  2.
26.24.
y"3 y'2 y  3  4 x   e 2x ;
y0  0,
y' 0  0.
26.25.
y"4 y'20 y  16 x  e 2x ;
y0  1,
y' 0  2.
26.26.
y" y  14  16 x   e  x ;
y0  0,
y' 0  1.
26.27.
y"5 y'6 y  52 sin 2 x;
y0  2,
y' 0  2.
70
1
.
27
26.28.
y"4 y  8  e 2x ;
y0  1,
y' 0  8.
26.29.
y"3 y'2 y   sin x  7 cos x;
y0  2,
y' 0  7.
26.30.
y"9 y'18 y  26 cos x  8sin x;
y0  0,
y' 0  2.
12. SONLI VA DARAJALI QATORLAR
27-topshiriq. Ishoralari almashinuvchi qatorlarni yaqinlashishga va absolyut
yaqinlashishga tekshiring.
(1) n
;
27.1. 1)  2
n

1
n 1


27.2. 1)
n
n 1

27.3. 1)

n 1

27.4. 1)
 1

n 1

;
2)
n
;
2)
;
2)
n 1

27.8. 1)
n 1
  1
n 1
1
.
n  2n
n1
1
.
ln  n  1
  1
n 1
n 1


n4 n
n
n
n 
2)   1 
 .
2
n

1


n 1
n

1
;
n 1
1
1
.
n 1
2
n2  1
2)   1
.
n3
n1
n5
27.6. 1)   1
;
n
3
n 1
n 1
  1

n
27.7. 1)   1
n 1
n 1
n
1

27.5. 1) 
;
n
n 1  2n  1  3

  1

n 1
n!
2n
.
n4
n 1


  1
n 1
n 1
n 1
n5
 1
2)
n
n
 1

2)

;
2)
  1
n 1
71
n1
1
n1
1
.
n  ln n
  1  n  1! .
n 1

27.9. 1)
  1
n1
n 1

27.10. 1)
  1
n 1
n 1

27.11. 1)
  1
n 1
n 1

27.12. 1)
  1
n 1
n 1

27.13. 1)
  1
n 1
n 1

27.14. 1)
  1
n 1
n 1

27.15. 1)
  1
n1
n1

27.16. 1)
n 1
  1
n 1

27.17. 1)
  1
n1
n 1

27.18. 1)
  1
n1
n 1

27.19. 1)
  1
n 1
n1

27.20. 1)
  1
n1

n
;
5n  7
n 1
2)
  1
n1
  1
n 1

;
2)

2)
1
;
n2
2)
  1
ln n
.
7n
n 1
1
.
n 1
2
n 1

n 1
n
.
9n  1
n1
3n
.
2n  3
  1
n 1

n
;
6n  5
2)
  1
n1

1
;
n
2)
  1
n 1
n 1

1
;
n  2n  1
2)
  1

2)
n
;
n
3  n  1
2)
  1
n
.
12n
n 1
n3
.
n2  1
n 1

  1
n1
n 1

1
;
ln  n  3
2)
72
2n  1
.
n  n  1
n1
n 1
3
;
ln  n  1
1
.
n  ln 2 n
n 1
  1
n 1
1
;
2n  1
.
n 

 .
 2n  1 

2)
n
n
n 1 
n1
n5
;
3n
n 4 n5
 ln  n  1 

2)
1
n 1
n1
2n  1
;
n  n  1
1
  1
2n  1
.
n  n  2
1
  1  n  1 n  4 .
n1
n1

27.21. 1)
  1
 2n  1n
n 1

27.22. 1)
  1

n 1
  1
n 1

27.24. 1)
  1

27.25. 1)
  1
1
n3 4 n
n1
n 1
2)
2)
;
2)
n 1
27.29. 1)  (1)
n 1

27.30. 1)
 (1)
n 1
n 1

sin n
6n
2)
2n  1
n1
  1

6n
;
5n  2
  1
n
n 1 
4n 

 .
5
n

1


1
n 1
3
n 1
n
4
.
(1) n1
2) 
.
n 1 2n  1


2)
 (1)
n 1
n 1

n
;
3n  1
2)
2n  1
;
n
 (1)
n
n 1

2)
 (1)
n
n 1

1
;
(2n  1)n
2)
 (1)
n 1
n
n
.
6n  5
n5
.
3n
3
.
ln( n  1)
3
.
ln( n  1)
28-topshiriq. Sonli qatorlarni yaqinlashishga tekshiring.
2n
28.1. a)  2 ;
n 1 n

 n 1
28.2. a)  

n2  n  1 

.
  1  n  1 n  2 .
n1
n1
(1) n1
;
27.27. 1) 
ln n
n 1

n1


n 1
  1

2)
(1) n1
;
27.26. 1) 
n!
n 1
n 1
27.28. 1)  (1)
n 1
n 1


1
  1  n  5 n  1 .
n1

n6
;
5n
n 1
n 1
;
1
;
n5
n 1
n 1
27.23. 1)

3n
n1

b)
n 1
n ( n 1)
(2n  1)!
 (3n  4)3
n
.
(n  3)2 n1
b) 
.
7n
n 1

;
73
n
 2n 2  1 
28.3. a)   2
 ;
n 1  3n  1 


b)
n2
(n  1) 2
b)  n
.
2 n!
n 1

n2
1  1
28.5. a)  n 1   ;
n
n 1 2 


b)
n2

28.7. a)

2n  1
n 1
n  2n
2
n 1
 n 1
28.6. a)  3 n 
 ;
n 1
 n 

.
3n
n 1
n 2
28.4. a)  3n 1 
 ;
n2
 n3

n

n!
.
1
n
2n
b) 
.
2
n 1 ( n  2)

2 n  n!
.

nn
n 1

;
b)
n  3n  2
28.8. a)  n ;
5
n 1
n n 1
b) 
.
n 1 ( n  1)!
n3
28.9. a)  n ;
n 1 e
b)




1
n5
n 1
n
 4n 2  5n  2 
28.10. a)   2
 ;
n 1  7 n  2n  1 

 2n  4 
28.11. a)  

n 1  2n  7 


b)
6
n 1
n2
n

;
b)
n 1
.
n!
.
(n  2)!
2n
 (n  3)! .
n 1
2n
28.12. a)  5 ;
n 1 n
4n
b) 
.
n 1 ( n  1)!
3n
28.13. a) 
;
n 1 ( n  1)!
b)




28.14. a)

n 1
2
 n(
n 1

n
n

;
b)
1
2)n
.
3
 (n  1)! .
n 1
2
2n
28.15. a)  2 ;
n 1 n

2n(n  1)
.
5n
n 1

b)
74

1
28.16. a) 
;
n
n 1 ( n  1)7
3n
b) 
.
n 1 ( n  1)(n  2)
2n
28.17. a) 
;
2
n 1 (1  n)
n2
b)  n 1 .
n 1 5




n2
;

n
n 1 2

28.18. a)

b)
n 1
n2
 n 1 
28.19. a)  
 ;
n 1  n  2 


 n(1  2n
28.20. a)
1
)
n2
b)
b)
2

n 1
;
b)

b)
28.27.

n 1
n  n 1
2
n  7n
 n  1
n
2
n!
.
1
 5n  2 .

1
3n  1
.
n 1
b)
;
n
  2n  3! .
n 1

6n
;

n
n 1 5  2n  1


n 1

28.26.
1
 ln  n  2 
n 1
3n
;
28.24. 
n 1 n  1!
28.25.
2n  1

b)


2 n .
 3n2  5 .

1
1
n 1
3n
;

n 1 2 n !
n 1
 (2n  1)


 1  4n
3n
.

2
n 1 n
n 1
5n
28.21. a) 
;
n 1 3  n !
28.23.
.
n2

;


n

b)
n 1
28.22. a)
3
b)
1
  5n  2 ln 5n  2 
n1

;
b)

n 1
75
1
 ln  n  3 
n
.


5n
28.28. 
;
n 1 4  n!

28.29.
b)
n 1

1

 4n  33
n 1
;
b)
5n
 3  n !. .
n 1


28.30.
1
 n  1  ln n  1 . .
n3  3
  n  1! ;
n 1
b)
8
n 1
29-topshiriq. Darajali
qatorlarning
1
 4  9n 
.
5
yaqinlashish
intervallari
topilsin.
Intervallar chegaralarida aiohida tekshirilsin.
nx n
29.1.  n .
n 1 5
3n n
29.2.  x .
n 1 n
xn
29.4. 
.
n 1 n
xn
29.5.  n .
n 1 2
xn
29.7.  n .
n 1 n5
xn
29.8.  n .
n 1 7
29.9.
xn
29.10.  n 1 .
n 1 n7
xn
29.11. 
.
n 1 n  2
xn
29.12.  n .
n 1 n7
xn
29.13. 
.
n 1 n  5
29.14.
nx n
29.16.  n .
n 1 2
nx n
29.17.  n .
n 1 5








n 1

29.22.
n
n
n 1
n 1 n
x .
n
n 1 8

xn .
 n  1
n 1
 nx n .

n!
n

29.20.


n
x .
1
5
n 1
n
xn .

xn
.

n 1
n 1 n  7



7
nx n
29.6.  n .
n 1 3



29.19.
2n x n
29.3. 
.
n 1 n  1

29.15.
xn
29.18.  n .
n 1 3


2n
xn .


n

n

1
n 1
29.21. 

3n
2n
.x n . 29.24.  n
29.23.  n
xn .
4


5

n

1
3

n
n 1
n 1
76

n
29.25.  n
x n . 29.26.
n 1 4  n  1
n  x n1
29.28.  n1 n .
3
n 1 2


6
n 1
5n
n
 n
2n  x n
29.27.  2
.
n 1 n  1

n
x .


xn
29.29. 
.
n 1 nn  1
xn
29.30. 
n 1 n 2n  1
30-topshiriq. f x  funksiyani Makloren qatoriga yoying.
30.1. f x   x 3  arctgx
30.2. f  x  
x2
30.3. f  x  
1 x
2x3
30.4. f  x   cos
3
30.5. f  x  
1
30.9. f  x  
30.8. f  x  
2
1
30.11. f x   e  x
30.12. f  x  
4


x 1
sin 3x
30.15. f  x  
x
30.17. f  x  
30.19. f  x   3
1
1  x3
30.10. f x   shx
ex
30.13. f x   ln
1
x4
30.6. f  x  
x2  1
30.7. f x   e  2 x
cos x
x2
1
x3  1
1
x2  1
cos x 2
x
30.14. f x   x cos x
cos x 4
30.16. f  x  
x2
30.18. f  x   3
1
x 1
cos 3 x
30.20. f  x  
x
77
30.21. f x   ln 3x  1
30.23. f  x   3
30.25. f  x  
1
5x  1
arctg 3x
x2
30.22. f  x  
cos x
x2
30.24. f x   arcsin x 2
2
30.26. f x   sin x
1 x
1 x
x
30.27. f x   xe
30.28. f x   ln
30.29. f x   ln(2  3x  x 2 )
2
30.30. f x   ln(1  x  x )
31-topshiriq. Integral ostidagi funksiyani
qatorga yoying, soʻngra uni
hadma – had intervallab 0,001 aniqlikda hisoblang.
1
31.1.  sin x dx .
2
0
1
31.2.  e
31.3.  xarctgxdx .
0
1
31.5.  cos xdx .
dx .
ln(1  x 2 )
31.4. 
dx .
x
0
0,5
0 ,1
31.6.  cos(100 x 2 )dx .
0
0
0,5
0,5
x
 xe dx .
31.8.
0
 arctgx
2
dx .
0
0,5
31.9.
3
0
0,5
31.7.
 x2
2
 x ln(1  x )dx .
0
0, 2
31.11.  cos(25 x 2 )dx .
0, 4
31.12.
0
1
sin x 2
0 x 2 dx .
0,5
31.10.
e
3 x 2
4
dx .
0
arctgxdx
31.13. 
.
x
0
ln(1  x)
dx .
x
0
0 ,1
2
31.14.
78

0,5
31.15.

1  x 3 dx .
0
3 x
 e dx .
2
31.18.
arctg ( x 2 )
31.19. 
dx .
x
0
x
 x ln(1 
1
2
31.20.

cos 3xdx .
 ln 1 
31.22.
0,5
x
x e dx .
31.24.
0
31.25.  sin x 2 dx.


 ln 1  x dx .
2
0
0, 5
31.29.

0


x dx .
x2
1
31.26.

e

x2
2 dx .
1
31.28.  cos
0
arctgx
dx .
x
dx. .
0
0
31.27.
.
arctg x 2
0
1
0,5
1 x3
0
0, 2

x )dx .
0, 25
2
0
31.23.
xdx
2
0
0,5
31.21.
4
0
0
1

1
0, 2
31.17.
1  cos x
dx .
x2
0
0,5
31.16.
0,5
31.30.

0
x2
dx. .
4
sin x 2
dx .
x
13. FURE QATORI
32-topshiriq. 32.1 – 32.20 misollarda berilgan f (x)
kesmada kosinuslar bo’yicha Fure qatoriga yoying.
32.1. f ( x)  x  2 .
32.2. f ( x)  1  2 x .
32.3. f ( x)  3x .
32.4. f ( x)  2 x  1 .
32.5. f ( x)  1  x .
32.6. f ( x)   x  1.
79
funksiyani [0, ]
32.7. f ( x)  2 x  1.
32.8. f ( x)    x .
32.9. f ( x)    2 x .
32.10. f ( x) 
32.11. f ( x)  3x  1 .
32.12. f ( x)   
32.13. f ( x)  2 x  3 .
32.14. f ( x)  7 x  1 .
32.15. f ( x)  x  2 .
32.16. f ( x)  x  1.
32.17. f ( x)  x 

2
32.19. f ( x)  8 x 
32.18. f ( x) 
.

2
.

2

2
 x.
1
x.
4
 2x .
32.20. f ( x)  3x  8 .
32.21 – 32.30 misollarda davriy f (x) funksiyani berilgan oraliq-davrda Fure
qatoriga yoying.
32.21.
f x   x  1,
  ,  .
32.22.
f x   x 2 ,
 1,1.
32.23.
2x
f x    ,
0
   x  0,
32.24.
32.25.
32.26.
0  x  .
 3  x  0,
 1
f x    ,
5
0  x  3.
 1  x  0,
x  1
f x   
,
0

0  x  1.
   x  0,
0
f x    ,
x
0  x  .
80
32.27.
32.28.
f x   x  1,
 1,1.
2
f x    ,
1
   x  0,
0  x  .
32.29.
f x   x ,
  ,  .
32.30.
f x   2  x,
 2, 2.
14. КOMLEKS SОNLAR
33-topshiriq. z1 va z 2 kompleks sonlar berilgan. Topish kerak:
1) z1 - komleks sonning moduli va argumentini;
2) z1  z2 ;
z1
;
z2
z1n -toping.
3) n z2 -toping.
33.1. n=3, z1  3  2i, z2  1  5i.
33.2. n=4, z1  5  2i, z2  3  i.
33.3. n=2, z1  6  8i, z2  8  6i.
33.4. n=5, z1  3  4i, z2  3  3i.
33.5. n=3, z1  2  3i, z2  4  3i.
33.6. n=7, z1  3  i, z2  3  i.
33.7. n=3, z1  2  2i, z2  5i.

33.8. n=2, z1  3  4i, z2  1  i.
2
33.9. n=5, z1  2  3i, z2  7  6i.
33.10. n=6, z1  4  3i, z2  6  2i.
81
1
2
33.11. n=7, z1  2i, z2  
1
2
33.12. n=3, z1  
3
i.
2
3
i, z2  i.
2

33.13. n=2, z1  2  3i, z2  i.
4
33.14. n=2, z1  1  i, z2  3  3i.
33.15. n=4, z1  5  3i, z2  3  4i.
33.16. n=3, z1  i, z2  5  6i.
33.17. n=5, z1  1  i, z2  1  i.
33.18. n=4, z1  2  3i, z2  7  5i.
33.19. n=6, z1  1  i, z2  6  2i.
33.20. n=2, z1  2  2i, z2  7  5i.
33.21. n=3, z1  13  i, z 2  7  6i.
33.22. n=4, z1  5  2i, z 2  2  5i.
33.23. n=3, z1  3  3i, z 2  2  4i.
33.24. n=5, z1  2  2i, z 2  1  3i.
33.25. n=6, z1 = 1 + i, z2 =  3  i .
33.26. n=5, z1  2  2 3 i , z 2   2  2i
33.27. n=4, z1  1  i , z 2  5  3i .
33.28. n=2, z1  9  7i , z 2  3  3i .
33.29. n=3, z1  11  2i , z 2  3  9i .
33.30. n=4, z1  1  2i , z 2  3  i .
82
O‘TILGAN MAVZULARNING O‘ZLASHTIRILISHINI
TEKSHIRISH UCHUN SAVOLLAR
Chiziqli algebra elementlari
1. Determinant deb nimaga aytiladi? Uning asosiy xossalarini keltiring.
2. Determinantning minori va algebraik toʻldiruvchilari deganda nimani
tushunasiz?
3. Determinantlarni hisoblash usullarini bilasizmi?
4. Matritsa deganda nimani tushunasiz? Matritsalar ustidagi chiziqli amallar
qanday bajariladi? Ularning asosiy xossalarini ayting.
5. Birlik matritsa deb qanday matritsaga aytiladi?
6. Teskari matritsa deb qanday matritsaga aytiladi va u qanday topiladi?
7. Chiziqli tenglamalar sistemasining yechimlari deganda nimani tushunasiz?
8. Tenglamalar sistemasini yechishdagi Kramer formulasi va uni qanday
hollarda qoʻllab boʻladi?
9. Qanday shart bajarilganda chiziqli tenglamalar sistemasi yagona yechimga
ega boʻladi?
10. Agar asosiy determinant 0 ga teng boʻlsa, chiziqli tenglamalar sistemasi
haqida nima deyish mumkin?
11. Qanday shart bajarilganda bir jinsli tenglamalar sistemasi noldan farqli
yechimga ega boʻladi?
12. Chiziqli tenglamalar sistemasini yechishda Gauss usulining ma’nosi
nimadan iborat?
13. Tenglamalar sistemasini matritsa usuli bilan yechish.
Tekislikdagi analitik geometriya
1. Chiziqning tenglamasini qanday tuzish mumkin?
2. Toʻgʻri chiziqning burchak koeffitsiyenti deb nimaga aytiladi?
3. Toʻgʻri chiziqning burchak koeffitsiyentli va umumiy tenglamalarini
yozing.
140
4. Toʻgʻri chiziqning kesmalardagi tenglamasi qanday koʻrinishda boʻladi?
5. Toʻgʻri chiziqlar dastasining tenglamasi. Ikki toʻgʻri chiziq orasidagi
burchaklar bissektrisalarining tenglamalarini yozing.
6. Ikki nuqtadan oʻtuvchi toʻgʻri chiziq tenglamasini qanday hosil qilasiz?
7. Toʻgʻri chiziqning normal tenglamasini va umumiy tenglamasini normal
koʻrinishga qanday keltiriladi?
8. Berilgan nuqtadan toʻgʻri chiziqqacha boʻlgan masofa qanday aniqlanadi?
9. Ikki toʻgʻri chiziq orasidagi burchak qanday hisoblanadi?
10. Aylana deb qanday egri chiziqqa aytiladi? Uning tenglamalarini yozing.
11. Ellips deb qanday egri chiziqqa aytiladi? Ellipsning fokuslari va
ekssentrisiteti qanday aniqlanadi?
12. Giperbola deb qanday nuqtalarning geometrik oʻrniga aytiladi?
13. Parabola deb qanday nuqtalarning geometrik oʻrniga aytiladi?
14. Ikkinchi tartibli egri chiziqlarning qutb koordinatalaridagi tenglamalarini
yozing.
Vektorlar algebrasi
1. Vektor va uning moduli deb nimaga aytiladi?
2. Qanday vektorlarga kollinear, komplanar, teng vektorlar deyiladi?
3. Modullari teng boʻlgan ikki vektor oʻzaro teng boʻlmasligi mumkinmi?
Agar teng boʻlmasa, farqi nimada?
4. Vektorlar ustida qanday algebraik amallar bajarish mumkin? Nol vektor
deb qanday vektorga aytiladi? Vektorlar ustida kiritilgan amallar uchun qanday
xossalar oʻrinli?
5. Tekislikda, fazoda basis deb qanday vektorlarga aytiladi? Qanday basis
ortonormal basis deyiladi?
6. Qanday vektorlarga chiziqli bogʻliq vektorlar deyiladi?
7. Dekart koordinatalar sistemasi qanday tanlanadi?
8. Vektorning komponentalari, uning boshlangʻich va oxirgi nuqtalarining
koordinatalari orqali qanday ifodalanadi?
9. Kesmani berilgan nisbatda boʻlishni koʻrsating.
141
10. Uchburchak ogʻirlik markazining koordinatalarini uning uchlarining
koordinatalari orqali ifodalang.
11. Nuqtaning va kesmaning oʻqdagi proyeksiyasi deb nimaga aytiladi?
12. Ikki vektorning skalyar koʻpaytmasi deb nimaga aytiladi? Uning xossalari.
Proyeksiyalari bilan berilgan ikki vektorning skalyar koʻpaytmasini qanday topasiz?
13. Vektorning uzunligini skalyar koʻpaytma orqali ifodalang.
14. Ikki vektorning vektor koʻpaytmasi deb nimaga aytiladi? Uning xossalari
va berilgan vektorlarning proyeksiyalari orqali ifodasi.
15. Uchta vektorning aralash koʻpaytmasi deb nimaga aytiladi? Uning
xossalari va geometrik ma’nosini aytib bering.
16. Uchta vektorning komplanarlik shartini ifodalang.
Fazodagi analitik geometriya
1. Qanday parametrlar berilganda fazoda tekislikning oʻrni aniqlangan
boʻladi?
2. Tekislik tenglamalarini (normal, umumiy, kesmalar boʻyicha; berilgan bitta
nuqtadan, uchta nuqtadan oʻtuvchi) yozing.
3. Ikki tekislik orasidagi burchakni qanday aniqlaysiz? Ikki tekislikning
parallellik va perpendikulyarlik shartlarini yozing.
4. Berilgan nuqtadan berilgan tekislikkacha boʻlgan masofa qanday topiladi?
5. Fazoda ikki tekislik kesishish chizigʻidan oʻtuvchi tekisliklar dastasining
tenglamasini yozing. Toʻgʻri chiziqning proyeksiyalar boʻyicha tenglamalarini
yozing.
6. Toʻgʻri chiziqning yoʻnaltiruvchi vektori deb qanday vektorga aytiladi?
Toʻgʻri chiziqning kanonik va parametrik tenglamalarini yozing. Berilgan ikki
nuqtadan oʻtuvchi toʻgʻri chiziq tenglamasini yozing.
7. Toʻgʻri chiziq bilan tekislik orasidagi burchak deb qanday burchakka
aytiladi va u qanday aniqlanadi? Toʻgʻri chiziq va tekislikning parallellik va
perpendikulyarlik shartlarini yozing.
8. Toʻgʻri chiziq bilan tekislikning kesishish nuqtasini qanday topasiz?
9. Ikki toʻgʻri chiziqning bir tekislikda yotish shartini yozing.
142
10. Sfera tenglamasini yozing.
11. Yasovchisi Oz oʻqiga parallel silindrik sirt tenglamasini yozing.
12. Aylanish sirtini qanday hosil qilasiz? Konus sirtlar tenglamasini yozing.
Matematik analiz
1. Funksiyaga taʻrif bering. Funksiyaning aniqlanish sohasi deb nimaga
aytiladi?
2. Qanday funksiyaga davriy funksiya deyiladi? Misol bilan tushuntiring.
Monoton funksiyalar, chegaralangan, chegaralanmagan funksiyalar.
3. Murakkab funksiya deb qanday funksiyaga aytiladi?
4. Qanday funksiyalarga elementar funksiyalar deyiladi?
5. Funksiyaning limiti deb nimaga aytiladi?
6. Funksiyaning chap va oʻng limiti deganda nimani tushunasiz?
7. Chegaralangan funksiya taʻrifini ayting. Qanday funksiyalar cheksiz kichik,
qanday funksiyalar cheksiz katta deyiladi?
8. Funksiya limiti haqidagi asosiy teoremalarni ayting va birini isbotlang.
9. Birinchi ajoyib limitni isbotlang.
10. 𝑒 soni. (Ikkinchi ajoyib limit).
11. Funksiyaning nuqtada uzluksizligini ta’riflang.
12. Kesmada uzluksiz funksiyalar xossalarini ayting.
13. Uzilish nuqtasi deb qanday nuqtaga aytiladi?
14. Cheksiz kichik miqdorga ta’rif bering, misol keltiring.
15. Funksiya hosilasi ta’rifini ayting. Uning fizik va geometrik ma’nosi
nimadan iborat?
16. Yigʻindi, koʻpaytma va boʻlinmaning hosilalari qanday topiladi? Misol
keltiring.
17. Murakkab funksiyaning hosilasi qanday topiladi?
18. Trigonometrik va logarifmik funksiyalarning hosilasi qanday topiladi?
19. Darajali va koʻrsatkichli funksiyalar hosilasi. Murakkab koʻrsatkichli
funksiyaga misol keltiring.
143
20. Teskari funksiya va teskari trigonometrik funksiyalar hosilasini qanday
topasiz? Parametrik tenglamalari bilan berilgan funksiya hosilasini qanday topasiz?
21. Funksiya differensiali deb nimaga aytiladi? Uning ma’nosi nimadan
iborat?
22. Yuqori tartibli hosila va differensialni qanday topasiz?
23. Roll teoremasini isbotlang. Uning geometrik ma’nosi nimadan iborat?
24. Lagranj teoremasining geometrik ma’nosini tushuntiring.
25. Qanday koʻrinishdagi aniqmasliklar uchun Lopital qoidasi qoʻllaniladi?
Misollar keltiring.
Funksiyani hosila yordamida tekshirish
1. Oʻsuvchi funksiya hosilasi kesmada musbat boʻlishini tushuntiring.
2. Funksiya ekstremumining zaruriy sharti nimadan iborat?
3. Funksiyani birinchi va ikkinchi tartibli hosilalar yordamida ekstremumga
tekshirishni koʻrsating.
4. Funksiya grafigining qavariq yoki botiqligini ikkinchi tartibli hosila
yordamida izohlang.
5. 𝑦 = 𝑓(𝑥) tenglama berilgan chiziq uchun vertikal va ogʻma asimptotalar
qanday aniqlanadi?
6. Funksiyani toʻla tekshirish sxemasi va grafigini chizishni bayon qiling.
Aniqmas integral
1. Boshlangʻich funksiya deb qanday funksiyaga aytiladi? Misol keltiring.
2. Biror funksiyaning aniqmas integrali deb nimaga aytiladi? Uning geometrik
ma’nosi.
3. Aniqmas integralning hosilasi va u nimaga teng? Misollar keltiring.
4. Asosiy integrallar jadvalini yozing.
5. Aniqmas integralning xossalari.
6. Aniqmas integralni oʻzgaruvchini almashtirish yoki oʻrniga qoʻyish usuli
bilan integrallash qanday bajariladi?
7. Boʻlaklab integrallash usuli formulasini yozing. Qaysi turdagi integrallarni
boʻlaklab integrallash qulaylik tugʻdiradi?
144
8. Kvadrat uchhad qatnashgan funksiyalar qanday integrallanadi?
9. Eng sodda ratsional kasrlarning birinchi, ikkinchi va uchinchi turlarini
integrallash qanday bajariladi va qanday funksiyalarni beradi?
10. Ratsional kasr maxrajining ildizi haqiqiy karrali va kompleks boʻlganda
qanday eng sodda kasrlar yigʻindisi etib yoziladi?
11. Trigonometrik funksiyalarni integrallash qanday usul bilan ratsional
funksiyalarni integrallashga keltiriladi? Misol keltiring.
12. Irratsional funksiyalar qanday integrallanadi?
Aniq integral
1. Quyi va yuqori integral yigʻindilar deb qanday yigʻindiga aytiladi?
2. [𝑎; 𝑏] kemada funksiyaning aniq integrali deb nimaga aytiladi? Aniq
integralning geometrik ma’nosini izohlang.
3. Aniq integralning xossalarini ayting.
4. [𝑎; 𝑏] kesmada juft va toq funksiyalarning integrali. Misol keltiring.
5. Aniq integralni hisoblash. Nyuton-Leybnis formulasini yozing.
6. Aniq integralda oʻzgaruvchini almashtirish qanday bajariladi?
7. Aniq integralni boʻlaklab integrallash formulasini yozing.
8. Aniq integralni taqiribiy hisoblash formulalarini yozing.
9. Jismning hajmini parallel kesimlar yuzlari boʻyicha qanday hisoblash
mumkin? Aylanish jismining hajmini-chi?
10. Aylanish jismining sirtini hisoblash formulasini yozing.
11. Aniq integral yordamida ishni qanday hisoblaysiz?
12. Tekis shaklning ogʻirlik markazi qanday aniqlanadi?
13. Qanday integralga xosmas integral deyiladi? Qachon xosmas integral
mavjud yoki yaqinlashuvchi deyiladi?
Koʻp oʻzgaruvchili funksiya
1. Koʻp oʻzgaruvchili funksiyaning berilish usullari.
2. Koʻp oʻzgaruvchili funksiyaning aniqlanish sohasi deb nimaga aytiladi?
Ochiq va yopiq sohaga misollar keltiring.
145
3. Skalyar maydonning sath chiziqlari deb nimaga aytiladi? Skalyar matdonda
funksiya grafigi ma’lum boʻlsa, sath chiziqlarini qanday hosil qilasiz? Sath chiziqlari
kesishadimi?
4. Qanday shart bajarilganda 𝑀0 (𝑥0 ; 𝑦0 ) nuqtada 𝑧 = 𝑓(𝑥, 𝑦) funksiya
uzluksiz deyiladi?
5. 𝑧 = 𝑓(𝑥, 𝑦) funksiyaning xususiy hosilalari qanday topiladi? Geometrik
ma’nosi.
6. Qachon 𝑧 = 𝑓(𝑥, 𝑦) funksiya berilgan nuqtada differensiallanuvchi
deyiladi? Berilgan nuqtada funksiyaning toʻliq differensiali deb nimaga aytiladi?
Toʻliq differensial taqribiy hisoblashda qanday qoʻllaniladi?
7. Murakkab 𝑧 = 𝑓(𝑢, 𝑣), 𝑢 = 𝜑(𝑥, 𝑦), 𝜗 = 𝜃(𝑦, 𝑥) funksiyaning xususiy
hosilalari qanday topiladi?
8. 𝑧 = 𝑓(𝑢, 𝑣), 𝑢 = 𝑢(𝑥), 𝑣 = 𝑣(𝑥) boʻlganda hosilani qanday topasiz?
9. Funksiya 𝐹(𝑥, 𝑦) = 0 tenglama bilan oshkormas shaklda berilganda hosila
qanday topialdi?
10. Yuqori tartibli xususiy hosilalar qanday topiladi? Ikki oʻzgaruvchili
funksiyaning aralash hosilalari.
11. 𝑢 = 𝑢(𝑥, 𝑦, 𝑧) funksiyaning nuqtada vektor yoʻnalishi boʻyicha hosilasi
deb nimaga aytiladi?
12. Berilgan 𝑀(𝑥, 𝑦, 𝑧) nuqtada 𝑢 = 𝑢(𝑥, 𝑦, 𝑧) skalyar maydonning gradient
deb nimaga aytiladi?
13. Ikki oʻzgaruvchili funksiya ekstremumga ega boʻlishining zaruriy va
yetarli shartlari nimadan iborat? Minimaks yoki ekstremum nuqtasi deb qanday
nuqtaga aytiladi?
14. 𝑧 = 𝑓(𝑥, 𝑦) funksiyaning shartli ekstremumi deb nimaga aytiladi va u
qanday topiladi?
Differensial tenglamalar
1. Differensial tenglama deb qanday tenglamaga aytiladi? Birinchi tartibli
differensial tenglama umumiy koʻrinishda qanday yoziladi?
146
2. Differensial tenglamaning yechimi deb nimaga aytiladi? Integral egri chiziq
nimani bildiradi?
3. Birinchi tartibli differensial tenglamaning umumiy yechimi deb nimga
aytiladi? Qanday qilib umumiy yechimdan xususiy yechim topiladi?
4. Boshlangʻich shart nimani bildiradi va uning geometrik maʻnosi nimadan
iborat? Koshi masalalasi nimadan iborat va qanday yechiladi?
5. Qanday birinchi tartbili differensial tenglamalar oʻzgaruvchilari ajralgan va
ajralmagan differensial tenglamalar deyiladi?
6. Qachon funksiya 𝑥 va 𝑦 oʻzgaruvchilarga nisbatan oʻlchovli bir jinlsi
funksiya deyiladi? Misol keltiring.
7. Bir jinsli differensial tenglama va uni yechish usuli.
8. Birinchi tartibli chiziqli differensial tenglamalar va ularni yechish usuli.
9. Bernulli tenglamasi qanday yechiladi?
10. Toʻliq differensialli tenglama va uni yechish usuli.
11. Tartibini pasaytirish mumkin boʻlgan 𝑦 (𝑛) = 𝑓(𝑥) tenglamaning yechimi
qanday topiladi?
12. Noma’lum 𝑦 funksiyani oshkor holda oʻz ichiga olmagan ikkinchi tartibli
differensial tenglama qanday yechiladi?
13. 𝑥 erkli oʻzgaruvchini oshkor holda oʻz ichiga olmagan ikkinchi tartibli
differensial tenglama yechimi qanday aniqlanadi?
14. Ikkinchi kosmik tezlik haqidagi masala qanday yechiladi?
15. Ikkinchi tartibli chiziqli differensial tenglamaning umumiy koʻrinishi (bir
jinsli boʻlmagan va bir jinsli boʻlgan).
16. Qachon ikkita funksiya oʻzaro chiziqli bogʻliq va qachon chziqli bogʻliq
boʻlmagan funksiyalar deyiladi?
17. Ikkinchi tartibli chiziqli differensial tenglamaning umumiy yechimi
qanday aniqlanadi?
18. Oʻzgarmas koeffitsiyentli ikkinchi tartibli chiziqli bir jinsli differensial
tenglamaning umumiy koʻrinishini yozing. Uning xarakteristik tenglamasi deb
qanday tenglamaga aytiladi?
147
19. Oʻzgarmas koeffitsiyentli ikkinchi tartibli, chiziqli, bir jinsli differensial
tenglamaning umumiy yechimini yozing:
1) xarakteristik tenglamaning yechimlari – a) haqiqiy, b) kompleks son
boʻlganda;
2) xarakteristik tenglamaning yechimlari haqiqiy karrali boʻlgan holda.
20. Chiziqli, bir jinsli boʻlmagan ikkinchi tartibli differensial tenglamaning
umumiy yechimi nimadan iborat?
21. Bir jinsli boʻlmagan differensial tenglamaning xususiy yechimini
topishning aniqmas koeffitsiyentlar usuli va ixtiyoriy oʻzgarmasni variatsiyalash
usuli nimadan iborat?
22. Erkin va majburiy tebranishlar tenglamasini yozing.
23. Oʻzgarmas koeffitsiyentli, chiziqli differensial tenglamalar sistemasining
xarakteristik tenglamasi qanday tuziladi?
24. Differensial tenglamalarning normal sistemasi va uni integrallash qanday
bajariladi?
Qatorlar
1. Sonli qator deb nimaga aytiladi?
2. Qatorning xususiy yigʻindisi deb qanday yigʻindiga aytiladi?
3. Qatorning yigʻindisi deb nimaga aytiladi?
4. Sonli qatorning yaqinlashuvchiligi va uzoqlashuvchiligi deganda nimani
tushunasiz?
5. Yaqinlashuvchi qatorning xossalarini ayting.
6. Qator yaqinlashuvchiligining zaruriy sharti nimadan iborat?
7. Garmonik qator deb nimaga aytiladi?
8. Musbat hadli qatorlarni taqqoslash deganda nimani tushunasiz?
9. Musbat hadli qatorlar uchun Dalamber alomati nimadan iborat?
10. Musbat hadli qatorlar uchun Koshi alomati qanday?
11. Musbat hadli qatorlar uchun Koshining integral alomati nimadan iborat?
12. Ishoralari almashinuvchi qator deb qanday qatorga aytiladi?
13. Leybnis teoremasini isbotlang.
148
14. Oʻzgaruvchan ishorali qator deb qanday qatorga aytiladi?
15. Qanday qatorga absolyut va shartli yaqinlashuvchi qator deyiladi?
16. Funksional qator deb qanday qatorga aytiladi?
17. Funksional qatorning yaqinlashish sohasi qanday aniqlanadi?
18. Funksional qatorning yigʻindisi va uning qoldiq hadi nimadan iborat?
19. Funksional qatorning kesmada tekis yaqinlashish shartlari nimadan
iborat?
20. Qator yigʻindisining uzluksizligini ayting.
21. Qanday funksional qatorni integrallash va differensiallash mumkin?
22. Darajali qator deb qanday qatorga aytiladi?
23. Teylor qatorini yozing.
24. Makloren qatorini yozing.
25. sin 𝑥 , cos 𝑥 , 𝑒 𝑥 funksiyalarni Makloren qatoriga yoying.
26. Aniq integrallarni qator yordamida hisoblang.
27. Differensial tenglamalarni qator yordamida yechish.
28. Furye qatori (trigonometrik qator) deb qanday qatorga aytiladi?
29. Furye koeffitsiyentlarini yozing.
30. Juft va toq funksiyalarni Furye qatoriga yoyilmasini yozing.
31. Davri 2𝑙 boʻlgan funksiyalar uchun Furye qatorini yozing.
32. Davriy boʻlmagan funksiyalarning Furye qatoriga yoyilmasini yozing.
Kompleks sonlar
1. Qanday ifodaga kompleks son deyiladi?
2. Kompleks sonning trigonometrik shaklini yozing. Uning moduli va
argumenti deb nimaga aytiladi?
3. Kompleks sonlar ustida qoʻshish, ayirish, koʻpaytirish va ildiz chiqarish
amallari qanday bajariladi? Muavr formulasini yozing. Misol keltirinng.
4. Haqiqiy sonni trigonometrik shaklda qanday tasvirlash mumkin?
149
JAVOBLAR
2.1. ( -2;1;-1 ).
2.2. (-1; 2. 2 ).
2.9.(3; -1; 2).
2.10.(-2; 3; -4).
2.14.(2; -1; -3).
2.15.(1; -1; 2).
2.12.(1; 1; 1). 2.13.(2; 3; 1).
2
3
4
3
3
5
5
5

. ln
.
a
b
13.13. 0 .
13.26.
.
1
5 cos x
.

16.6.
 2 ctg x  1  C .
1
arctg 4 x  C .
4
.

1
ln e 3 x  a 2  C .
3
16.23. 
1
16( x  3) 8
2
13.16. e . 13.17. 1 . 13.18. 1 . 13.19
2
13.22. e . 13.23.
a
b
. 13.24. e 2 .
13.25. 2 .

5
 x4  2
16
16.21.
C.

4
5
C .
16.7. 3  3 sin x  C .
3
2
arcsin x  2  C .
3

16.18.
3
e
1 x2
e  C . 16.11. 2e
2
1
ln x 2 10  C .
2
16.17.

16.10.

16.3.
1 8
tg x  C .
8
16.14.
a
1
. 13.4. . 13.5.
b
3
13.9. 1 . 13.10. 0 . 13.11.  . 13.12.
2
9
1 3
x 5 C.
27
C.
2.27.(2; 3; 5). 2.28.(1; 0; 2).
1
2
13.28. 1 . 13.29. a . 13.30. . 6 . 16.1.
x3  5  C
13.27. 1 .

3 a
13.15. 1 .
1
3
16.2.
5
13.14. 1 .
2.23.( 0; 1; -2).
. 13.2. 0. 13.3.
6
m mn
.
a
n
. 13.20.  . 13.21. 1.
2

e
2
2.30.(2; 1; 0 ). 13.1.
13.6. 2 . 13.7. 1 . 13.8.
2.19.(1; 2; 3).
2.22.(8; 4; 2).
). 2.25.(2; -2; 0). 2.26.(1; 0; -1).
2.29.(3; -2; -5).
4a 2
2.18.(1; 0; 2).
2.21.(2; -2; 3).
2
2.24.( − , ;
2.
2.17.(-1; 0; 1).
; 2; ).
3
2.8.( 1; -2;1).
2.11.(1; 1; 1).
1
3
2.5.( ; 1; ).
2.7.( 3; 0; -1).
2.20.(
1
2.4.( 3; 1;-1).
2.6.(-1; 2; 2).
2.16. (5; 6; 10).
1
2.3.( 2;-1;1 ).
x
C.
16.15. 

1
ln 2 x  4 x  1  C .
ln 2
16.24.  ln cos2 x  3  C .
150
1 6
s in x  C .
6
16.8. 
16.12.
1
4 arcsin
1
ln x 3  5  C .
3

16.4.
4
x
1
C.
2 tg 2 x
1 7
ln x  C .
7
16.5.
16.9.
16.13.
 C . 16.16. sin e x  C
16.19. ln e x  4  C .
16.22.

16.20.

1
ln x 4  1  x 8  C .
4
16.25. ln ln x  C .
16.26.
1 4
ln x  C .
4
16.27.  e cos x  C . 16.28.  1 e 5 x  C . 16.29.  1 e  x  C . 16.30.
5
5


ln e x  1  e 2 x  C .
76
.
3
19.1.
1
е
19.2. e  1 .
4
3
5
19.3.
2
e
27
.
4
19.6. е   2 . 19.7. 2  . 19.8. 2e  . 19.9.
. 19.12. 2 . 19.13.

.
8
1
e
19.14. 6 . 19.25.
1
2
3
2
19.29. 12 .
sin x
.
2 cos x  C
25.22.a)
19.27. e  . 19.28. 1  ln .
.
yx n  ax  C ;
y  Ce x 
y2 
c) y 
5
.
4
19.4. 13.
9
.
2
19.10.
8  13

13 1 .

27  8

9
.
2
19.5.
19.11. 6
19.26.  lntg

8
19.30. 25.21.а) cos y  C cos x ; b)
2 y  ln y  2 x  C ;
b)


1
cos x  sin x  ; c) x2  C 2  2Cy . 25.23. a) x 2  4  C ; b) y  xn e x  C ; c)
2
1 2y
1
1  Ce x
2
.
25.24. a)
1
 C ( y 2  2) ;
2
x 1
c)
ln Cx  e

y
x
.
25.25. a)
2
2e x
y3  9e x  C .
1  y  arcsin x  C ; b) y 
2 ; c)
Cx
25.26. a) y 2  ln( x2  3)  C ; b)
3
2
y  2 x  Cx3 ( y  x) . 25.27.a)
x 1  y 2   y 1  x2  C ; b) 2 y  x  1  C x  1 ;
4
1

e x
1
2
2

x
. 25.28.a) y  x 1  Ce  ; b) y  x  C  1  x ; c) y 
y  3
Cx
Cx  x 2



2
y 2  2 xy   x2  C ; b) arctg
25.30. b) y   x 2 ln Cx


y
 ln C x 2  y 2 ;
x
.
151

c) y  1
2
c)
25.29. a)
ln Ctg
cos x
x
2
.
ILOVALAR
Ilova А. Ba’zi elementar funksiyalar.
Xususiy hollar:
Xususiy hollar:
x  ex
x  (,)
ln x  x ( x  0)
y  sin x, x  (,)
y  cos x, x  (,)



y  tg x  x   n, n  N 
2


y  ctg x x  n, n  N 
152
y  arcsin x,
x   1,1
y  arctg x, x  (,)
y  arccos x,
x   1,1
y  arc ctg x, x  (,)
arcsin x  arccos x 

arctg x  arcctg x 
2

2
Ilova В. Ba’zi trigonometrik formulalar.
cos 2   sin 2   1
sin 2  
1  cos 2 x
2
1  tg 2  
cos 2  
1
cos 2 
1  cos 2
2
153
1  ctg 2  
1
.
sin 2 
1
sin   cos   sin 2 .
2
sin   cos  
1
sin      sin    
2
cos   cos  
1
cos     cos   
2
sin   sin  
1
cos     cos   
2
sin(    )  sin  cos   sin  cos 
cos(   )  cos  cos   sin  sin 
Ilova C. Logarifmlarning ba’zi xossalari.
ln x
1) e  x
2) a
3) ln x  ln y  ln xy
4) ln x  ln y  ln
log a x
x
x
y
5) n ln x  ln x n
Ilova D. Daraja koʻrsatkichlar.
1) x  x  x
n
k
 
3) x n
k
xn
2) k  x n  k
x
 xnk
xn  x 
5) n   
y
 y
7)
nk
1
x
xn
4) x n  y n  ( x y) n
n
6)
n
x
1
xn
n
Ilova E. Asosiy elementar funksiyalar hosilalari jadvali.
1.
(c)' = 0
2.
(x)' = x – 1
3.
(ax)' = axln a, (a > 0, a ≠ 1)
4.
(ex)' = ex
154
5.
1
(loga x)' =
, (a > 0; a ≠ 1)
x ln a
6.
(ln x)' =
7.
(sin x)' =cos x
8.
(cos x)' = – sin x
9.
(tg x)' =
1
x
1
cos 2 x
1
sin 2 x
10. (ctg x)' = –
11. (arcsin x)' =
1
1  x2
12. (arccos x)' = –
13. (arctg x)' =
1
1  x2
1
1  x2
14. (arcctg x)' = 
1
1 x2
Ilova I. Asosiy integrallar jadvali.
x α 1
 C , α  1 .
1.  x dx 
α 1
2. 
ax
C.
3.  a dx 
ln a
x
x
4.  e dx  e  C .
5.  sin xdx   cos x  C .
5.  cos xdx  sin x  C .
α
x
6.
8.
dx
 cos 2 x  tg x  C .
7.
dx
 ln | x |  C .
x
dx
 sin 2 x  ctg x  C .
dx
1
x

arct
g
 C, a  0 .
 a2  x2 a
a
155
9.

10. 
11. 
dx
a2  x2
dx
x2  a2
dx
x2  a2
 arcsin

x
 C , a  0,  a  x  a .
a
1
xa
ln
 C , a  0; x  a .
2a x  a
 ln x  x 2  a 2  C .
156
Foydalanilgan adabiyotlar
1.Пискунов Н.C. Дифференциальное и интегральное исчисление. Т., I, II.
– М., 1973.
2.Минорский В.П. Сборник задач по высшей математике. – М., «Наука»,
1969.
3.Привалов И.И. Аналитическая геометрия. «Наука», – М., 1966.
4. Задачи и упражнения по математическому анализу. Под ред.
Б.П.Демидовича. – М., 1974.
5.Файзибоев Э.Ф., Цирмиракс Н.М. Интеграл ҳисоб курсидан амалий
машғулотлар. Тошкент, “Ўқитувчи”, 1982.
6.Хudoyorov B.A. Oliy matematikadan hisob-grafik ishlarini bajarish uchun
uslubiy korsatma. 2012. 65 b.
7.Жураев Т. ва бошқ. Олий математика асослари. “Ўзбекистон”,
Тошкент, 1994.
8.Fayziboyev E., Suleymenov Z.I., Xudoyorov B.A. Oliy matematikadan
misol va masalalar to‘plami. Toshkent, “Oʻqituvchi”, 2005.
9. Гусак А.А. Сборник задач и упражнений по высшей математике.
Минск, «Вышэйшая школа», 1967.
10. Сборник задач по математике для втузов. Линейная алгебра и основы
математического анализа. Под ред. А.Ф.Ефимова, Б.П.Демидовича. – М.,
«Наука», 1974.
11. Берман Г.Н. Сборник задач по курсу математического анализа. – М.,
«Наука», 1965.
157
MUNDARIJA
SO‘ZBOSHI …………………………………………………………….. 3
1.
OLIY ALGEBRA ……………………………………………….
5
1-topshiriq………………………………………………………….. 5
2-Topshiriq ………………………………………………………… 7
3-topshiriq …………………………………………………………. 10
4-topshiriq………………………………………………………….. 12
2.
TEKISLIKDAGI ANALITIK GEOMETRIYA ……………….. 15
5-topshiriq………………………………………………………….. 15
3.
VEKTORLAR ALGEBRASI……………………………………. 16
6-topshiriq …………………………………………………………. 16
7-topshiriq …………………………………………………………. 18
4.
IKKINCHI TARTIBLI EGRI CHIZIQLAR …………………..
21
8-topshiriq …………………………………………………………. 21
9-topshiriq …………………………………………………………. 24
5.
6.
7.
8.
FAZODAGI ANALITIK GEOMETRIYA ……………………..
27
10-topshiriq ……………………………………………………….
27
FUNKSIYA LIMITI ……………………………………………..
29
11-topshiriq ………………………………………………………..
FUNKSIYA HOSILASI …………………………………………
29
34
12-topshiriq ………………………………………………………..
13-topshiriq ………………………………………………………..
34
40
HOSILA YORDAMIDA FUNKSIYANI TEKSHIRISH ……. 42
14-topshiriq ………………………………………………………..
9.
42
ANIQMAS VA ANIQ INTEGRAL …………………………… 44
15-topshiriq ………………………………………………………..
16-topshiriq ………………………………………………………..
17-topshiriq ………………………………………………………..
18-topshiriq ………………………………………………………..
19-topshiriq ………………………………………………………..
20-topshiriq ………………………………………………………..
21-topshiriq ………………………………………………………..
158
44
46
47
52
55
57
59
22-topshiriq ………………………………………………………..
10.
KOʻP ARGUMENTLI FUNKSIYALAR ………………………. 62
23-topshiriq ………………………………………………………..
24-topshiriq ………………………………………………………..
11.
60
62
63
DIFFERENSIAL TENGLAMALAR …………………………… 65
25-topshiriq ………………………………………………………... 65
26-topshiriq ………………………………………………………... 69
12.
SONLI VA DARAJALI QATORLAR………………………… 71
27-topshiriq…………………………………………………………
28-topshiriq…………………………………………………………
29-topshiriq ………………………………………………………..
30-topshiriq
31-topshiriq…………………………………………………………
13.
71
73
76
77
78
FURE QATORI…………………………………………………… 79
32-topshiriq………………………………………………………… 79
14.
KOMPLEKS SONLAR …………………………………………. 81
33-topshiriq…………………………………………………………
Matematikadan misollarni yechish namunalari…………………….
O‘TILGAN MAVZULARNING O‘ZLASHTIRILISHINI
TEKSHIRISH UCHUN SAVOLLAR …………………………….
Javoblar…………………………………………………………….
ILOVALAR ……………………………………………………….
Foydalanilgan adabiyotlar ………………………………………….
159
81
82
140
150
152
157
ОГЛАВЛЕНИЕ
ПРЕДИСЛОВИЕ ……………………………………………………… 3
1.
2.
3.
4.
5.
ВЫСШАЯ АЛГЕБРА …………………………………………..
5
Задание 1 ………………………………………………………….
Задание 2…………………………………………………………..
5
7
Задание 3…………………………………………………………..
Задание 4…………………………………………………………..
10
12
АНАЛИТИЧЕСКАЯ ГЕОМЕТРИЯ НА ПЛОСКОСТИ …..
15
Задание 5………………………………………………………….
15
ВЕКТОРНАЯ АЛГЕБРА ………………………………………
16
Задание 6………………………………………………………….
Задание 7………………………………………………………….
16
18
КРИВЫЕ ВТОРОГО ПОРЯДКА ……………………………
21
Задание 8………………………………………………………….
Задание 9………………………………………………………….
21
24
АНАЛИТИЧЕСКАЯ ГЕОМЕТРИЯ В ПРОСТРАНСТВЕ
27
Задание 10…………………………………………………………. 27
6.
ПРЕДЕЛЫ ФУНКЦИЙ ………………………………………...
7.
Задание 11…………………………………………………………. 29
ПРОИЗВОДНАЯ ФУНКЦИИ ………………………………… 34
29
Задание 12…………………………………………………………. 34
Задание 13…………………………………………………………. 40
8.
ИССЛЕДОВАНИЕ ФУНКЦИЙ С ПОМОЩЬЮ
ПРОИЗВОДНОЙ ………………………………………………..
42
Задание 14…………………………………………………………. 42
9.
НЕОПРЕДЕЛЕННЫЙ И ОПРЕДЕЛЕННЫЙ ИНТЕГРАЛ
44
Задание 15………………………………………………………….
Задание 16………………………………………………………….
Задание 17………………………………………………………….
Задание 18………………………………………………………….
Задание 19………………………………………………………….
44
46
47
52
55
160
Задание 20…………………………………………………………. 57
Задание 21…………………………………………………………. 59
Задание 22…………………………………………………………. 60
10.
ФУНКЦИИ МНОГИХ ПЕРЕМЕННЫХ …………………… 62
Задание 23…………………………………………………………. 62
Задание 24…………………………………………………………. 63
11.
ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ …………………… 65
Задание 25…………………………………………………………. 65
Задание 26…………………………………………………………. 69
12.
13.
ЧИСЛОВЫЕ И СТЕПЕННЫЕ РЯДЫ ………………………
71
Задание 27………………………………………………………….
Задание 28………………………………………………………….
Задание 29………………………………………………………….
Задание 30………………………………………………………….
Задание 31………………………………………………………….
71
73
76
77
78
РЯДЫ ФУРЬЕ …………………………………………………… 79
Задание 32…………………………………………………………. 79
14.
КОМПЛЕКСНЫЕ ЧИСЛА ……………………………………. 81
Задание 33………………………………………………………….
Образцы решения заданий по математике …………...…………
ВОПРОСЫ ДЛЯ САМОПРОВЕРКИ…………………………….
Ответы …………………………………………………………….
ПРИЛОЖЕНИЯ ………………………………………….……….
Использованная литература …………………………….………..
161
81
82
140
150
152
157
CONTENTS
PREFACE ………………………………………………………………. 3
1.
HIGHER ALGEBRA …………………………………………….
5
Task 1………………………………………………………………. 5
Task 2………………………………………………………………. 7
Task 3………………………………………………………………. 10
Task 4………………………………………………………………. 12
2.
PLANE ANALYTICAL GEOMETRY………………………….
15
Task 5………………………………………………………………. 15
3.
VECTOR ALGEBRA …………………………………………...
16
Task 6………………………………………………………………. 16
Task 7………………………………………………………………. 18
4.
SECOND-ORDER CURVES…………………………………….
21
Task 8………………………………………………………………. 21
Task 9………………………………………………………………. 24
5.
6.
7.
8.
9.
ANALYTICAL GEOMETRY IN SPACE ……………………...
27
Task 10……………………………………………………………..
27
LIMITS OF FUNCTIONS ………………………………………
29
Task 11…………………………………………………………….. 29
DERIVATIVE OF A FUNCTION ……………………………… 34
Task 12 …………………………………………………………….
Task 13……………………………………………………………..
34
40
APPLICATION OF DERIVATIVE IN INVESTIGATION OF
FUNCTIONS
42
Task 14……………………………………………………………..
42
INDEFINITE AND DEFINITE INTEGRALS
44
Task 15…………………………………………………………….
Task 16…………………………………………………………….
Task 17…………………………………………………………….
Task 18…………………………………………………………….
Task 19…………………………………………………………….
44
46
47
52
55
162
Task 20…………………………………………………………….
Task 21…………………………………………………………….
Task 22…………………………………………………………….
10.
11.
12.
13.
14.
57
59
60
FUNCTIONS OF SEVERAL VARIABLES …………………… 62
Task 23…………………………………………………………….
Task 24…………………………………………………………….
62
63
DIFFERENTIAL EQUATIONS ………………………………..
65
Task 25……………………………………………………………..
Task 26……………………………………………………………..
65
69
NUMERICAL AND POWER SERIES …………………………. 71
Task 27……………………………………………………………..
Task 28……………………………………………………………..
Task 29……………………………………………………………..
Task 30……………………………………………………………..
Task 31……………………………………………………………..
71
73
76
77
78
FOURIER SERIES ………………………………………………
79
Task 32 …………………………………………………………….
79
COMPLEX NUMBERS …………………………………………. 81
Task 33 ……………………………………………………………..
Solutions of sample problems of higher mathematics ……..………
Self-evaluation quiz ………………………………………………..
Answers ……………………………………………………………
Appendices …………………………………………………………
References……………………………………………………...…..
163
81
82
140
150
152
157