7-bob. Differensial hisob

2. Funksiyani differensiallash qoidalari. Hosilalar jadvali

1. 
   Differensiallash qoidasi qayerda xato ko‘rsatilgan ?
   

A) (Cu)=Cu (C-const.). B) (uv)= uv. C) (uv)= uv+uv.
D) . E) (f(u))= f (u)u.

2. 
   Diffrentsiallanuvchi u va v funksiyalar u/v nisbatining hosilasini hisoblash formulasi to‘g‘ri yozilgan javobni ko‘rsating.
   

A) . B) . C) .
D) E) .

3. 
   y=x² /sinx funksiyaning y¢ hosilasini hisoblang.
   

A) y¢= x² /cosx. B) y¢= 2x /sinx. C) y¢= 2x /cosx.
D) y¢= x(2sinx +xcosx)/sin²x. E) y¢= x(2sinx- xcosx)/ sin²x .

4. 
   Diffrentsiallanuvchi u va v funksiyalar u∙v ko‘paytmasining hosilasini hisoblash formulasi qayerda to‘g‘ri yozilgan ?
   

A) u′v′ . B) u′v′+uv . C) u′v+uv′ . D) u′v–uv′ . E) u′v′–uv .

5. 
   y=x²sinx funksiyaning y¢ hosilasini hisoblang.
   

A) y¢= x²cosx. B) y¢= x(xsinx-2cosx). C) y¢= 2xsinx.
D) y¢= x(xcosx+2sinx). E) y¢= x(xcosx-2sinx) .

6. 
   Agar y=f(x), u=u(x) differensiallanuvchi funksiyalar bo‘lsa, y=f(u)
   

murakkab funksiya hosilasini hisoblash formulasini ko‘rsating.
A) y¢=f ¢(u) . B) y¢=f (u¢) . C) y¢=f ¢(u) .
D) y¢=f ¢(u)u¢ . E) y¢=f(u¢)u¢ .

7. 
   f(x)=sinx, u(x)=lnx funksiyalar bo‘yicha tuzilgan y=f(u)=sinlnx murakkab funksiya hosilasini hisoblang.
   

A) y¢=coslnx. B) y¢=sin(1/x). C) y¢=(sinlnx)/x.
D) y¢=(coslnx)/lnx. E) y¢=(coslnx)/x .

8. 
   y=sinarcsinx (–1 ≤ x ≤ 1) murakkab funksiya hosilasini hisoblang.
   

A) y¢=cosarcsinx. B) y¢=1. C) y¢= sinarccosx.
D) y¢= cosarccosx. E) y¢=x.

9. 
   y=cos(x²+1) funksiyaning y¢ hosilasini hisoblang.
   

A) y¢= sin(x²+1). B) y¢= −sin(x²+1). C) y¢= sin2x.
D) y¢=2xsin(x²+1). E) y¢=-2xsin(x²+1).

10. 
    funksiyaning hosilasi to‘g‘ri yozilgan javobni toping.
    

A) . B) . C) .
D) . E) .
3. Differensiallanuvchi funksiyalar haqidagi asosiy teoremalar

1. 
   Roll teoremasida y=f(x) funksiyaga qaysi shart qo‘yilmaydi ?
   

A) Biror [a,b] kesmada aniqlangan.
B) [a,b] kesmada uzluksiz.
C) [a,b] kesma ichida differensiallanuvchi.
D) [a,b] kesma chegaralarida f(a)=f(b).
E) Keltirilgan barcha shartlar qo‘yiladi .

2. 
   Roll teoremasining tasdig’ini ko‘rsating: Agar y=f(x) funksiya [a,b] kesmada uzluksiz va uning ichki nuqtalarida differensiallanuvchi hamda chegaraviy nuqtalarda f(a)=f(b) bo‘lsa, u holda bu kesma ichida kamida bitta shunday x=c nuqta topiladiki, unda ..... bo‘ladi.
   

A) f ′(c)>0. B) f ′(c)<0. C) f ′(c)=0. D) f ′(c)≠0. E) f ′(c)= f (c).

3. 
   f(x)=xsinx funksiya uchun Roll teoremasi qaysi kesmada o‘rinli bo‘ladi?
   

A) [0, 1]. B) [0, π/2]. C) [0, π]. D) [2, π]. E) [π/2, π] .

4. 
   f(x)=x²–7x+9 funksiya uchun Roll teoremasining shartlari bajariladigan [a,b] kesmalarning umumiy ko’rinishini aniqlang.
   

A) [1–α, 1+α], α>0 . B) [2–α, 2+α], α>0 .
C) [3–α, 3+α], α>0 . D) [4–α, 4+α], α>0 .
E) bunday kesmalar mavjud emas .

5. 
   f(x)=x²–7x+9, x[1,5], funksiya uchun Roll teoremasining tasdig’i o‘rinli bo‘ladigan c nuqtani toping.
   

A) c=1.5 . B) c=2 . C) c=2.5 . D) c=3 . E) c=4.5 .

6. 
   funksiya uchun [0,8] kesmada Roll teoremasining tasdig’i o‘rinli bo‘ladigan c nuqtani toping.
   

A) c=2 . B) c=3 . C) c=4 . D) c=5 . E) c=7 .

7. 
   Lagranj teoremasida y=f(x) funksiyadan qaysi shart talab etilmaydi ?
   

A) Biror [a,b] kesmada aniqlangan .
B) [a,b] kesmada uzluksiz .
C) [a,b] kesma ichida differensiallanuvchi .
D) [a,b] kesma chegaralarida f(a)=f(b) .
E)Keltirilgan barcha shartlar talab etiladi .

8. 
   Lagranj teoremasining tasdig’ini ko‘rsating: Agar y=f(x) funksiya [a,b] kesmada uzluksiz va uning ichki nuqtalarida differensiallanuvchi bo‘lsa, u holda bu kesma ichida kamida bitta shunday x=c nuqta topiladiki, unda ..... tenglik o‘rinli bo‘ladi.
   

A) . B) .
C) . D) .
E) .

9. 
   f(x)=xcosx funksiya uchun Lagranj teoremasi o‘rinli bo‘lmaydigan kesmani aniqlang .
   

A) [π/2,π] . B) [0,π/2] . C) [0,π] . D) [π,2π] .
E) Barcha kesmalarda Lagranj teoremasi o‘rinli bo’ladi .

10. 
    f(x)=x²–2x+9 funksiya uchun [a,b] kesmada Lagranj teoremasining tasdig’i o‘rinli bo‘ladigan c nuqtani toping.
    

A) c=a+b/2. B) c= b−a/2. C) c=(a+b)/2. D) c=(b−a)/2.
E) umuimy holda c nuqtani aniqlab bo’lmaydi.


4. Funksiya differensiali. Yuqori tartibli hosila va differensiallar

1. 
   Agar funksiya orttirmasi ∆f=A∆x+α(∆x)∆x (∆x→0 => α(∆x)→0) ko‘rinishda bo‘lsa, uning df differensiali qanday aniqlanadi ?
   

A) df= ∆x. B) df=A+∆x. C) df=A∆x. D) df=A–∆x. E) df= α(∆x)∆x.

2. 
   Agar y=f(x) funksiya hosilasi f (x) mavjud va chekli bo‘lsa, quyidagi hollardan qaysi birida df differensial mavjud bo‘ladi ?
   

A) f (x)>0. B) f (x)<0. C) f (x)=0. D) f (x)≠0.
E) barcha hollarda df differensial mavjud bo‘ladi.

3. 
   Differensiallanuvchi y=f(x) funksiyaning f (x) hosilasi df differensial orqali qanday ifodalanadi ?
   

A) . B) . C) .
D) . E) .

4. 
   Agar y=f(x) funksiyaning hosilasi f (x) mavjud va chekli bo‘lsa, uning df
   

differensiali qanday topiladi ?
A) df= f (x)+dx . B) df= f (x)–dx C) df= f (x)/dx .
D) df= f (x)dx . E) df= f (x) .

5. 
   Differensiallanuvchi y=f(x) funksiya argumentining orttirmasi x kichik bo‘lganda uning differensiali df va orttirmasi f orasida doimo qaysi munosobat o’rinli bo’ladi?
   

A) . B) . C) . D) . E) .

6. 
   Funksiyani differensiallash qoidasi qayerda noto‘g‘ri ko‘rsatilgan ?
   

A) . B) . C) .
D) . E) .

7. 
   y=cos(3x+4) funksiya differensiali dy qayerda to‘g‘ri ko‘rsatilgan ?
   

A) dy= sin(3x+4)dx. B) dy= 4sin(3x+4)dx. C) dy= –3sin(3x+4)dx.
D) dy= –4sin(3x+4)dx. E) dy= 3sin(3x+4)dx.

8. 
   y=xlnx funksiyaning dy differensialini toping .
   

A) dy= xdx. B) dy= lnxdx. C) dy=(1/x)dx.
D) dy= (1+lnx)dx. E) dy=(1–lnx)dx.

9. 
   Funksiyani differensial yordamida taqribiy hisoblash formulasi qayerda
   

to‘g‘ri ifodalangan ?
A) f(x+∆x)≈ f(x)df . B) f(x+∆x)≈ f(x)+df . C) f(x+∆x)≈ f(x)/df . D) f(x+∆x)≈ df / f(x) . E) f(x+∆x)≈ f(x)±df .

10. 
    Qaysi funksiyaning n-tartibli hosilasi noto‘g‘ri yozilgan ?
    

A) . B) . C) .
D) . E) barcha hosilalar to‘g‘ri yozilgan.
5. Funksiyani I tartibli hosila yordamida tekshirish

1. 
   Differensiallanuvchi y=f(x) funksiyaning kamayish sohasi uning f (x) hosilasi yordamida qanday munosabatdan topiladi?
   

A) f (x)=0 . B) f (x)≠0 . C) f (x)>0 . D) f (x)<0 . E) f (x)<∞ .

2. 
   f(x)=x³3x funksiyaning kamayish oralig‘ini toping.
   

A) (1, 1). B) (1, ). C) (1, ). D) (,1). E) (, 1).

3. 
   y=xlnx funksiyaning kamayish sohasi qayerda to‘g‘ri ko‘rsatilgan ?
   

A) (0, e). B) (, 1/e). C) (0, ). D) (0, 1/e). E) (1/e, ).

4. 
   Differensiallanuvchi y=f(x) funksiyaning o’sish sohasi uning f (x) hosilasi
   

yordamida qanday munosabatdan topiladi?
A) f (x)=0 . B) f (x)≠0 . C) f (x)>0 . D) f (x)<0 . E) f (x)<∞ .

5. 
   f(x)=xe^x funksiyaning o’sish oralig‘ini toping.
   

A) (1, 1). B) (1, ). C) (1, ). D) (,1). E) (, 1).

6. 
   y=xlnx funksiya o’sish sohasi qayerda to‘g‘ri ko‘rsatilgan ?
   

A) (0, e). B) (, 1/e). C) (0, ). D) (0, 1/e). E) (1/e, ).

7. 
   y=f(x) funksiya x₀ nuqta atrofida aniqlangan bo‘lib, unda lokal maksimumga ega. Agar argument orttirmasi x yetarlicha kichik bo‘lsa, quyidagi tasdiqlardan qaysi biri o‘rinli bo‘lmaydi ?
   

A) f(x₀+x)<f(x₀). B) f(x₀–x)<f(x₀). C) f<0.
D) f(x₀+x)+f(x₀x)<2f(x₀). E) barcha tasdiqlar o‘rinli bo‘ladi.

8. 
   y=f(x) funksiya x₀ nuqta atrofida aniqlangan bo‘lib, unda lokal minimumga ega. Agar argument orttirmasi x yetarlicha kichik bo‘lsa, quyidagi tasdiqlardan qaysi biri o‘rinli bo‘lmaydi ?
   

A) f(x₀+x)>f(x₀). B) f(x₀–x)>f(x₀). C) f>0.
D) f(x₀+x)+f(x₀x)>2f(x₀). E) barcha tasdiqlar o‘rinli bo’ladi.

9. 
   Differensiallanuvchi y=f(x) funksiya x₀ nuqtada lokal ekstremumga ega bo‘lsa, quyidagi shartlardan qaysi biri o‘rinli bo‘ladi ?
   

A) f (x₀)>0. B) f (x₀)<0. C) f (x₀)=0.
D) f (x₀)0. E) f (x₀) mavjud emas.

10. 
    f(x)=x³3x funksiyaning kritik nuqtalarini toping.
    

A) ±1 . B) 0 va 1 . C) –1 va 0 . D) 2 va 3 . E) ±2 .