Oliy ta’lim muassalari talabalari uchun

Topshiriqni yuklab oling

1. Agar $n\in\mathbb{N}$ bo‘lsa, u holda quyidagi tengliklarni isbotlang:

\[ \int\limits_0^{\frac{\pi}{2}}\frac{\sin(2n-1)x}{\sin x}dx=\frac{\pi}{2}, \]

\[ \int\limits_0^{\frac{\pi}{2}}\frac{\sin 2nx}{\sin x}dx=2\sum\limits_{k=1}^n\frac{(-1)^{k-1}}{2k-1}. \]

2. Sirkul va chizg‘ich yordamida teng tomonli uchburchakni shunday bo‘laklarga bo‘lingki, ulardan kvadrat hosil qilib bo‘lsin.

3. Ixtiyoriy $(a,b)\subset \mathbb{R}$ da kontinium uzulishga hamda kontinium uzluksiz nuqtaga ega bo‘lgan $f:\mathbb{R}\to\mathbb{R}$ funksiya quring.

4. Agar $p>3$ tub son hamda

\[ \frac{n}{m}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{p} \]

bo‘lsa, u holda $n-m$ qiymat $p^3$ ga bo‘linishini isbotlang.

5. Agar $\{{{x}_{n}}\}$ ketma-ketlik uchun ${{x}_{1}}=1$ va ${{x}_{n+1}}={{x}_{n}}+\frac{1}{{{x}_{n}}}$ shartlar bajarilsa, $\lim\limits_{n\to\infty}\frac{{{x}_{n}}}{\sqrt{n}}$ ni toping.

6. Agar $f\in {{C}^{1}}[a,b]$ va

\[ \int\limits_{a}^{b}{f(x)dx}=\int\limits_{a}^{b}{f^\prime (x)dx}=0 \]

bo‘lsa, u holda

\[ |f(x)| \le \frac{1}{2}\int\limits_{a}^{b}{|f^\prime (x)|dx},\,\,\,\,\forall x\in [a,b] \]

tengsizlikni isbotlang.

Javoblarni junatish muddati: 10.05.2017
Javoblaringizni quyidagi imkoniyatlar orqali yuborishingiz mumkin:
  1. Email: yoshmatematiklar@mail.ru
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Topshiriqni yuklab oling

1. Umumiy hadi

\[ a_n=\left[\frac{1\cdot3\cdot5\cdot...\cdot(2n-1)}{2\cdot4\cdot6\cdot \ldots \cdot2n}\right]^2 \]

bo‘lgan $\sum\limits_{n=1}^\infty a_n$ qatorni yaqinlashuvchanlikka tekshiring.

2. Limitni hisoblang:

\[ \lim\limits_{x\to0}\frac{\mathrm{tg}(\mathrm{tg} x)-\sin(\sin x)}{\mathrm{tg} x-\sin x}. \]

3. $y=\sqrt{ax+b}+c$ parabolaning fokusi koordinatalarini toping.

4. Faraz qilaylik

\[ A=\begin{bmatrix} a & b\\0 & c\end{bmatrix} \]

bo‘lsin, bu yerda $a,b,c$ lar haqiqiy sonlar. Berilgan matritsaning biror darajasi birlik matritsa, ya’ni $\exists n\in \mathbb{N}, A^n=E$, bo‘ladigan $a,b,c$ larni toping.

5. Agar $\vec{a}_1,\vec{a}_2,\vec{b}_1,\vec{b}_2$ uch o‘lchovli fazodagi vektorlar bo‘lsa, quyidagini isbotlang:

\[ \begin{vmatrix}(\vec{a}_1\vec{b}_1)&(\vec{a}_1\vec{b}_2)\\ (\vec{a}_2\vec{b}_1)&(\vec{a}_2\vec{b}_2)\end{vmatrix}=([\vec{a}_1\vec{a}_2][\vec{b}_1\vec{b}_2]), \]

bu yerda $(\vec{x}\vec{y})$ va $[\vec{x}\vec{y}]$ lar mos ravishda $\vec{x},\vec{y}$ vektorlarnig skalyar va vektor ko‘paytmalari.

6. Quyidagi

\[ y^2=1+x+x^2+x^3+x^4 \]

tenglamani butun sonlarda yeching.

Javoblarni junatish muddati: 15.04.2017
Javoblaringizni quyidagi imkoniyatlar orqali yuborishingiz mumkin:
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  3. Shaxsiy kabinet
Topshiriqni yuklab oling

1. Tenglikni isbotlang:

\[ \sup_{x\in\mathbf{R}}\left| \left( \frac{\sin x}{x} \right)^n \right| = \frac{1}{n+1}. \]

2. $f(x)$ funksiyaning davri $\pi$ va $g(x)$ funksiyaning davri $e$ ga teng bo‘lsa $f(x) + g(x)$ funksiya davriy emasligini isbotlang.

3. $A \in M_n(\mathbf{R})$ matritsa uchun $A^3 = 4I_n - 3A$ tenglik bajarilsa

\[ \det(A + I_n) \]

qiymatini toping.

4. $\{a_n\}_{n=1}^\infty$ ketma-ketlik uchun $a_1 = 1$ va $n \in \mathbf{N}$ uchun

\[ a_{n+1} = a_n + \frac{1}{a_n^{2017}} \]

tenglik bajarilsa

\[ \sum\limits_{n=1}^\infty \frac{1}{na_n} \]

qator yaqinlashuvchi bo‘ladimi?

5. $f_n: \mathbf{R} \to \mathbf{R}$ funksiyalar ketma-ketligi va barcha $x \in \mathbf{R}$ uchun

\[ \lim\limits_{n \to \infty} f_n(x) = f(x) \]

tenglik bajarilsa, u holda

\[ \{x: f(x) < c\} = \bigcup\limits_{k=1}^\infty \bigcap\limits_{n=1}^\infty \bigcap\limits_{m>n}^\infty \left\{ x: f_n(x) < c - \frac{1}{k} \right\} \]

tenglikni isbotlang.

Yechimlarni qabul qilish muddati 25 martgacha uzaytirildi.
Javoblarni junatish muddati: 25.03.2017
Javoblaringizni quyidagi imkoniyatlar orqali yuborishingiz mumkin:
  1. Email: yoshmatematiklar@mail.ru
  2. Telegram: @ms_nuu
  3. Shaxsiy kabinet