Hosila ta'rifi va geometrik ma'nosi

Bu nima va nega kerak

Avtomobil spidometri ko'rsatadigan tezlik — bu butun yo'ldagi o'rtacha tezlik emas, balki ayni lahzadagi oniy tezlikdir.

Vaqt oralig'i $\Delta t \to 0$ ga intilganda, yo'l orttirmasi $\Delta s$ ning vaqt orttirmasi $\Delta t$ ga nisbatining chekli limiti oniy tezlikni beradi:

$v(t) = \lim_{\Delta t \to 0} \frac{\Delta s}{\Delta t}.$

Matematikada bu jarayon hosila tushunchasi orqali ifodalanadi:

• Fizikada: tezlik — yo'lning hosilasi, tezlanish — tezlikning hosilasi;
• Geometriyada: egri chiziqqa o'tkazilgan urinmaning burchak koeffitsiyenti;
• DTM testlarida: funksiya o'sish-kamayish oraliqlari va ekstremumlarini aniqlash asosi.

Ushbu darsda hosila ta'rifi, uning geometrik va mexanik ma'nosini o'rganamiz.

Asosiy g'oya

1. Argument va funksiya orttirmasi

Argument $x_0$ dan $x$ ga o'zgarganda:

• $\Delta x = x - x_0$ — argument orttirmasi ($x = x_0 + \Delta x$).
• $\Delta y = \Delta f = f(x_0 + \Delta x) - f(x_0)$ — funksiya orttirmasi.

2. Hosila ta'rifi

Funksiya orttirmasining argument orttirmasiga nisbatining $\Delta x \to 0$ dagi chekli limiti $y = f(x)$ funksiyaning $x_0$ nuqtadagi hosilasi deyiladi va $f'(x_0)$ deb belgilanadi:

$f'(x_0) = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = \lim_{\Delta x \to 0} \frac{f(x_0 + \Delta x) - f(x_0)}{\Delta x}$

3. Eng sodda funksiyalarning hosilalari

Funksiya $f(x)$	Hosilasi $f'(x)$
$C$ (o'zgarmas son)	$0$
$x$	$1$
$kx + b$	$k$
$x^2$	$2x$
$x^3$	$3x^2$

4. Hosilaning geometrik ma'nosi

Funksiya grafigiga $(x_0; f(x_0))$ nuqtada o'tkazilgan urinmaning $Ox$ o'qi musbat yo'nalishi bilan hosil qilgan burchagi $\alpha$ bo'lsa:

$k = \text{tg}\,\alpha = f'(x_0)$

• $\alpha$ o'tkir burchak bo'lsa ($0 < \alpha < 90^\circ$), $f'(x_0) > 0$ (funksiya o'suvchi).
• $\alpha$ o'tmas burchak bo'lsa ($90^\circ < \alpha < 180^\circ$), $f'(x_0) < 0$ (funksiya kamayuvchi).
• Urinma $Ox$ o'qiga parallel bo'lsa ($\alpha = 0^\circ$), $f'(x_0) = \text{tg}\,0^\circ = 0$.

5. Hosilaning mexanik ma'nosi

To'g'ri chiziqli $s(t)$ harakatda:

• Oniy tezlik: $v(t) = s'(t)$.
• Oniy tezlanish: $a(t) = v'(t) = s''(t)$.

Yechilgan misollar

1-misol. Ta'rif bo'yicha hosila hisoblash

$f(x) = x^2 + 3x$ funksiyaning $x_0 = 2$ nuqtadagi hosilasini ta'rif bo'yicha hisoblang.

Yechish:

1. $f(2) = 2^2 + 3 \cdot 2 = 10$.
2. $f(2 + \Delta x) = (2 + \Delta x)^2 + 3(2 + \Delta x) = 10 + 7\Delta x + (\Delta x)^2$.
3. $\Delta f = 7\Delta x + (\Delta x)^2$.
4. $\dfrac{\Delta f}{\Delta x} = 7 + \Delta x$.
5. Limit: $f'(2) = \lim_{\Delta x \to 0}(7 + \Delta x) = 7$.

Javob: $7$.

2-misol. Urinmaning og'ish burchagi

$f(x) = 2x^2 - 7x + 1$ parabola grafigiga $x_0 = 2$ nuqtada o'tkazilgan urinmaning $Ox$ o'qi bilan hosil qilgan burchagini toping.

Yechish:

1. $f'(x) = 4x - 7$.
2. $x_0 = 2$ da: $f'(2) = 4 \cdot 2 - 7 = 1$.
3. $k = \text{tg}\,\alpha = 1 \implies \alpha = 45^\circ$.

Javob: $45^\circ$.

3-misol. To'g'ri chiziqli harakat va tezlik

Nuqta $s(t) = 2t^3 - 9t^2 + 12t - 5$ qonun bo'yicha harakatlanmoqda ($s$ — metr, $t$ — soniya). a) $t = 3\text{ s}$ dagi oniy tezlikni toping. b) Qaysi paytlarda nuqta to'xtagan ($v = 0$)?

Yechish: a) $v(t) = s'(t) = 6t^2 - 18t + 12$. $t = 3$ da: $v(3) = 6(9) - 18(3) + 12 = 12\text{ m/s}$. b) $v(t) = 0 \implies 6(t^2 - 3t + 2) = 0 \implies t_1 = 1\text{ s}, t_2 = 2\text{ s}$.

Javob: a) $12\text{ m/s}$; b) $t = 1\text{ s}$ va $t = 2\text{ s}$.

4-misol. Argument va funksiya orttirmasi

$f(x) = 3x^2$ funksiya uchun $x_0 = 1$ va $\Delta x = 0{,}1$ bo'lganda $\Delta f$ va $\dfrac{\Delta f}{\Delta x}$ ni toping hamda $f'(1)$ bilan solishtiring.

Yechish:

1. $f(1) = 3$.
2. $f(1{,}1) = 3(1{,}1)^2 = 3{,}63$.
3. $\Delta f = 3{,}63 - 3 = 0{,}63$.
4. $\dfrac{\Delta f}{\Delta x} = \dfrac{0{,}63}{0{,}1} = 6{,}3$.
5. $f'(x) = 6x \implies f'(1) = 6$. $\Delta x \to 0$ da $6{,}3 \to 6$ bo'ladi.

Javob: $\Delta f = 0{,}63$, $\dfrac{\Delta f}{\Delta x} = 6{,}3$, $f'(1) = 6$.

Mashq

1. $f(x) = 7x - 4$ funksiyaning $x_0 = 3$ nuqtadagi hosilasini toping.
2. $f(x) = x^2$ funksiyaning $x_0 = 4$ nuqtadagi hosilasini ta'rif orqali hisoblang.
3. $y = 3x^2 - 5x + 2$ parabolaga $x_0 = 1$ nuqtada o'tkazilgan urinmaning burchak koeffitsiyentini toping.
4. $s(t) = 4t^2 - 6t + 1\text{ (m)}$ harakatda $t = 2\text{ s}$ dagi oniy tezlikni toping.
5. $y = f(x)$ grafigiga o'tkazilgan urinma $Ox$ o'qi bilan $\alpha = 135^\circ$ burchak hosil qilsa, $f'(x_0)$ nimaga teng?
6. $f(x) = 2x^2 + 5$ funksiyada $x_0 = 1$ va $\Delta x = 0{,}1$ bo'lganda $\Delta f$ ni hisoblang.
7. $s(t) = t^3 - 12t$ qonun bilan harakatlanayotgan jism qachon to'xtaydi ($t > 0$)?
8. $y = x^2 - 8x + 12$ parabolaning qaysi nuqtasida urinma $Ox$ o'qiga parallel bo'ladi?
9. $f(x) = \dfrac{2}{x}$ funksiyaning $x_0 = 1$ nuqtadagi hosilasini ta'rif orqali toping.
10. $s(t) = 2t^3 - 3t^2 + 4t$ harakatda $t = 2\text{ s}$ dagi tezlanishni toping.

Javoblar va yechim yo'llari

1. Chiziqli funksiyada $(kx + b)' = k \implies f'(3) = 7$. Javob: $7$.

2. $\Delta f = (4 + \Delta x)^2 - 16 = 8\Delta x + (\Delta x)^2 \implies \frac{\Delta f}{\Delta x} = 8 + \Delta x \to 8$. Javob: $8$.

3. Burchak koeffitsiyenti $k = f'(1) = 6(1) - 5 = 1$. Javob: $1$.

4. $v(t) = s'(t) = 8t - 6 \implies v(2) = 8(2) - 6 = 10\text{ m/s}$. Javob: $10\text{ m/s}$.

5. $f'(x_0) = \text{tg}\,135^\circ = -1$. Javob: $-1$.

6. $f(1) = 7$, $f(1{,}1) = 7{,}42 \implies \Delta f = 7{,}42 - 7 = 0{,}42$. Javob: $0{,}42$.

7. $v(t) = 3t^2 - 12 = 0 \implies t = 2\text{ s}$ ($t > 0$). Javob: $2\text{ s}$.

8. $y' = 2x - 8 = 0 \implies x_0 = 4$. $y_0 = 4^2 - 8(4) + 12 = -4$. Nuqta: $(4; -4)$. Javob: $(4; -4)$.

9. $\Delta f = \frac{2}{1 + \Delta x} - 2 = \frac{-2\Delta x}{1 + \Delta x} \implies \frac{\Delta f}{\Delta x} \to -2$. Javob: $-2$.

10. $v(t) = 6t^2 - 6t + 4 \implies a(t) = 12t - 6$. $a(2) = 18\text{ m/s}^2$. Javob: $18\text{ m/s}^2$.

Keng tarqalgan xatolar

1. Funksiya qiymati $f(x_0)$ bilan hosila qiymati $f'(x_0)$ ni chalkashtirish. ✗ $f(x) = x^2$ funksiyaning $x = 3$ dagi hosilasini $f(3) = 3^2 = 9$ deb yozish. ✓ $f(3) = 9$ — funksiya qiymati. Hosilasi $f'(x) = 2x$, shuning uchun $f'(3) = 2 \cdot 3 = 6$.

2. Urinmaning burchak koeffitsiyentini sinus yoki kosinus deb olish. ✗ Burchak $\alpha = 60^\circ$ bo'lsa, $k = \sin 60^\circ = \dfrac{\sqrt{3}}{2}$ deb hisoblash. ✓ Urinmaning burchak koeffitsiyenti tangensga teng: $k = \text{tg}\,60^\circ = \sqrt{3}$.

3. Orttirma nisbatida limitga o'tishni unutish. ✗ $\Delta f = 6\Delta x + (\Delta x)^2$ bo'lganda, nisbat $\dfrac{\Delta f}{\Delta x} = 6 + \Delta x$ deb yakunlash. ✓ Hosila bu limit: $\Delta x \to 0$ da $6 + \Delta x \to 6$. Javob aniq son $6$ bo'ladi.

4. Gorizontal urinmada hosila yo'q deb o'ylash. ✗ Urinma $Ox$ o'qiga parallel bo'lsa, hosila mavjud emas deb hisoblash. ✓ Urinma $Ox$ ga parallel bo'lsa, $\alpha = 0^\circ$ va $k = \text{tg}\,0^\circ = 0$. Demak, hosila mavjud va $f'(x_0) = 0$.

DTM uslubidagi test

Ushbu mavzu bo'yicha tayyorlangan 5 ta test savoli platformada interaktiv tarzda ishlaydi: o'z javobingizni tanlang va to'liq yechimni ko'ring.

Keyingi qadam

Biz hosila ta'rifi bilan tanishdik va sodda funksiyalar uchun uni limit orqali topdik. Keyingi darsda hosila hisoblashni osonlashtiruvchi Hosila hisoblash qoidalari va hosilalar jadvalini o'rganamiz.