Mafunso ndi Zitsanzo za Ntchito Zowonjezera, Ntchito Zochepetsa, ndi Ntchito Zosasinthasintha
Ntchito za masamu ndi nkhani yozama kwambiri ndipo ili ndi makhalidwe osiyanasiyana, imodzi mwa izo ndi momwe zingasankhidwire pankhani ya kuwonjezeka, kuchepa, kapena kukhazikika. Kudziwa ngati ntchito ikuwonjezeka, kuchepa, kapena kukhazikika pa nthawi inayake ndikofunikira kwambiri pakugwiritsa ntchito masamu osiyanasiyana, kuphatikizapo zachuma, fizikisi, ndi uinjiniya. Nkhaniyi ifotokoza zitsanzo ndi zokambirana zawo zokhudzana ndi ntchito zowonjezeka, kutsika, ndi kukhazikika.
Kodi ntchito zowonjezera, ntchito zocheperako, ndi ntchito zosasinthasintha ndi ziti?
1. Ntchito Yowonjezera: Ntchito \( f(x) \) imanenedwa kuti ikuwonjezeka pa interval \( I \) ngati pa \( x_1 \) ndi \( x_2 \) mu \( I \) ndi \( x_1 < x_2 \), tili ndi \( f(x_1) \leq f(x_2) \). 2. Ntchito Yochepetsa: Mosiyana ndi zimenezi, ntchito \( f(x) \) imanenedwa kuti ikuchepa pa interval \( I \) ngati pa \( x_1 \) ndi \( x_2 \) mu \( I \) ndi \( x_1 < x_2 \), tili ndi \( f(x_1) \geq f(x_2) \). 3. Ntchito Yosasinthasintha: Ntchito \( f(x) \) imanenedwa kuti ndi yosasinthasintha pa interval \( I \) ngati pa \( x \) iliyonse mu \( I \), ntchitoyo ili ndi mtengo womwewo, womwe ndi \( f(x) = c \) pa \( x \) iliyonse mu \( I \), pomwe c ndi chosasintha. Chitsanzo Funso 1: Kudziwa Interval ya Ntchito Yowonjezereka Popeza ntchito \( f(x) = 2x^3 - 3x^2 - 12x + 5 \). Dziwani interval pomwe ntchitoyo ikuwonjezeka! Kukambirana: Kuti tidziwe interval pomwe ntchitoyo ikuwonjezeka, tiyenera kupeza derivative yoyamba ya ntchitoyo kenako nkuwunika chizindikiro cha derivative. 1. Gawo 1: Pezani chochokera choyamba: \[ f'(x) = d/dx (2x^3 - 3x^2 - 12x + 5) \] \[ f'(x) = 6x^2 - 6x - 12 \] 2. Gawo 2: Dziwani mfundo zofunika: Mfundo zofunika ndi mfundo zomwe chochokera choyamba chili zero kapena chosafotokozedwa. \[ 6x^2 - 6x - 12 = 0 \] Gawani equation yonse ndi 6: \[ x^2 - x - 2 = 0 \] Timawerengera equation iyi ya quadratic: \[ (x-2)(x+1) = 0 \] Chifukwa chake, mfundo zofunika ndi \( x = 2 \) ndi \( x = -1 \). 3. Gawo 3: Dziwani chizindikiro cha chochokera choyamba pa nthawi yopangidwa ndi mfundo zofunika: Tipanga tebulo la chizindikiro cha \( f'(x) \) pa nthawi \( (-\infty, -1) \), \( (-1, 2) \), ndi \( (2, \infty) \). - Pa \( x \in (-\infty, -1) \): Tengani \( x = -2 \) \[ f'(-2) = 6(-2)^2 - 6(-2) - 12 = 24 + 12 - 12 = 24 \] Popeza \( f'(-2) > 0 \), ndiye \( f(x) \) ikuwonjezeka pa nthawi \( (-\infty, -1) \).
– Kwa \( x \in (-1, 2) \): Tengani \( x = 0 \)
\[ f'(0) = 6(0)^2 – 6(0) – 12 = -12 \]
Popeza \( f'(0) < 0 \), ndiye \( f(x) \) ikuchepa pa nthawi \( (-1, 2) \). - Kwa \( x \in (2, \infty) \): Tengani \( x = 3 \) \[ f'(3) = 6(3)^2 - 6(3) - 12 = 54 - 18 - 12 = 24 \] Popeza \( f'(3) > 0 \), ndiye \( f(x) \) ikuwonjezeka pa nthawi \( (2, \infty) \).
Kotero, ntchito \( f(x) \) ikuwonjezeka pa nthawi \( (-\infty, -1) \cup (2, \infty) \).
Chitsanzo Chachiwiri: Kudziwa Nthawi Yochepa Yogwira Ntchito
Popeza ntchito \( g(x) = 4x^4 – 8x^3 + 2 \). Dziwani nthawi zomwe ntchitoyo imachepa!
Kukambirana:
1. Gawo 1: Pezani chochokera choyamba:
\[ g'(x) = d/dx (4x^4 – 8x^3 + 2) \]
\[ g'(x) = 16x^3 – 24x^2 \]
2. Gawo 2: Dziwani mfundo yofunika kwambiri:
\[ 16x^3 – 24x^2 = 0 \]
\[ 8x^2(2x – 3) = 0 \]
Kotero mfundo zofunika kwambiri ndi \( x = 0 \) ndi \( x = \frac{3}{2} \).
3. Gawo 3: Dziwani chizindikiro cha chochokera choyamba pa nthawi:
– Kwa \( x \in (-\infty, 0) \): Tengani \( x = -1 \)
\[ g'(-1) = 16(-1)^3 – 24(-1)^2 = -16 – 24 = -40 \]
Popeza \( g'(-1) < 0 \), ndiye \( g(x) \) ikuchepa pa interval \( (-\infty, 0) \). - Kwa \( x \in (0, \frac{3}{2}) \): Tengani \( x = 1 \) \[ g'(1) = 16(1)^3 - 24(1)^2 = 16 - 24 = -8 \] Popeza \( g'(1) < 0 \), ndiye \( g(x) \) ikuchepa pa interval \( (0, \frac{3}{2}) \). - Kwa \( x \in (\frac{3}{2}, \infty) \): Tengani \( x = 2 \) \[ g'(2) = 16(2)^3 - 24(2)^2 = 128 - 96 = 32 \] Popeza \( g'(2) > 0 \), ndiye \( g(x) \) ikuwonjezeka pa nthawi \( (\frac{3}{2}, \infty) \).
Kotero, ntchito \( g(x) \) imachepa pa interval \( (-\infty, 0) \cup (0, \frac{3}{2}) \).
Chitsanzo Funso 3: Kudziwa Nthawi Yogwira Ntchito Yosasinthasintha
Popeza ntchito \( h(x) = 7 \), dziwani nthawi zomwe ntchitoyo ili yosasinthasintha!
Kukambirana:
Ntchito yosasinthika monga \( h(x) = 7 \) ili ndi chiyambi cha zero pa zonse \( x \):
\[ h'(x) = 0 \]
Popeza chochokera choyamba nthawi zonse chimakhala zero, ntchitoyo imakhala yosakhazikika pa domain yonse, kotero tinganene kuti ntchito \( h(x) = 7 \) imakhala yosakhazikika pa manambala onse enieni, omwe mu interval notation ndi \( (-\infty, \infty) \).
Mapeto
Kumvetsetsa nthawi zomwe ntchito ikukula, kuchepa, komanso kukhazikika kwake ndi gawo lofunikira pa kusanthula kwa ntchito. Kudzera mu zitsanzo zomwe zili pamwambapa, tafotokoza mfundo ndi masitepe ofunikira kuti tipeze nthawi zimenezi. Chidziwitsochi n'chothandiza kwambiri pakugwiritsa ntchito masamu m'njira zosiyanasiyana zothandiza komanso zamaganizo.
