Chitsanzo cha funso lokambirana pa kugwiritsa ntchito mankhwala ochokera ku zinthu zina
Chochokera ku chinthu ichi ndi lingaliro lofunikira la calculus lomwe limagwiritsidwa ntchito kwambiri m'moyo watsiku ndi tsiku komanso m'magawo ena asayansi, monga fizikisi, zachuma, zamoyo, ndi uinjiniya. M'nkhaniyi, tikambirana mavuto angapo a zitsanzo ndikukambirana za momwe zinthu zochokera ku chinthuchi zimagwiritsidwira ntchito, makamaka pankhani yokonza bwino ndi kusanthula magwiridwe antchito.
Chiyambi cha Mapulogalamu Ochokera
Chochokera ku ntchito chimapereka chidziwitso chokhudza kusintha kwa ntchitoyo poyerekeza ndi chosinthika chake chodziyimira pawokha. Chitsanzo chosavuta ndi liwiro, lomwe ndi chochokera ku malo poyerekeza ndi nthawi. Mwachidule, zochokera ku ntchito zingagwiritsidwe ntchito kupeza ma values apamwamba komanso ochepera a ntchito, kudziwa nthawi zomwe ntchitoyo ikukwera kapena kutsika, ndikupereka chidziwitso chokhudza makhalidwe ndi mawonekedwe a ntchitoyo.
Chitsanzo Funso 1: Kupeza Miyezo Yaikulu ndi Yocheperako
Funso:
Dziwani mfundo zazikulu ndi zochepa za ntchito \( f(x) = x^3 – 3x^2 + 4 \).
Kukambirana:
1. Kupeza chochokera choyamba:
Kuti tipeze mfundo zofunika, tifunika kupeza chiyambi cha ntchitoyo ndikuchiyerekeza ndi zero.
\[
f'(x) = 3x^2 – 6x
\]
\[
3x^2 – 6x = 0
\]
2. Konzani equation:
Timaganizira equation:
\[
3x(x – 2) = 0
\]
Chifukwa chake, timapeza mfundo zofunika pa \( x = 0 \) ndi \( x = 2 \).
3. Unikani chochokera chachiwiri:
Kuti tidziwe ngati mfundo zofunika kwambiri ndi maxima kapena minima, tifunika kupeza chochokera chachiwiri cha ntchitoyo:
\[
f”(x) = 6x – 6
\]
Kuwunika pa mfundo zofunika kwambiri:
\[
f”(0) = 6(0) – 6 = -6 \, (\text{negative, so\ } x = 0 \text{\ is a local maximum})
\]
\[
f”(2) = 6(2) – 6 = 6 \, (\text{positive, so\ } x = 2 \text{\ ndi chiwerengero chochepa chapafupi})
\]
4. Werengani mitengo yayikulu komanso yocheperako:
Sinthani mfundo zofunika kwambiri mu ntchito yoyambirira:
\[
f(0) = 0^3 – 3 \cdot 0^2 + 4 = 4 \, (\text{maximum})
\]
\[
f(2) = 2^3 – 3 \cdot 2^2 + 4 = 8 – 12 + 4 = 0 \, (\text{minimum})
\]
Kotero, ntchito \( f(x) = x^3 – 3x^2 + 4 \) ili ndi maximum yapafupi pa \( (0, 4) \) ndi maximum yapafupi pa \( (2, 0) \).
Chitsanzo Funso 2: Kukonza ndi Zopinga
Funso:
Mlimi akufuna kumanga bwalo lamakona anayi m'mphepete mwa mtsinje. Popeza mpanda uli ndi mamita 100, dziwani kukula kwa bwalo kuti malo ake akhale okwanira.
Kukambirana:
1. Pangani equation:
Tiyerekeze kuti kutalika kwa mpanda wofanana ndi mtsinje ndi mamita 1 (x) ndipo m'lifupi ndi mamita 1 (y). Popeza mbali imodzi imadutsa mtsinje, mpanda wofunikira ndi wa mbali zitatu.
\[
2y + x = 100
\]
2. Pezani malo okwanira:
Dera la khola \( A \) ndi:
\[
A = x \cdot y
\]
Kuchokera ku equation ya mpanda, tikhoza kufotokoza \( y \) motsatira \( x \):
\[
y = \frac{100 – x}{2}
\]
Kotero, equation ya dera imakhala:
\[
A(x) = x \cdot \frac{100 – x}{2} = 50x – \frac{x^2}{2}
\]
3. Kupeza chochokera choyamba:
Kuti tipeze mtengo wapamwamba kwambiri, timapeza chochokera choyamba cha \( A(x) \):
\[
A'(x) = 50 – x
\]
Kufanana ndi zero:
\[
50 – x = 0 \amatanthauza x = 50
\]
4. Werengani mtengo wa \( y \):
Lowetsani \( x = 50 \) mu equation:
\[
y = \frac{100 – 50}{2} = 25
\]
Choncho, miyeso ya khola yomwe imapereka malo ochulukirapo ndi mamita 50 kutalika ndi mamita 25 m'lifupi.
Chitsanzo Funso 3: Kudziwa Liwiro Lalikulu
Funso:
Tinthu tating'onoting'ono timayenda molunjika ndi malo omwe amafotokozedwa ngati nthawi \( s(t) = t^3 – 6t^2 + 9t + 1 \). Dziwani liwiro lalikulu la tinthu tating'onoting'ono.
Kukambirana:
1. Dziwani liwiro (lochokera pamalo):
Liwiro la tinthu ndi chiyambi cha malo poyerekeza ndi nthawi:
\[
v(t) = \frac{ds}{dt} = 3t^2 – 12t + 9
\]
2. Dziwani chochokera chachiwiri:
Kuti tipeze mfundo zazikulu, timapeza chochokera chachiwiri:
\[
a(t) = \frac{dv}{dt} = 6t – 12
\]
3. Kupeza mfundo yofunika kwambiri:
Kuyerekeza chiyambi cha liwiro ndi zero:
\[
3t^2 – 12t + 9 = 0
\]
Gawani ndi 3:
\[
t^2 – 4t + 3 = 0
\]
Kuwerengera:
\[
(t – 3)(t – 1) = 0
\]
Kotero, mfundo zofunika kwambiri ndi \( t = 1 \) ndi \( t = 3 \).
4. Unikani kuthamanga kuti mupeze kuchuluka kwakukulu:
\[
a(1) = 6(1) – 12 = -6 \amatanthauza t = 1 \malemba{\ ndi chiwerengero chapamwamba chapafupi}
\]
\[
a(3) = 6(3) – 12 = 6 \amatanthauza t = 3 \malemba{\ ndi osachepera am'deralo}
\]
5. Kuwerengera liwiro lalikulu:
Lowetsani \( t = 1 \) mu equation ya liwiro:
\[
v(1) = 3(1)^2 – 12(1) + 9 = 3 – 12 + 9 = 0 \, (\malemba{osasangalatsa})
\]
Chongani malire ena oyenera kapena malo olumikizirana kuti muwonetsetse kuti yankho labwino kwambiri.
Ndi njira izi, tikhoza kupanga njira yothetsera mavuto osiyanasiyana omwe ali pamwambapa pogwiritsa ntchito njira zochokera kuzinthu zomwe zili pamwambapa.
Mapeto
Zitsanzo zomwe zili pamwambapa zikusonyeza momwe ma derivatives angagwiritsidwire ntchito kuthetsa mavuto m'njira zosiyanasiyana. Kupeza mfundo zazikulu ndi zochepa, kukonza bwino zinthu, ndi kusanthula kayendedwe ndi zina mwa njira zomwe zimagwiritsidwa ntchito poganizira ma derivatives. Kudziwa bwino njira ndi njirazi ndikofunikira kwa iwo omwe amaphunzira masamu apamwamba ndi maphunziro ena ofanana.
