Mienzaniso yeMibvunzo Inokurukura Kushandiswa kweIntegrals Mumunda weEconomics neBusiness
Pendauluan
Ma-integrals ipfungwa huru mukuverenga uye ane mashandisirwo akawanda muzvikamu zvakasiyana-siyana, kusanganisira economics nebhizinesi. Munyaya iyi, ma-integrals anowanzo shandiswa kuongorora purofiti yese, mutengo, mari inowanikwa, uye mashandiro ekushandisa nekugadzira. Kunzwisisa mashandisirwo ema-integrals muhupfumi nebhizinesi hakungobatsiri chete kugadzirisa matambudziko ehunyanzvi asiwo kunopa ruzivo rwakadzama pamusoro pekuchinja kwemusika, kuita sarudzo, uye kuronga zvine chekuita nehurongwa.
Mashandisirwo Akabatana muEconomics neBhizinesi
1. Verenga Mari Yose Inowanikwa
Kuti tiverenge mari yose inowanikwa, tinowanzo fanira kubatanidza mari shoma inowanikwa kubva mukutengeswa kwezvikamu zvega zvechigadzirwa. Kana mutengo wechigadzirwa uchisiyana zvichienderana nehuwandu hwakatengeswa, saka basa remutengo-huwandu rinofanira kubatanidzwa kuti rione mari yose inowanikwa.
Muenzaniso wematambudziko:
Ngatitii mutengo \( p \) wechinhu unoenderana nehuwandu hwechigadzirwa \( q \) chakatengeswa, icho chinopiwa nebasa rinotevera:
\[ p(q) = 100 – 2q \]
Verenga mari yose inowanikwa kana zvinhu gumi zvakatengeswa.
Mhinduro:
Mari yose inowanikwa \(R \) ipfungwa inokosha yemutengo pamusoro pehuwandu hunobva pa0 kusvika \(Q \) mayuniti.
\[ R = \int_{0}^{Q} p(q) \, dq \]
Ne \( p(q) = 100 – 2q \) uye \( Q = 10 \):
\[ R = \int_{0}^{10} (100 – 2q) \, dq \]
Saka, tinoverenga chikamu chakakosha:
\[ R = \kuruboshwe[ 100q – q^2 \kurudyi]_{0}^{10} \]
Ongorora miganhu yechinhu chakakosha:
\[ R = \kuruboshwe(100 \cdot 10 – 10^2 \kurudyi) – \kuruboshwe(100 \cdot 0 – 0^2 \kurudyi) \]
\[ = 1000 – 100 \]
\[ = 900 \]
Saka, mari yose inowanikwa kana zvinhu zvikatengeswa i900.
2. Verenga Mutengo Wose
Kushandiswa kwezvikamu zvekubatanidza mukuverenga mari yose yekugadzira kunobatsira zvikuru, kunyanya kana mari shoma isiri yenguva dzose uye zvichienderana nehuwandu hunogadzirwa. Mari shoma inogona kutsanangurwa seyakabva pamari yose, uye kuti tiwane mari yose tinofanira kuibatanidza.
Muenzaniso wematambudziko:
Kana mutengo wepakati \(MC \) wekugadzira \(q \) mayuniti echinhu ukapihwa na:
\[ MC(q) = 50 + 3q^2 \]
Verengai mutengo wose kana mayunitsi mashanu ezvinhu akagadzirwa muchitora mitengo yakatarwa \(C \) ye200.
Mhinduro:
Mutengo wose \(TC \) ndiwo musanganiswa wemutengo wepakati pamwe nemutengo wakatarwa:
\[ TC = \int_{0}^{Q} MC(q) \, dq + C \]
Ne \( MC(q) = 50 + 3q^2 \) uye \( Q = 5 \):
\[ TC = \int_{0}^{5} (50 + 3q^2) \, dq + 200 \]
Tinoverenga chikamu chakakosha:
\[ TC = \kuruboshwe[ 50q + q^3 \kurudyi]_{0}^{5} + 200 \]
Ongorora miganhu yechinhu chakakosha:
\[ TC = \kuruboshwe( 50 \cdot 5 + 5^3 \kurudyi) – \kuruboshwe( 50 \cdot 0 + 0^3 \kurudyi) + 200 \]
\[ = \kuruboshwe( 250 + 125 \kurudyi) + 200 \]
\[ = 375 + 200 \]
\[ = 575 \]
Saka, mari yose yekugadzira zvinhu zvishanu i575.
3. Kuverenga Kushandiswa Kwezviwanikwa
MaIntegrals anoshandiswawo kuverenga kushandiswa kana kushandiswa kwechiwanikwa mukati menguva yakatarwa. Izvi zvinonyanya kukosha mubhizinesi rine zviwanikwa zvakaita sesimba, zvinhu, kana vanhu.
Muenzaniso wematambudziko:
Mwero wekushandiswa kwesimba zuva nezuva \(E \) mufekitori unotevera basa re exponential rinotevera:
\[ E(t) = 10e^{0.1t} \]
Verenga simba rose rinoshandiswa kwemazuva gumi.
Mhinduro:
Kushandiswa kwesimba rose \(C \) munguva [0, T] ndicho chinhu chinobatanidzwa muhuwandu hwesimba rinoshandiswa:
\[ C = \int_{0}^{T} E(t) \, dt \]
Na \( E(t) = 10e^{0.1t} \) uye \( T = 10 \):
\[ C = \int_{0}^{10} 10e^{0.1t} \, dt \]
Kuti tiverenge chinhu chakakosha, tinogona kushandisa nzira yekutsiva:
Regai \( u = 0.1t \), wobva \( du = 0.1 \, dt \), kana \( dt = \frac{du}{0.1} \),
\[ C = \int_{0}^{1} 10e^{u} \frac{du}{0.1} \]
\[ = zana \int_{0}^{1} e^{u} \, du \]
\[ = 100 \kuruboshwe[ e^{u} \kurudyi]_{0}^{1} \]
Ongorora miganhu yechinhu chakakosha:
\[ C = 100 \kuruboshwe( e^{1} – e^{0} \kurudyi) \]
\[ = 100 \kuruboshwe( e – 1 \kurudyi) \]
Na \( e \approx 2.718 \):
\[ C \inenge 100 (2.718 - 1) \]
\[ = 100 \kawa 1.718 \]
\[ = 171.8 \]
Saka, simba rose rinoshandiswa kwemazuva gumi i171.8 units yesimba.
Mhedziso
Pfungwa ye "integrals" inokosha muhupfumi nebhizinesi, sezvo ichibvumira vaongorori nevanoita sarudzo kuverenga nekufanotaura zvinhu zvakakosha zvakaita semari inowanikwa, mitengo, uye mashandisirwo. Kunzwisisa mashandisirwo e "integrals" mumamiriro ezvinhu akasiyana aya kunogona kupa mukana wekukwikwidzana uye kunzwisisa zviri nani mashandiro ebhizinesi. Tinovimba kuti matambudziko aya emuenzaniso achakubatsira kunzwisisa mashandisirwo anoshanda e "integrals" muhupfumi nebhizinesi.
