Kushandisa Mafomura Ekuwedzera
Fomura ye exponential ipfungwa yemasvomhu inowanzoitika muhupenyu hwezuva nezuva, kunyangwe isu tisingazvizivi. Patinotaura nezvekuwedzera kwevanhu, kuwedzera mubereko pamaakaundi ekuchengetedza, kupararira kwemavhairasi, kuora kwezvinhu zvine mwaranzi, uye kunyange kukura kwevashandisi vemapurogiramu edhijitari, zvese izvi zvinogona kutevedzera uchishandisa patani yakafanana: shanduko "dzinowanda" nekufamba kwenguva. Patani iyi ndiyo hunhu hukuru hwe exponentials. Chinyorwa chino chichakurukura tsananguro yefomura ye exponential, chimiro chayo chakajairika, mashandisirwo ayo, pamwe chete nemienzaniso yekushandisa uye mazano ekudzivirira zvikanganiso zvekuverenga.
1. Chii chinonzi exponential?
Muchidimbu, exponentiation inzira yekuverenga inosanganisira masimba. Kana tikanyora \(a^n\), ipapo \(a\) inonzi base uye \(n\) inonzi exponent kana simba. Muenzaniso uri nyore: \(2^3 = 8\), zvinoreva kuti 2 yakawedzerwa neyo pachayo katatu: \(2 \times 2 \times 2\).
Zvisinei, kana tichitarisa kutevedzera, "fomura yekutsanangura" inowanzoreva basa rine kukosha kunowedzera kana kuderera nechinhu chakagadziriswa mukati menguva yakatarwa. Semuenzaniso, nhamba inoramba ichikwira ne10% gore rega rega zvinoreva kuti kukosha kwayo kunowedzerwa ne1,10 gore rega rega. Iyi ipatani yekutsanangura - kwete kuwedzera nehuwandu hwakagadziriswa, asi nechikamu chakagadziriswa.
2. Chimiro chakazara chefomura ye exponential
Kune mhando mbiri dzemafomura e exponential anowanzoshandiswa:
1) Kukura/kuora kweganda (zvichienderana nenguva yakatarwa)
\[
N(t) = N_0 \kawa pa^t
\]
Information:
– \(N(t)\): kukosha panguva \(t\)
– \(N_0\): kukosha kwekutanga
– \(a\): kuwanda kwechikamu chimwe nechimwe (semuenzaniso 1,10 yekuwedzera kwe10%; 0,90 yekudzikira kwe10%)
– \(t\): nhamba yenguva (semuenzaniso makore, mwedzi, mazuva)
2) Kukura/kuora kunoramba kuripo (muenzaniso wehuwandu hunoramba huripo)
\[
N(t) = N_0 \nguva e^{rt}
\]
Information:
– \(e\) ndiyo nhamba yaEuler (inenge 2,71828)
– \(r\) mwero wekukura unoenderera mberi (unogona kuva wakanaka pakukura, asi usina kunaka pakudzikira)
– \(t\) nguva
Muzviitiko zvakawanda zvechikoro kana mashandisirwo ari nyore, chimiro chakasiyana \(N(t)=N_0 a^t\) chakakwana. Chimiro chinoenderera mberi chinowanzo shandiswa mukuongorora kwakadzama, semuenzaniso mukuverenga, fizikisi, kana kutevedzera epidemic.
3. Matanho ekushandisa fomura ye exponential
Kuti udzivise kuvhiringidzika, tevera matanho aya paunenge uchishanda pamatambudziko ekuwedzera:
1) Sarudza kukosha kwekutanga kwe \(N_0\)
Iyi ndiyo huwandu huri pakutanga kwekucherechedza (gore 1, zuva 0, zvichingodaro).
2) Kuziva kana kuri kukura kana kuora.
– Kukura: kukosha kunowedzera (factor \(a > 1\) kana \(r>0\))
– Kuora: kukosha kunowedzera kuderera (factor \(0)
5) Verenga mhedzisiro Shandisa karukureta kana exponent yakakura kana ine madecimal. 4. Mienzaniso yekushandiswa kwekukura kwe exponential Muenzaniso 1: Mubereko uri nyore we compound Munhu anochengetedza Rp. 5.000.000 nechiyero chemubereko che8% pagore, mubereko unobhadharwa wowedzerwa kune yasara pakupera kwegore rega rega (compound interest). Ndeipi yasara mushure memakore mashanu? Zvapiwa: - \(N_0 = 5.000.000\) - \(a = 1 + 0,08 = 1,08\) - \(t = 5\) Fomura: \[ N(5) = 5.000.000 \times (1,08)^5 \] Kukosha \((1,08)^5 \approx 1,4693\) Saka: \[ N(5) \approx 5.000.000 \times 1,4693 = 7.346.500 \] Yakaenzana neRp7.346.500 (zvichienderana nekutenderera). Muenzaniso 2: Kukura kwemushandisi weApplication App ine vashandisi 20.000. Huwandu hwevashandisi hunokura ne25% pamwedzi. Vangani vashandisi vachavapo mushure memwedzi mitanhatu? - \(N_0 = 20.000\) - \(a = 1,25\) - \(t = 6\) \[ N(6) = 20.000 \times 1,25^6 \] Sezvo \(1,25^6 \inenge 3,8147\), ipapo: \[ N(6) \inenge 20.000 \inenge 3,8147 = 76.294 \] Saka vashandisi vanenge 76.294 mushure memwedzi mitanhatu. 5. Mienzaniso yekushandiswa kwekuora kwe exponential Muenzaniso 3: Kuderera kwemutengo wezvinhu (kuderera kwemutengo) Mota ine mutengo weRp. 18.000.000 inowana kudzikira kwemutengo kwe12% pagore. Kukosha kwayo chii mushure memakore mana? - \(N_0 = 18.000.000\) - yakaderera ne12% → \(a = 0,88\) - \(t = 4\) \[ N(4) = 18.000.000 \times 0,88^4 \] Sezvo \(0,88^4 \approx 0,5997\), ipapo: \[ N(4) \approx 18.000.000 \times 0,5997 = 10.794.600 \] Kukosha kwacho kuri Rp10.794.600. Muenzaniso 4: Kuora kwechinhu Ngatitii chinhu chinoderera ne5% awa yega yega. Kana pakutanga chaive magiramu 200, chinosara zvakadii mushure memaawa gumi?
- \(N_0=200\) - \(a=0,95\) - \(t=10\) \[ N(10)=200 \times 0,95^{10} \] Kubva \(0,95^{10} \approx 0,5987\): \[ N(10)\approx 200 \times 0,5987 = 119,74 \] Zvasara magiramu angangoita 119,74. 6. Kuziva nguva (kutsvaga \(t\)) nema logarithms Dzimwe nguva mubvunzo hausi wekupedzisira, asi "zvinotora nguva yakareba sei". Panyaya iyi, tinoda ma logarithms. Kana: \[ N(t) = N_0 a^t \] zvino: \[ \frac{N(t)}{N_0} = a^t \] Tora log: \[ t = \frac{\log\left(\frac{N(t)}{N_0}\right)}{\log(a)} \] Muenzaniso unokurumidza: Mari inochengetwa inowedzera ne10% pagore. Ichapetwa kaviri riini? - \(a=1,10\) - \(\frac{N(t)}{N_0}=2\) \[ t = \frac{\log(2)}{\log(1,10)} \approx \frac{0,3010}{0,0414} \approx 7,27 \] Saka anenge makore 7,27 . 7. Zvikanganiso zvinowanzoitwa pakushandisa maformula e exponential 1) Kushandura mapercentage zvisirizvo kuita ma factors e10% pachinzvimbo che 0,10 semuwandisi mukuru, asi kuita 1,10 yekukura. 2) Nguva isiriyo mayuniti Kana chikamu chiri pamwedzi, usashandise \(t\) mumakore pasina shanduko. 3) Kukanganisa kukura kwe exponential nekukura kwe linear Linear inowedzera "huwandu hwakatarwa," semuenzaniso, +5 nguva yega yega. Exponential inowedzera "chikamu chakatarwa," saka kuwedzera kunowedzera nekufamba kwenguva. 4) Kutenderera nekukurumidza Edza kuchengetedza mamwe manhamba pakati pekuverenga, uchiatenderera pakupera. 8. Mhedziso Kushandisa maformula e exponential kunotibatsira kunzwisisa zviitiko zvinoshanduka nekukurumidza nekufamba kwenguva. Nekuziva kukosha kwekutanga \(N_0\), kuona kukura/kuora kwechinhu \(a\), uye nguva \(t\), tinogona kufanotaura kukosha kweramangwana kana kuverenga kuti zvichatora nguva yakareba sei kuti tisvike pachinangwa. Fomura iyi yakakosha zvikuru muhupfumi, sainzi, tekinoroji, uye zvinhu zvakawanda zvehupenyu chaihwo. Chinokosha ndechekuenderana kwemayuniti uye kururama mukushandura maperesenti kuita zvinhu. Kana wangoziva zvekutanga, matambudziko ekuwedzera achaita seari nyore uye ane musoro. Kana uchida, ndinogona kuwedzera matambudziko ekudzidzira netsananguro, kana kugadzirisa chinyorwa ichi kuvadzidzi vepuraimari, yepakati, kana yesekondari.
