Cov Lus Nug Piv Txwv Sib Tham Txog Kev Loj Hlob Exponential
Kev loj hlob sai yog lub tswv yim tseem ceeb hauv ntau qhov chaw, suav nrog kev lag luam, biology, kev lag luam, thiab physics. Hauv lub neej txhua hnub, peb feem ntau ntsib cov xwm txheej uas muaj kev loj hlob sai, xws li kev loj hlob ntawm tib neeg, kev poob qis ntawm cov nqi nominal vim yog kev nce nqi, kev sib kis ntawm cov kab mob, thiab ntau lwm yam. Tsab xov xwm no yuav tham txog ntau yam piv txwv teeb meem thiab kev daws teeb meem ntsig txog kev loj hlob sai, pab cov nyeem ntawv nkag siab zoo dua txog lub tswv yim no.
Cov Ntsiab Lus Tseem Ceeb ntawm Kev Loj Hlob Exponential
Kev loj hlob exponential yog txhais tias yog kev loj hlob uas tshwm sim ntawm tus nqi tas li ib chav ntawm lub sijhawm. Hauv kev lej, kev loj hlob exponential tuaj yeem txhais los ntawm cov qauv:
\[ N(t) = N_0 \cdot e^{kt} \]
Qhov twg:
- \( N(t) \) yog qhov sib ntxiv ntawm lub sijhawm \( t \)
- \( N_0 \) yog tus lej pib
- \( k \) yog tus nqi loj hlob
- \( t \) yog lub sijhawm
– \( e \) yog tus lej Euler (kwv yees li 2.718)
Yog tias tus nqi ntawm \(k\) yog qhov zoo, ces peb muaj kev loj hlob exponential. Ntawm qhov tod tes, yog tias \(k\) yog qhov tsis zoo, peb muaj kev shrinkage exponential.
Piv txwv lus nug 1: Kev loj hlob ntawm cov pej xeem
Lo lus nug
Ib lub nroog muaj cov pej xeem thaum pib ntawm 100,000 tus neeg. Yog tias cov pej xeem hauv nroog loj hlob sai heev ntawm 2% hauv ib xyoos, cov pej xeem hauv nroog yuav yog dab tsi tom qab 10 xyoo?
Kev daws teeb meem
Ua ntej, peb yuav tsum txheeb xyuas cov nqi paub:
– \( N_0 \) = 100,000
– \( k \) = 2% ib xyoos = 0.02
– \( t \) = 10 xyoo
Peb yuav siv cov mis exponential growth formula:
\[ N(t) = N_0 \cdot e^{kt} \]
Ntsaws cov nqi paub rau hauv cov mis:
\[ N(10) = 100,000 e^{0.02 \cdot 10} \]
\[ N(10) = 100,000 e^{0.2} \]
Tus nqi kwv yees ntawm \( e^{0.2} \) yog li ntawm 1.2214.
\[ N(10) = 100,000 1.2214 \]
\[ N(10) = 122,140 \]
Yog li, cov pej xeem ntawm lub nroog tom qab 10 xyoo yog li ntawm 122,140 tus neeg.
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Piv txwv Lo Lus Nug 2: Tus Nqi Peev Xwm
Lo lus nug
Koj tso $5,000 rau hauv ib tus account uas them paj laum 5% txhua xyoo. Koj cov nyiaj khaws cia yuav muaj nqis npaum li cas tom qab 15 xyoos?
Kev daws teeb meem
Txheeb xyuas cov nqi:
– \( N_0 \) = 5,000
– \( k \) = 5% ib xyoos = 0.05
– \( t \) = 15 xyoo
Siv cov qauv kev loj hlob exponential:
\[ N(t) = N_0 \cdot e^{kt} \]
Sau cov nqi uas paub lawm:
\[ N(15) = 5,000 e^{0.05 \cdot 15} \]
\[ N(15) = 5,000 e^{0.75} \]
Qhov kwv yees \( e^{0.75} \) yog li ntawm 2.117.
\[ N(15) = 5,000 2.117 \]
\[ N(15) = 10,585 \]
Yog li, tus nqi ntawm koj cov nyiaj khaws cia tom qab 15 xyoo yog kwv yees li $ 10,585.
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Piv txwv 3: Kev poob nqi ntawm lub tsheb
Lo lus nug
Tus nqi ntawm lub tsheb tshiab $30,000 yuav txo qis sai heev ntawm tus nqi 12% hauv ib xyoos. Lub tsheb yuav muaj nqis li cas tom qab 5 xyoos?
Kev daws teeb meem
Txheeb xyuas cov nqi:
– \( N_0 \) = 30,000
– \( k \) = -12% ib xyoos = -0.12
– \( t \) = 5 xyoo
Siv cov qauv kev loj hlob exponential:
\[ N(t) = N_0 \cdot e^{kt} \]
Sau cov nqi uas paub lawm:
\[ N(5) = 30,000 e^{-0.12 5} \]
\[ N(5) = 30,000 e^{-0.6} \]
Qhov kwv yees \( e^{-0.6} \) yog li ntawm 0.5488.
\[ N(5) = 30,000 0.5488 \]
\[ N(5) = 16,464 \]
Yog li, tus nqi ntawm lub tsheb tom qab 5 xyoos yog kwv yees li $ 16,464.
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Piv txwv lus nug 4: Kev kis tus kab mob
Lo lus nug
Ib lub nroog uas muaj cov pej xeem thawj zaug ntawm 200 tus neeg tau kis tus kab mob nrog tus nqi loj hlob exponential ntawm 15% ib hnub. Tag nrho cov neeg kis tus kab mob tom qab 7 hnub yog pes tsawg?
Kev daws teeb meem
Txheeb xyuas cov nqi:
– \( N_0 \) = 200
– \( k \) = 15% ib hnub = 0.15
– \( t \) = 7 hnub
Siv cov qauv kev loj hlob exponential:
\[ N(t) = N_0 \cdot e^{kt} \]
Sau cov nqi uas paub lawm:
\[ N(7) = 200 e^{0.15 \cdot 7} \]
\[ N(7) = 200 e^{1.05} \]
Qhov kwv yees \( e^{1.05} \) yog li ntawm 2.857.
\[ N(7) = 200 2.857 \]
\[ N(7) = 571.4 \]
Yog li, tag nrho cov neeg kis tus kab mob tom qab 7 hnub yog kwv yees li 571 tus neeg.
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Cov no yog cov piv txwv ntawm cov teeb meem thiab kev sib tham txog kev loj hlob sai. Kev nkag siab txog yuav ua li cas daws cov teeb meem kev loj hlob sai tuaj yeem muaj txiaj ntsig zoo heev rau ntau yam kev siv hauv ntiaj teb tiag. Yog tias muaj kev xyaum txaus, lub tswv yim no tuaj yeem yooj yim ua tus tswv thiab siv tau raws li qhov xav tau.
