Tekanyo ea Mohala oa Tangent ho ea ho Curve
Lipalong tsa lipalo, equation ea tangent le curve e bapala karolo ea bohlokoa ho utloisiseng liketsahalo tse fapaneng, ka bobeli saenseng le boenjiniere, le lits'ebetsong tsa letsatsi le letsatsi. Tangent le curve ke mola o amang curve ntlheng e le 'ngoe feela e itseng. Ho utloisisa mohopolo ona haholoanyane, re hloka ho utloisisa tlhaloso, lits'ebetso le lipalo tsa tangent equation.
Pendahuluan
Mokokotlo o ka har'a sefofane sa coordinate ke setšoantšo se bonahalang sa equation kapa mosebetsi oa lipalo. Ho sa le joalo, mola o otlolohileng ke mola o otlolohileng o amang mokokotlo ntlheng e le 'ngoe 'me o na le moepa o tšoanang le mokokotlo ntlheng eo. Moelelong oa jeometri ea tlhahlobo, mola o otlolohileng o ka sebelisoa ho fumana moepa (gradient) oa mokokotlo ntlheng e itseng.
Tlhaloso ea Mohala oa Tangent
Tlhaloso ea motheo ea mola o kobehileng ke mola o amang mothinya ntlheng e le 'ngoe feela ntle le ho o kopanya. Mola ona o na le litšobotsi tse peli tse ka sehloohong:
1. Mola o nang le tangent o na le gradient e tshwanang le curve ntlheng ya tangency.
2. Mola o tangent o kopana le mothinya feela ntlheng e le 'ngoe e itseng.
Moedi kapa gradient ya mola o tangent ho moedi o fanwa ke derivative ya pele ya mosebetsi e hlalosang moedi ntlheng e itseng.
Mehopolo ea Motheo ea ho Bala Litekanyo tsa Mela ea Tangent
Ho bala equation ea mola o kobehileng ho ea ho kobeho, mehato e latelang e tlameha ho nkuoa:
1. Fumana Mosebetsi le Ntlha ea Tangency:
\( y = f(x) \) ke mosebetsi o hlalosang mothinya, mme re hloka ho fumana mola o tangent ntlheng \( (a, f(a)) \).
2. Bala Tlhahiso ea Pele ea Mosebetsi:
Ntho ea pele e nkiloeng \( f'(x) \) e fana ka moepa oa mola o otlolohileng ho mothinya ntlheng ka 'ngoe \( x \).
3. Ho Nkela Lintlha Sebakeng sa Li-Derivative:
Moedi wa mola o tangent ho \( x = a \) ke \( f'(a) \).
4. Ngola equation ea mola oa tangent:
Ka ho sebedisa foromo ya ntlha-lekgaba ya mola \( y – y_1 = m(x – x_1) \), moo \( m \) e leng lekgaba mme \( (x_1, y_1) \) e le ntlha moleng, jwale equation ya mola o nang le tangent e ka ngolwa tjena:
\[
y – f(a) = f'(a)(x – a)
\]
Mohlala oa Palo ea Mohala oa Tangent
A re re re na le mothapo o hlalositsweng ke equation \( y = x^2 \), mme re batla ho fumana mola o tangent ntlheng \( (1, 1) \).
1. Mesebetsi le Lintlha tse Tangent:
Mosebetsi ke \( y = f(x) = x^2 \) mme ntlha ya tangency ke \( (1, 1) \).
2. Tšimoloho ea Pele ea Mosebetsi:
\[
f'(x) = 2x
\]
3. Ho theoha ha maemo:
\[
f'(1) = 2 \makgetlo a 1 = 2
\]
4. Tekanyo ea Mola o Tangent:
Ka ntlha \( (1, 1) \) le leralla \( m = 2 \):
\[
y – 1 = 2(x – 1)
\]
Kahoo, equation ea mola oa tangent ke:
\[
y = 2x - 1
\]
Ka hona, equation ea mola o tangent ho ea ho curve \( y = x^2 \) ntlheng \( (1, 1) \) ke \( y = 2x – 1 \).
Litšebeliso tsa Mohala oa Tangent
Tekanyo ea mola o kobehileng le mothapo o nang le litšebeliso tse fapaneng tse sebetsang masimong a fapaneng:
1. Fisiks le Mekaniki:
– Ka mohlala, ho fisiks tlhahlobong ea motsamao, lebelo la ntho ka nako e itseng le ka fumanoa ka ho fumana tangent ho curve ea boemo le nako.
2. Moruo le Lichelete:
– Moruong, litšenyehelo tse ka thoko li ka hlahlojoa ho sebelisoa mohopolo oa mela e tangent, moo derivative ea mosebetsi oa litšenyehelo tsohle e fanang ka litšenyehelo tse ka thoko.
3. Boenjiniere:
– Baenjiniere ba tsa meaho le tsa mechini hangata ba sebedisa di-tangent ho bala kgatello le kabo ya kgatello hodima sebopeho se fanoeng.
4. Bongaka:
– Tlhahlobong ea lintlha tsa bongaka, likhutlo tseo lintlha tsa bakuli li thathamisitsoeng ho tsona hangata li hloka li-tangent ho fumana sekhahla sa phetoho kapa mokhoa oa kholo.
Mathata a Tloaelehileng le Litharollo tsa 'Ona
Mathata a ho fumana equation ea mola oa tangent a ka hlaha haeba:
1. Derivative ha e eo kapa ha e hlalosoe:
Ka linako tse ling, derivative ea mosebetsi e kanna ea se be teng. Sena se ka etsahala lintlheng tsa mabota kapa likhutlong tse kobehileng.
2. Likhutlo tse Rarahaneng:
Mesebetsi e rarahaneng haholo kapa e ke keng ea aroloa ka tlhahlobo e ka hloka mokhoa oa lipalo ho fumana tangent.
Tharollo:
1. Moeli oa Tšebeliso:
Haeba derivative e tobileng e sa fumanehe, mohopolo oa meeli o ka sebelisoa ho hakanya leralla la mola oa tangent.
2. Phapang ea Lipalo:
Mekhoa ea lipalo joalo ka mokhoa oa phapang e lekanyelitsoeng e ka sebelisoa ho hakanya li-derivatives.
Qetello
Tekanyo ea tangent le curve ke mohopolo oa motheo lipalo tse nang le lits'ebetso tse pharaletseng masimong a fapaneng. Ho utloisisa mokhoa oa ho bala tangent le curve ho hloka kutloisiso ea li-derivatives le mekhoa e fapaneng. Ka kutloisiso ena, re ka sekaseka le ho bolela esale pele liketsahalo tse fapaneng tsa tlhaho, tsa moruo le tsa botekgeniki ka nepo le ka katleho. Ho sebelisa mohopolo ona ho re lumella ho qapa lipatlisiso, ho theha mahlale a macha le ho rarolla mathata a letsatsi le letsatsi.
