Mehlala ea Lipotso le Likarabo Mabapi le Meeli
Meeli ke e 'ngoe ea likhopolo tsa bohlokoa ka ho fetisisa lipalo, haholo-holo ho calculus. Meeli e re lumella ho utloisisa boitšoaro ba mosebetsi ha boleng ba oona bo fetohang bo atamela palo e itseng, ebang ke ho tloha ka letsohong le letšehali kapa le letona. Khopolo ena e theha motheo oa lipuisano tsa li-derivatives le li-integrals. Sehloohong sena, u tla fumana tlhaloso e khutšoanyane, hammoho le mehlala e 'maloa ea lipotso le likarabo mabapi le meeli e hlahang khafetsa boikoetlisong le litlhahlobong.
Kutloisiso e Khutšoanyane ea Meeli
Ka kakaretso, moedi o emela boleng boo mosebetsi o "atamelang" ha phetoho ya wona e ntse e atamela boleng bo itseng. E ngotswe tjena:
\[
\lim_{x \ho isa ho a} f(x)
\]
Ke hore, re batla ho tseba boleng ba \( f(x) \) ha \( x \) e atamela \( a \). Hopola hore moedi ha se kamehla o tshwanang le boleng ba mosebetsi ntlheng eo (mosebetsi o kanna wa ba wa se hlaloswe ntlheng eo), empa moedi o ntse o ka ba teng.
Mefuta e Tloaelehileng ea Mathata a Moeli
Mefuta e meng ea lipotso tsa moeli tse atisang ho ithutoa e kenyelletsa:
1. Moeli oa ho nkela sebaka ka ho toba (haeba mosebetsi o tsoela pele ntlheng eo).
2. Meeli ea sebopeho se sa lekanyetsoang joalo ka \( \frac{0}{0} \) kapa \( \frac{\infty}{\infty} \).
3. Meeli e amanang le metso.
4. Meeli ea Trigonometri.
5. Moeli o ea ho sa feleng.
Ho utloisisa hamolemo, ha re kene lipotsong tsa mohlala le puisanong ea tsona.
-
Mohlala oa 1: Meeli e nang le Phetolo e Tobileng
Potso:
Fumana boleng:
\[
\lim_{x \ho isa ho 2} (3x + 5)
\]
Karabo:
Kaha foromo ea mosebetsi e mola 'me ha e hlahise libopeho tse sa tsejoeng, re ka e bala ka ho e nkela sebaka ka ho toba.
\[
\lim_{x \ho isa ho 2} (3x + 5) = 3(2) + 5 = 6 + 5 = 11
\]
Qetello: Boleng ba moedi ke 11.
-
Mohlala oa 2: Moeli oa Sebopeho se sa Fetoheng \( \frac{0}{0} \) (Factorization)
Potso:
Palo:
\[
\lim_{x \to 3} \frac{x^2 – 9}{x – 3}
\]
Karabo:
Haeba o nkela sebaka ka ho toba \( x = 3 \), sephetho ke:
\[
\frac{9 – 9}{3 – 3} = \frac{0}{0}
\]
Ena ke foromo e sa hlakang, kahoo e hloka ho nolofatswa. Bala palo ya dinomoro ka ho e lekanya:
\[
x^2 – 9 = (x – 3)(x + 3)
\]
Kahoo:
\[
\frac{(x – 3)(x + 3)}{x – 3} = x + 3
\]
Jwale bala moedi:
\[
\lim_{x \ho isa ho 3} (x + 3) = 3 + 3 = 6
\]
Qetello: Boleng ba moedi ke 6.
-
Mohlala oa 3: Meeli e nang le Metso (Ho Lokisa Maikutlo)
Potso:
Etsa qeto:
\[
\lim_{x \to 4} \frac{\sqrt{x} – 2}{x – 4}
\]
Karabo:
Phetoho e tobileng e hlahisa:
\[
\frac{2 – 2}{4 – 4} = \frac{0}{0}
\]
Ho beha mabaka ka ho atisa metsoalle:
\[
\frac{\sqrt{x} – 2}{x – 4} \cdot \frac{\sqrt{x} + 2}{\sqrt{x} + 2}
= \frac{x – 4}{(x – 4)(\sqrt{x} + 2)}
\]
Nolofatsa:
\[
= \frac{1}{\sqrt{x} + 2}
\]
Jwale nka sebaka sa \( x = 4 \):
\[
\frac{1}{\sqrt{4} + 2} = \frac{1}{2 + 2} = \frac{1}{4}
\]
Qetello: Boleng ba moedi ke 1/4.
-
Mohlala oa Potso ea 4: Meeli ea Motheo ea Trigonometric
Potso:
Palo:
\[
\lim_{x \to 0} \frac{\sin x}{x}
\]
Karabo:
Moeli ona ke moedi o tsebahalang haholo oa motheo oa trigonometry:
\[
\lim_{x \ho isa ho 0} \frac{\sin x}{x} = 1
\]
E ka pakoa ka ho sebelisa jeometri kapa letoto la Taylor, empa maemong a sekolo hangata ho lekane ho e hopola e le foromo ea mantlha.
Qetello: Boleng ba moedi ke 1.
-
Mohlala oa Potso ea 5: Meeli ea Trigonometric ka ho Fetola
Potso:
Etsa qeto:
\[
\lim_{x \to 0} \frac{1 – \cos x}{x^2}
\]
Karabo:
Moeli ona o boetse o kenyelletsa meeli ea motheo e sebelisoang khafetsa:
\[
\lim_{x \ho isa ho 0} \frac{1 – \cos x}{x^2} = \frac{1}{2}
\]
Haeba o batla ho e theola, o ka sebelisa boitsebiso:
\[
1 – \cos x = 2\sin^2\left(\frac{x}{2}\right)
\]
E le hore:
\[
\frac{1 – \cos x}{x^2} = \frac{2\sin^2(x/2)}{x^2}
= 2 \left(\frac{\sin(x/2)}{x}\right)^2
\]
Fetola hanyane:
\[
\frac{\sin(x/2)}{x} = \frac{\sin(x/2)}{x/2} \cdot \frac{1}{2}
\]
Kahoo moedi ke:
\[
2 \left(1 \cdot \frac{1}{2}\right)^2 = 2 \cdot \frac{1}{4} = \frac{1}{2}
\]
Qetello: Boleng ba moedi ke 1/2.
-
Mohlala oa Potso ea 6: Mekhoa ea ho Fokotsa Moeli
Potso:
Palo:
\[
\lim_{x \to \infty} \frac{5x^2 + 3x}{2x^2 – 7}
\]
Karabo:
Bakeng sa moedi wa mosebetsi o utlwahalang ha \( x \ho isa ho \infty \), bapisa di-degree tse hodimo ka ho fetisisa. Kaha ka bobedi ba tsona ke tsa degree ya 2, moedi ke karolelano ya di-coefficient tse hodimo ka ho fetisisa:
\[
\lim_{x \to \infty} \frac{5x^2 + 3x}{2x^2 – 7} = \frac{5}{2}
\]
Qetello: Boleng ba moedi ke 5/2.
-
Mohlala Potso ea 7: Meeli e nang le Phetolo le ho Nolofatsa Likaroloana
Potso:
Etsa qeto:
\[
\lim_{x \to 1} \frac{x^3 – 1}{x – 1}
\]
Karabo:
Phetolo e tobileng e fana ka sebopeho \( \frac{0}{0} \). Ntlha:
\[
x^3 – 1 = (x – 1)(x^2 + x + 1)
\]
Nolofatsa:
\[
\frac{(x – 1)(x^2 + x + 1)}{x – 1} = x^2 + x + 1
\]
Phetolo \( x = 1 \):
\[
1^2 + 1 + 1 = 3
\]
Qetello: Boleng ba moedi ke 3.
-
Ho koala
Meeli ea ho ithuta e tla ba bonolo haholo haeba u tseba mekhoa ea motheo: hore na u lokela ho sebelisa sebaka se tobileng neng, hore na u lokela ho etsa mabaka afe, hore na u lokela ho fana ka mabaka afe, le hore na u lokela ho sebelisa mekhoa ea moeli oa trigonometric neng. Ka ho itloaetsa mathata khafetsa joalo ka mohlala o kaholimo, u tla tloaela ho sebetsana le mefuta e fapaneng ea meeli, esita le e bonahalang e rarahane.
Haeba o batla, nka boela ka etsa sephutheloana sa ho itlhakisa sa dipotso tse 20–30 tse nang le dipuisano tsa mohato ka mohato (ho tloha ho tsa motheo ho isa ho tse tswetseng pele), kapa ka se fetola ho latela lenaneothuto la sekolo se phahameng/sekolo sa mosebetsi wa matsoho/UTBK.
