Mehlala ea Lipotso tse Buisanang ka Khopolo-taba ea Motheo ea Calculus
Calculus ke lekala la bohlokoa la lipalo le kenyeletsang likhopolo tsa meeli, lintho tse tsoang ho eona, le lintho tse kopaneng. Theorem ea Motheo ea Calculus (FDTC) ke e 'ngoe ea likhopolo-taba tsa motheo tse hokahanyang likhopolo tsena. Sehloohong sena, re tla hlahloba tlhaloso le ts'ebeliso ea Theorem ea Motheo ea Calculus ka letoto la mathata le lipuisano tsa mehlala.
Ho utloisisa Khopolo-taba ea Motheo ea Calculus
Theorem ea Motheo ea Calculus e na le likarolo tse peli tse kholo:
1. Karolo ea Pele: Haeba \( f \) e le mosebetsi o tsoelang pele karolong ea \([a, b]\), 'me \( F \) e le antiderivative ea \( f \) karolong eo ea karohano, joale:
\[ \int_a^bf(x) \, dx = F(b) – F(a) \]
2. Karolo ea Bobeli: Haeba \( f \) e le mosebetsi o tsoelang pele karolong \([a, b]\), 'me re hlalosa mosebetsi \( F \) ka:
\[ F(x) = \int_a^xf(t) \, dt \]
ebe \( F \) ke antiderivative ya \( f \), e leng:
\[ F'(x) = f(x) \]
Kamora ho utloisisa mohopolo oa motheo, ha re feteleng ka ho toba lipotsong tse ling tsa mehlala le lipuisanong tsa tsona ho hlakisa ts'ebeliso ea Theorem ea Motheo ea Calculus.
Lipotso tsa Mehlala ea Puisano
Mohlala Bothata ba 1: Ho Sebelisa Karolo ea Pele ea Thuto ea Motheo ea Calculus
Potso:
Ha ho fanoe ka mosebetsi \( f(x) = 3x^2 \). Bala karolo e sa lekanyetsoang ea \( f(x) \) ho tloha \( x = 1 \) ho isa \( x = 4 \).
Puisano:
Ho rarolla bothata bona, re hloka ho fumana antiderivative \( F(x) \) ya \( f(x) \).
Mohato oa 1: Fumana antiderivative \( F(x) \) ea \( f(x) = 3x^2 \).
\[ \int 3x^2 \, dx = x^3 + C \]
Kahoo, \( F(x) = x^3 \).
Mohato oa 2: Bala boleng ba \( F(x) \) ho latela meeli e fanoeng ea integral.
\[ \int_1^4 3x^2 \, dx = F(4) – F(1) \]
\[ = 4^3 – 1^3 \]
\[ = 64 – 1 \]
\[ = 63 \]
Kahoo, boleng ba bohlokoa ke 63.
Mohlala oa Potso ea 2: Ho Sebelisa Karolo ea Bobeli ea Thuto ea Motheo ea Calculus
Potso:
Haeba \( F(x) = \int_2^x (2t + 1) \, dt \), fumana derivative ya \( F(x) \).
Puisano:
Ho ya ka karolo ya bobedi ya Motheo wa Thuto ya Motheo ya Calculus, haeba \( F(x) = \int_a^xf(t) \, dt \), ebe \( F'(x) = f(x) \).
Ho latela boemo bo fanoeng:
\[ F(x) = \int_2^x (2t + 1) \, dt \]
Ebe derivative ea \( F(x) \) ke:
\[ F'(x) = 2x + 1 \]
Mohlala oa 3: Ho Sebelisa Theorem ea Motheo ea Calculus ka Mesebetsi e Rarahaneng Haholoanyane
Potso:
Fanoeng \( f(x) = \sqrt{x} \). Bala karolo e sa lekanyetsoang ea \( f(x) \) ho tloha \( x = 0 \) ho isa \( x = 4 \).
Puisano:
Mohato oa 1: Fumana antiderivative \( F(x) \) ea \( f(x) = \sqrt{x} \).
\[ \int \sqrt{x} \, dx = \int x^{1/2} \, dx \]
Sebelisa melao ea motheo ea li-integral:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
Kahoo:
\[ \int x^{1/2} \, dx = \frac{x^{3/2}}{3/2} + C \]
\[ = \frac{2}{3} x^{3/2} + C \]
Kahoo, \( F(x) = \frac{2}{3} x^{3/2} \).
Mohato oa 2: Bala boleng ba \( F(x) \) ho latela meeli e fanoeng ea integral.
\[ \int_0^4 \sqrt{x} \, dx = F(4) – F(0) \]
\[ = \left( \frac{2}{3} \cdot 4^{3/2} \right) – \left( \frac{2}{3} \cdot 0^{3/2} \right) \]
\[ = \frac{2}{3} \cdot 8 – 0 \]
\[ = \frac{16}{3} \]
Kahoo, boleng ba karolo e kopaneng ke \( \frac{16}{3} \).
Mohlala oa Potso ea 4: Kopanyo le Mesebetsi ea Karolo
Potso:
Kopanya \( f(x) = \frac{2}{x} \) ho tloha \( x = 1 \) ho isa \( x = 3 \).
Puisano:
Mohato oa 1: Fumana antiderivative \( F(x) \) ea \( f(x) = \frac{2}{x} \).
\[ \int \frac{2}{x} \, dx = 2 \int \frac{1}{x} \, dx \]
Rea tseba hore:
\[ \int \frac{1}{x} \, dx = \ln |x| +C\]
Kahoo:
\[ \int \frac{2}{x} \, dx = 2 \ln |x| +C\]
Le \( F(x) = 2 \ln |x| \).
Mohato oa 2: Bala boleng ba \( F(x) \) ho latela meeli e fanoeng ea integral.
\[ \int_1^3 \frac{2}{x} \, dx = F(3) – F(1) \]
\[ = 2 \ln |3| – 2 \ln | 1| \]
\[ = 2 \ln 3 – 2 \ln 1 \]
\[ = 2 \ln 3 – 0 \]
\[ = 2 \ln 3 \]
Kahoo, boleng ba karolo e kopaneng ke \( 2 \ln 3 \).
Mohlala oa Potso ea 5: Mesebetsi e Kopanetsoeng ea Trigonometric
Potso:
Kopanya \( f(x) = \sin x \) ho tloha \( x = 0 \) ho isa \( x = \pi \).
Puisano:
Mohato oa 1: Fumana antiderivative \( F(x) \) ea \( f(x) = \sin x \).
\[ \int \sin x \, dx = -\cos x + C \]
Le \( F(x) = -\cos x \).
Mohato oa 2: Bala boleng ba \( F(x) \) ho latela meeli e fanoeng ea integral.
\[ \int_0^\pi \sin x \, dx = F(\pi) – F(0) \]
\[ = -\cos(\pi) – (-\cos(0)) \]
\[ = -(-1) – (-1) \]
\[ = 1 – (-1) \]
\[ = 1 + 1 \]
\[ = 2 \]
Kahoo, boleng ba bohlokoa ke 2.
Qetello
Thuto ea Motheo ea Calculus ke sesebelisoa se matla thutong ea lipalo le lipalo ka kakaretso. Ka ho hokahanya li-derivatives le li-integral, thuto ena e re lumella ho bala sebaka se ka tlas'a kobeho le ho utloisisa phetoho ea mosebetsi ka tsela e tebileng haholoanyane. Ho utloisisa le ho tseba ts'ebeliso ea thuto ena ka ho ikoetlisa ke senotlolo sa ho ba le tsebo ea thuto ea lipalo. Sengoloa sena se qaqisa feela bokaholimo ba se ka finyelloang ka Thuto ea Motheo ea Calculus, empa re tšepa hore se tla fana ka setšoantšo se hlakileng sa mokhoa oa ho sebetsa le e 'ngoe ea likhopolo tsa motheo tsa lipalo.
