Su'aalo Tusaale ah oo Ka Hadlaya Aragtida Aasaasiga ah ee Xisaabta
Xisaabtu waa laan muhiim ah oo xisaabeed oo ku lug leh fikradaha xuduudaha, derivatives, iyo integrals. Aragtida Aasaasiga ah ee Xisaabtu (FDTC) waa mid ka mid ah aragtiyaha aasaasiga ah ee isku xira fikradahan. Maqaalkan, waxaan ku sahamin doonnaa qeexidda iyo adeegsiga Aragtida Aasaasiga ah ee Xisaabtu iyada oo loo marayo taxane ah dhibaatooyin iyo doodo tusaale ah.
Fahmidda Aragtida Aasaasiga ah ee Xisaabta
Aragtida Aasaasiga ah ee Xisaabtu waxay ka kooban tahay laba qaybood oo waaweyn:
1. Qaybta Koowaad: Haddii \( f \) uu yahay shaqo joogto ah oo ku taal muddada \([a, b]\), iyo \( F \) uu yahay mid ka soo horjeeda \( f \) muddadaas, markaa:
\[ \int_a^bf(x) \, dx = F(b) – F(a) \]
2. Qaybta Labaad: Haddii \( f \) uu yahay shaqo joogto ah oo ku taal muddada \([a, b]\), oo aan ku qeexno shaqo \( F \) iyadoo la adeegsanayo:
\[ F(x) = \int_a^xf(t) \, dt \]
markaas \( F \) waa lidka ku ah \( f \), kuwaas oo kala ah:
\[ F'(x) = f(x) \]
Ka dib markaan fahamno fikradda aasaasiga ah, aan si toos ah u guda galno su'aalo tusaale ah iyo doodahooda si aan u caddayno ku dhaqanka Aragtida Aasaasiga ah ee Xisaabta.
Tusaalaha Su'aalaha Doodda
Tusaale ahaan Dhibaatada 1aad: Adeegsiga Qaybta Koowaad ee Aragtida Aasaasiga ah ee Xisaabta
Su'aal:
Marka la eego shaqada \( f(x) = 3x^2 \). Xisaabi isku-darka aan xadidnayn ee \( f(x) \) laga bilaabo \( x = 1 \) ilaa \( x = 4 \).
Dood:
Si aan u xallino dhibaatadan, waxaan u baahanahay inaan helno anti-derivative \( F(x) \) ee \( f(x) \).
Tallaabada 1: Soo hel ka-hortagga \( F(x) \) ee \( f(x) = 3x^2 \).
\[ \int 3x^2 \, dx = x^3 + C \]
Markaa, \( F(x) = x^3 \).
Tallaabada 2: Xisaabi qiimaha \( F(x) \) xadka isku-dhafka ah ee la bixiyay.
\[ \int_1^4 3x^2 \, dx = F(4) – F(1) \]
\[ = 4^3 – 1^3 \]
\[ = 64 – 1 \]
\[ = 63 \]
Markaa, qiimaha isku-dhafka ah waa 63.
Tusaale Su'aal 2aad: Adeegsiga Qaybta Labaad ee Aragtida Aasaasiga ah ee Xisaabta
Su'aal:
Haddii \( F(x) = \int_2^x (2t + 1) \, dt \), hel tarjumaadda \( F(x) \).
Dood:
Sida ku cad qaybta labaad ee Aragtida Aasaasiga ah ee Xisaabta, haddii \( F(x) = \int_a^xf(t) \, dt \), markaas \( F'(x) = f(x) \).
Sida ku cad xaaladda la bixiyay:
\[ F(x) = \int_2^x (2t + 1) \, dt \]
Markaas ka-soo-horjeedka \( F(x) \) waa:
\[ F'(x) = 2x + 1 \]
Tusaale 3: Adeegsiga Aragtida Aasaasiga ah ee Xisaabinta oo leh Shaqooyin Kakan oo Dheeraad ah
Su'aal:
La siiyay \( f(x) = \sqrt{x} \). Xisaabi isku-darka aan xadidnayn ee \( f(x) \) laga bilaabo \( x = 0 \) ilaa \( x = 4 \).
Dood:
Tallaabada 1: Soo hel ka-hortagga \( F(x) \) ee \( f(x) = \sqrt{x} \).
\[ \int \sqrt{x} \, dx = \int x^{1/2} \, dx \]
Isticmaal xeerarka aasaasiga ah ee isku-dhafka:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
Markaa:
\[ \int x^{1/2} \, dx = \frac{x^{3/2}}{3/2} + C \]
\[ = \frac{2}{3} x^{3/2} + C \]
Markaa, \( F(x) = \frac{2}{3} x^{3/2} \).
Tallaabada 2: Xisaabi qiimaha \( F(x) \) xadka isku-dhafka ah ee la bixiyay.
\[ \int_0^4 \sqrt{x} \, dx = F(4) – F(0) \]
\[ = \left( \frac{2}{3} \cdot 4^{3/2} \right) – \left( \frac{2}{3} \cdot 0^{3/2} \right) \]
\[ = \frac{2}{3} \cdot 8 – 0 \]
\[ = \frac{16}{3} \]
Markaa, qiimaha isku-dhafka waa \( \frac{16}{3} \).
Su'aal Tusaale ah 4: Is-dhexgalka Hawlaha Jajabka
Su'aal:
Isku-darka \( f(x) = \frac{2}{x} \) laga bilaabo \( x = 1 \) ilaa \( x = 3 \).
Dood:
Tallaabada 1: Soo hel anti-derivative \( F(x) \) ee \( f(x) = \frac{2}{x} \).
\[ \int \frac{2}{x} \, dx = 2 \int \frac{1}{x} \, dx \]
Waan ognahay taas:
\[ \int \frac{1}{x} \, dx = \ln |x| +C\]
Markaa:
\[ \int \frac{2}{x} \, dx = 2 \ln |x| +C\]
Iyo \( F(x) = 2 \ln |x| \).
Tallaabada 2: Xisaabi qiimaha \( F(x) \) xadka isku-dhafka ah ee la bixiyay.
\[ \int_1^3 \frac{2}{x} \, dx = F(3) – F(1) \]
\[ = 2 \ln |3| – 2 \ln |1| \]
\[ = 2 \ln 3 – 2 \ln 1 \]
\[ = 2 \ln 3 – 0 \]
\[ = 2 \ ln 3 \]
Markaa, qiimaha isku-dhafka waa \( 2 \ ln 3 \).
Su'aal Tusaale ah 5: Isku-dhafka Shaqooyinka Trigonometric
Su'aal:
Isku-darka \( f(x) = \sin x \) laga bilaabo \( x = 0 \) ilaa \( x = \pi \).
Dood:
Tallaabada 1: Soo hel anti-derivative \( F(x) \) ee \( f(x) = \sin x \).
\[ \int \sin x \, dx = -\cos x + C \]
Iyo \( F(x) = -\cos x \).
Tallaabada 2: Xisaabi qiimaha \( F(x) \) xadka isku-dhafka ah ee la bixiyay.
\[ \int_0^\pi \sin x \, dx = F(\pi) – F(0) \]
\[ = -\cos(\pi) - (-\cos(0)) \]
\[ = -(-1) – (-1) \]
\[ = 1 – (-1) \]
\[ = 1 + 1 \]
\[ = 2 \]
Markaa, qiimaha isku-dhafka ah waa 2.
Gabagabo
Aragtida Aasaasiga ah ee Xisaabtu waa qalab awood badan oo ku jira xisaabinta iyo xisaabta guud ahaan. Marka la isku xiro waxyaabaha ka soo jeeda iyo kuwa isku dhafan, aragtidani waxay noo ogolaanaysaa inaan xisaabino aagga hoos yimaada qalooca oo aan si qoto dheer u fahanno isbeddelka shaqada. Fahmidda iyo barashada adeegsiga aragtidan iyada oo loo marayo ku dhaqanka ayaa fure u ah in lagu noqdo qof ku takhasusay xisaabinta. Maqaalkani wuxuu kaliya xoqayaa dusha sare ee waxa lagu gaari karo Aragtida Aasaasiga ah ee Xisaabtu, laakiin waxaan rajeynayaa inuu bixiyo sawir cad oo ku saabsan sida loola shaqeeyo mid ka mid ah fikradaha xisaabta ee ugu aasaasiga ah.
