Ngā Tauira Pātai e Matapaki ana i te Ariā Taketake o te Tātaitai
He peka nui te Tātaitai i roto i te pāngarau e pā ana ki ngā ariā o ngā rohe, ngā taupatupatu, me ngā taunga whakauru. Ko te Kaupapa Taketake o te Tātaitai (FDTC) tētahi o ngā kaupapa matua e hono ana i ēnei ariā. I roto i tēnei tuhinga, ka tūhuratia e mātou te whakamāramatanga me te whakamahinga o te Kaupapa Taketake o te Tātaitai mā roto i tētahi raupapa tauira rapanga me ngā kōrero.
Te Mārama ki te Kaupapa Taketake o te Tātaitai
E rua ngā wāhanga matua o te Kaupapa Taketake o te Tātaitai:
1. Wāhanga Tuatahi: Mena he mahi tonu a \( f \) i runga i te wā \([a, b]\), ā, he antiderivative a \( F \) o \( f \) i runga i taua wā, kāti:
\[ \int_a^bf(x) \, dx = F(b) – F(a) \]
2. Wāhanga Tuarua: Mena he mahi tonu a \( f \) i runga i te wā \([a, b]\), ā, ka tautuhia e tātou he mahi \( F \) mā:
\[ F(x) = \int_a^xf(t) \, dt \]
kātahi ko \( F \) te ārai-whakaputa o \( f \), arā:
\[ F'(x) = f(x) \]
Kia mārama tātou ki te ariā taketake, me haere tika tātou ki ētahi tauira pātai me ā rātou matapakinga hei whakamārama i te whakamahinga o te Ariā Taketake o te Tātaitai.
Tauira Pātai Kōrero
Tauira Raru 1: Te Whakamahi i te Wāhanga Tuatahi o te Ariā Taketake o te Tātaitai
Pātai:
I runga i te mahi \( f(x) = 3x^2 \). Tātaihia te taupū mutunga kore o \( f(x) \) mai i \( x = 1 \) ki \( x = 4 \).
Kōrero:
Hei whakaoti i tēnei raruraru, me kimi e tātou te ārai-tāpiritanga \( F(x) \) o \( f(x) \).
Hipanga 1: Kimihia te ātete-whakaputa \( F(x) \) o \( f(x) = 3x^2 \).
\[ \int 3x^2 \, dx = x^3 + C \]
Nō reira, \( F(x) = x^3 \).
Hipanga 2: Tātaihia te uara o \( F(x) \) i ngā rohe taupū kua hoatu.
\[ \int_1^4 3x^2 \, dx = F(4) – F(1) \]
\[ = 4^3 – 1^3 \]
\[ = 64 – 1 \]
\[ = 63 \]
Nō reira, ko te uara taupū ko te 63.
Tauira Pātai 2: Te Whakamahi i te Wāhanga Tuarua o te Ariā Taketake o te Tātaitai
Pātai:
Mena ko te F(x) = 2^x (2t + 1) dt, kimihia te pānga o te F(x)
Kōrero:
E ai ki te wāhanga tuarua o te Kaupapa Taketake o te Tātaitai, mēnā ko \( F(x) = \int_a^xf(t) \, dt \), ko \( F'(x) = f(x) \).
E ai ki te āhuatanga i hoatu:
\[ F(x) = \int_2^x (2t + 1) \, dt \]
Kātahi ka ko te pānga o \( F(x) \) ko:
\[ F'(x) = 2x + 1 \]
Tauira 3: Te Whakamahi i te Ariā Taketake o te Tātaitai me ngā Mahi Uaua Ake
Pātai:
Homai \( f(x) = \sqrt{x} \). Tātaihia te taupū mutunga kore o \( f(x) \) mai i \( x = 0 \) ki \( x = 4 \).
Kōrero:
Hipanga 1: Kimihia te ātete-whakaputa \( F(x) \) o \( f(x) = \sqrt{x} \).
\[ \int \sqrt{x} \, dx = \int x^{1/2} \, dx \]
Whakamahia ngā ture taketake o ngā taupū:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
Nā reira:
\[ \int x^{1/2} \, dx = \frac{x^{3/2}}{3/2} + C \]
\[ = \frac{2}{3} x^{3/2} + C \]
Nō reira, \( F(x) = \frac{2}{3} x^{3/2} \).
Hipanga 2: Tātaihia te uara o \( F(x) \) i ngā rohe taupū kua hoatu.
\[ \int_0^4 \sqrt{x} \, dx = F(4) – F(0) \]
\[ = \left( \frac{2}{3} \cdot 4^{3/2} \right) – \left( \frac{2}{3} \cdot 0^{3/2} \right) \]
\[ = \frac{2}{3} \cdot 8 – 0 \]
\[ = \frac{16}{3} \]
Nō reira, ko te uara o te taupū ko \( \frac{16}{3} \).
Tauira Pātai 4: Whakaurunga ki ngā Mahi Hautau
Pātai:
Whakauruhia \( f(x) = \frac{2}{x} \) mai i \( x = 1 \) ki \( x = 3 \).
Kōrero:
Hipanga 1: Kimihia te ātete-tāpiritanga \( F(x) \) o \( f(x) = \frac{2}{x} \).
\[ \int \frac{2}{x} \, dx = 2 \int \frac{1}{x} \, dx \]
E mōhio ana mātou:
\[ \int \frac{1}{x} \, dx = \ln |x| +C\]
Nā reira:
\[ \int \frac{2}{x} \, dx = 2 \ln |x| +C\]
Ā, \( F(x) = 2 \ln |x| \).
Hipanga 2: Tātaihia te uara o \( F(x) \) i ngā rohe taupū kua hoatu.
\[ \int_1^3 \frac{2}{x} \, dx = F(3) – F(1) \]
\[ = 2 \ln |3| – 2 \ln |1| \]
\[ = 2 \ln 3 – 2 \ln 1 \]
\[ = 2 \ln 3 – 0 \]
\[ = 2 \ln 3 \]
Nō reira, ko te uara o te taupū ko \( 2 \ln 3 \).
Tauira Pātai 5: Te Whakapūtanga o ngā Mahi Pāngatoru
Pātai:
Whakauruhia \( f(x) = \sin x \) mai i \( x = 0 \) ki \( x = \pi \).
Kōrero:
Hipanga 1: Kimihia te ātete-tāpiritanga \( F(x) \) o \( f(x) = \sin x \).
\[ \int \sin x \, dx = -\cos x + C \]
Ā, \( F(x) = -\cos x \).
Hipanga 2: Tātaihia te uara o \( F(x) \) i ngā rohe taupū kua hoatu.
\[ \int_0^\pi \sin x \, dx = F(\pi) – F(0) \]
\[ = -\cos(\pi) – (-\cos(0)) \]
\[ = -(-1) – (-1) \]
\[ = 1 – (-1) \]
\[ = 1 + 1 \]
\[ = 2 \]
Nō reira, ko te uara taupū ko te 2.
Whakamutunga
He taputapu kaha te Ariā Taketake o te Tātaitai i roto i te tātaitai me te pāngarau whānui. Mā te hono i ngā taupatupatu me ngā taunga whakauru, ka taea e tēnei ariā te tatau i te horahanga i raro i te kōpiko me te mārama ki te huringa o tētahi mahi i roto i te huarahi hōhonu ake. Ko te mārama me te matatau ki te whakamahinga o tēnei ariā mā te mahi te mea nui ki te matatau ki te tātaitai. He tīmatanga noa iho tēnei tuhinga ki ngā mea ka taea te whakatutuki mā te Ariā Taketake o te Tātaitai, engari ko te tumanako ka whakaratohia he pikitia mārama mō te mahi me tētahi o ngā ariā pāngarau tino taketake.
