Tauira Pātai me ngā Whakautu mō ngā Here
Ko ngā rohe tētahi o ngā ariā tino nui i roto i te pāngarau, inā koa i roto i te tātaitai. Mā ngā rohe ka taea e tātou te mārama ki te whanonga o tētahi mahi i te mea e tata ana tōna uara taurangi ki tētahi tau, mai i te taha maui, mai i te taha matau rānei. Ko tēnei ariā te pūtake mō ngā kōrero mō ngā taupatupatu me ngā taunga whakauru. I roto i tēnei tuhinga, ka kitea e koe he whakamārama poto, me ētahi tauira pātai me ngā whakautu mō ngā rohe e puta pinepine ana i roto i ngā mahi whakaharatau me ngā whakamātautau.
He Mārama Poto mō ngā Here
I te nuinga o te wā, ko te rohe te tohu o te uara e "tata atu" ana tētahi mahi i te whakatata atu o tana taurangi ki tētahi uara. Ka tuhia penei:
\[
\lim_{x \to a} f(x)
\]
Arā, e hiahia ana tātou ki te mōhio ki te uara o \( f(x) \) i te whakatata atu o \( x \) ki \( a \). Kia maumahara kāore te rohe e rite tonu ki te uara mahi i taua wāhi (tērā pea kāore te mahi i tautuhia i taua wāhi), engari ka taea tonu te noho o te rohe.
Ngā Momo Raru Herenga Noa
Ko ētahi momo pātai here e akohia pinepinetia ana ko:
1. Te rohe o te whakakapinga tika (mēnā he mahi tonu te mahi i taua pūwāhi).
2. Ngā rohe o te āhua mutunga kore pērā i te \( \frac{0}{0} \) me te \( \frac{\infty}{\infty} \).
3. Ngā rohe e pā ana ki ngā pakiaka.
4. Ngā rohenga pākoki.
5. Ka tae te rohe ki te mutunga kore.
Kia pai ake ai te mārama, me tahuri tātou ki ngā tauira pātai me ā rātou matapakinga.
-
Tauira 1: Ngā Herenga me te Whakakapinga Tika
Pātai:
Whakatauhia te uara:
\[
\lim_{x \to 2} (3x + 5)
\]
Whakautu:
I te mea he rārangi te āhua o te mahi, ā, kāore e puta ake he āhua kore-taurite, ka taea e tātou te tatau mā te whakakapinga tika.
\[
\lim_{x \to 2} (3x + 5) = 3(2) + 5 = 6 + 5 = 11
\]
Whakatau: Ko te uara rohe ko te 11.
-
Tauira 2: Te rohenga o te āhua mutunga kore \( \frac{0}{0} \) (Whakawehewehenga)
Pātai:
Tatau:
\[
\lim_{x \to 3} \frac{x^2 – 9}{x – 3}
\]
Whakautu:
Ki te whakakapi tika koe i te \( x = 3 \), ko te hua ko:
\[
\frac{9 – 9}{3 – 3} = \frac{0}{0}
\]
He āhua kore-taurite tēnei, nō reira me whakangawari. Tātaitia te taupū:
\[
x^2 – 9 = (x – 3)(x + 3)
\]
Nā reira:
\[
\frac{(x – 3)(x + 3)}{x – 3} = x + 3
\]
Tātaihia te rohe inaianei:
\[
\lim_{x \to 3} (x + 3) = 3 + 3 = 6
\]
Whakatau: Ko te uara rohe ko te 6.
-
Tauira 3: Ngā Herenga me ngā Pūtake (Whakataurite)
Pātai:
Whakatauhia:
\[
\lim_{x \to 4} \frac{\sqrt{x} – 2}{x – 4}
\]
Whakautu:
Ngā hua o te whakakapinga tika:
\[
\frac{2 – 2}{4 – 4} = \frac{0}{0}
\]
Te whakamārama mā te whakanui i ngā hoa:
\[
\frac{\sqrt{x} – 2}{x – 4} \cdot \frac{\sqrt{x} + 2}{\sqrt{x} + 2}
= \frac{x – 4}{(x – 4)(\sqrt{x} + 2)}
\]
Whakangāwaritia:
\[
= \frac{1}{\sqrt{x} + 2}
\]
Whakakapia inaianei \( x = 4 \):
\[
\frac{1}{\sqrt{4} + 2} = \frac{1}{2 + 2} = \frac{1}{4}
\]
Whakatau: Ko te uara rohe ko te 1/4.
-
Tauira Pātai 4: Ngā Herenga Pānga-toru Taketake
Pātai:
Tatau:
\[
\lim_{x \to 0} \frac{\sin x}{x}
\]
Whakautu:
He rohe tino rongonui tēnei mō te rohe taketake o te ine whārite:
\[
\lim_{x \to 0} \frac{\sin x}{x} = 1
\]
Ka taea te whakamatau mā te whakamahi i te āhuahanga, i te raupapa Taylor rānei, engari i te taumata kura he nui noa iho te maumahara hei tauira matua.
Whakatau: Ko te uara rohe ko te 1.
-
Tauira Pātai 5: Ngā Herenga Pānga-toru me te Whakarerekētanga
Pātai:
Whakatauhia:
\[
\lim_{x \to 0} \frac{1 – \cos x}{x^2}
\]
Whakautu:
Kei roto hoki i tēnei rohe ngā rohe taketake e whakamahia pinepinetia ana:
\[
\lim_{x \to 0} \frac{1 – \cos x}{x^2} = \frac{1}{2}
\]
Ki te hiahia koe ki te whakaiti i te uara, ka taea e koe te whakamahi i te tuakiri:
\[
1 – \cos x = 2\sin^2\left(\frac{x}{2}\right)
\]
Nō reira:
\[
\frac{1 – \cos x}{x^2} = \frac{2\sin^2(x/2)}{x^2}
= 2 \left(\frac{\sin(x/2)}{x}\right)^2
\]
Hurihia he iti:
\[
\frac{\sin(x/2)}{x} = \frac{\sin(x/2)}{x/2} \cdot \frac{1}{2}
\]
Nō reira, ko te rohe:
\[
2 \left(1 \cdot \frac{1}{2}\right)^2 = 2 \cdot \frac{1}{4} = \frac{1}{2}
\]
Whakatau: Ko te uara rohe ko te 1/2.
-
Tauira Pātai 6: Ngā Huarahi Herenga Mutunga Kore
Pātai:
Tatau:
\[
\lim_{x \to \infty} \frac{5x^2 + 3x}{2x^2 – 7}
\]
Whakautu:
Mō te rohe o tētahi mahi whaitake ina \( x \to \infty \), whakatairitea ngā nekehanga teitei rawa. Nā te mea he nekehanga 2 rāua, ko te rohe ko te ōwehenga o ngā tauwehenga teitei rawa:
\[
\lim_{x \to \infty} \frac{5x^2 + 3x}{2x^2 – 7} = \frac{5}{2}
\]
Whakatau: Ko te uara rohe ko te 5/2.
-
Tauira Pātai 7: Ngā Herenga me te Whakakapinga me te Whakangawari i ngā Hautau
Pātai:
Whakatauhia:
\[
\lim_{x \to 1} \frac{x^3 – 1}{x – 1}
\]
Whakautu:
Mā te whakakapinga tika ka puta te āhua \( \frac{0}{0} \). Tauwehe:
\[
x^3 – 1 = (x – 1)(x^2 + x + 1)
\]
Whakangāwaritia:
\[
\frac{(x – 1)(x^2 + x + 1)}{x – 1} = x^2 + x + 1
\]
Whakakapinga \( x = 1 \):
\[
1^2 + 1 + 1 = 3
\]
Whakatau: Ko te uara rohe ko te 3.
-
Te Katinga
Ka māmā ake te ako i ngā rohe mēnā ka mōhio koe ki ngā tauira taketake: te wā e whakamahia ai te whakakapinga tika, te wā e tauwehe ai, te wā e whakamārama ai, me te wā e whakamahia ai ngā tātai rohe pākoki. Mā te whakaharatau pinepine i ngā rapanga pēnei i te tauira i runga ake nei, ka taunga ake koe ki te whakahaere i ngā momo rohe, tae atu ki ngā mea e ahua uaua ana.
Ki te hiahia koe, ka taea hoki e au te waihanga i tētahi kete whakaharatau o ngā pātai herenga 20–30 me ngā kōrero taahiraa-i-te-taahiraa (mai i te taketake ki te matatau), ka taea rānei e au te whakarerekē ki te marautanga kura tuarua/kura mahi/UTBK.
