Nā nīnau a me nā pane hoʻohālike no nā palena
ʻO nā palena kekahi o nā manaʻo koʻikoʻi loa i ka makemakika, ʻoi aku hoʻi i ka calculus. Hiki i nā palena ke hoʻomaopopo i ke ʻano o kahi hana i ka wā e hoʻokokoke aku ai kona waiwai loli i kahi helu, mai ka hema a i ʻole ka ʻākau. ʻO kēia manaʻo ke kumu no nā kūkākūkā ʻana o nā derivatives a me nā integrals. Ma kēia ʻatikala, e ʻike ʻoe i kahi wehewehe pōkole, me kekahi mau nīnau a me nā pane hoʻohālike e pili ana i nā palena e ʻike pinepine ʻia i ka hana a me nā hoʻokolohua.
He Hoʻomaopopo Pōkole o nā Palena
Ma keʻano laulā, hōʻike ka palena i ka waiwai a kahi hana e "hoʻokokoke aku ai" i ka wā e hoʻokokoke aku ai kāna loli i kahi waiwai. Ua kākau ʻia penei:
\[
lim_{x `to a} f(x)
\]
ʻO ia hoʻi, makemake mākou e ʻike i ka waiwai o \( f(x) \) i ka wā e hoʻokokoke aku ai ʻo \( x \) iā \( a \). E hoʻomanaʻo ʻaʻole like mau ka palena me ka waiwai o ka hana ma ia wahi (hiki paha ke wehewehe ʻole ʻia ka hana ma ia wahi), akā hiki nō ke noho ka palena.
Nā ʻAno Maʻamau o nā Pilikia Palena
ʻO kekahi mau ʻano nīnau palena i aʻo pinepine ʻia:
1. Palena o ka pani pololei (inā he hoʻomau ka hana ma ia wahi).
2. Nā palena o ke ʻano palena ʻole e like me \( \frac{0}{0} \) a i ʻole \( \frac{\infty}{\infty} \).
3. Nā palena e pili ana i nā aʻa.
4. Nā palena trigonometric.
5. Hele ka palena a hiki i ka palena ʻole.
I mea e maopopo ai, e hele kākou i nā nīnau hoʻohālike a me kā lākou kūkākūkā ʻana.
-
Laʻana 1: Nā Palena me ka Hoʻololi Pololei
Nīnau:
E hoʻoholo i ka waiwai:
\[
\lim_{x \to 2} (3x + 5)
\]
Pane:
ʻOiai he linear ke ʻano o ka hana a ʻaʻole ia e hāʻawi i nā ʻano indeterminate, hiki iā mākou ke helu iā ia ma o ka hoʻololi pololei.
\[
lim_{x 2} (3x + 5) = 3(2) + 5 = 6 + 5 = 11
\]
Hopena: ʻO ka palena palena he 11.
-
Laʻana 2: Palena o ke ʻAno Pau ʻole \( \frac{0}{0} \) (Factorization)
Nīnau:
Helu:
\[
\lim_{x \to 3} \frac{x^2 – 9}{x – 3}
\]
Pane:
Inā ʻoe e pani pololei iā \( x = 3 \), ʻo ka hopena penei:
\[
\frac{9 – 9}{3 – 3} = \frac{0}{0}
\]
He ʻano indeterminate kēia, no laila pono e hoʻomaʻalahi ʻia. E hoʻohālikelike i ka helu:
\[
x^2 – 9 = (x – 3)(x + 3)
\]
No laila:
\[
\frac{(x – 3)(x + 3)}{x – 3} = x + 3
\]
I kēia manawa e helu i ka palena:
\[
\lim_{x \to 3} (x + 3) = 3 + 3 = 6
\]
Hopena: ʻO ka palena palena he 6.
-
Laʻana 3: Nā Palena me nā Aʻa (Rationalization)
Nīnau:
E hoʻoholo:
\[
\lim_{x \to 4} \frac{\sqrt{x} – 2}{x – 4}
\]
Pane:
Loaʻa nā hopena o ka hoʻololi pololei:
\[
\frac{2 – 2}{4 – 4} = \frac{0}{0}
\]
Ka hoʻākāka ʻana ma ka hoʻonui ʻana i nā hoaaloha:
\[
\frac{\sqrt{x} – 2}{x – 4} \cdot \frac{\sqrt{x} + 2}{\sqrt{x} + 2}
= \frac{x – 4}{(x – 4)(\sqrt{x} + 2)}
\]
E hoʻomaʻalahi:
\[
= \frac{1}{\sqrt{x} + 2}
\]
I kēia manawa, e pani i ka \( x = 4 \):
\[
\frac{1}{\sqrt{4} + 2} = \frac{1}{2 + 2} = \frac{1}{4}
\]
Hopena: ʻO ka palena waiwai he 1/4.
-
Laʻana Nīnau 4: Nā Palena Trigonometric Kumu
Nīnau:
Helu:
\[
\lim_{x \to 0} \frac{\sin x}{x}
\]
Pane:
He palena kumu kaulana loa kēia palena o ka trigonometry:
\[
\lim_{x \to 0} \frac{\sin x}{x} = 1
\]
Hiki ke hōʻoia ʻia me ka hoʻohana ʻana i ke geometry a i ʻole ka moʻo Taylor, akā ma ka pae kula ua lawa ka hoʻomanaʻo ʻana iā ia ma ke ʻano he haʻilula koʻikoʻi.
Hopena: ʻO ka palena palena he 1.
-
Laʻana Nīnau 5: Nā Palena Trigonometric me ka Hoʻopunipuni
Nīnau:
E hoʻoholo:
\[
\lim_{x \to 0} \frac{1 – \cos x}{x^2}
\]
Pane:
Hoʻokomo pū kēia palena i nā palena kumu i hoʻohana pinepine ʻia:
\[
\lim_{x \to 0} \frac{1 – \cos x}{x^2} = \frac{1}{2}
\]
Inā makemake ʻoe e hoʻohaʻahaʻa iā ia, hiki iā ʻoe ke hoʻohana i ke ʻano inoa:
\[
1 – \cos x = 2\sin^2\left(\frac{x}{2}\right)
\]
No laila:
\[
\frac{1 – \cos x}{x^2} = \frac{2\sin^2(x/2)}{x^2}
= 2 \left(\frac{\sin(x/2)}{x}\right)^2
\]
E hoʻololi iki:
\[
\frac{\sin(x/2)}{x} = \frac{\sin(x/2)}{x/2} \cdot \frac{1}{2}
\]
No laila, ʻo ka palena:
\[
2 \left(1 \cdot \frac{1}{2}\right)^2 = 2 \cdot \frac{1}{4} = \frac{1}{2}
\]
Hopena: ʻO ka palena waiwai he 1/2.
-
Laʻana Nīnau 6: Nā Palena Hoʻokokoke Pau ʻole
Nīnau:
Helu:
\[
\lim_{x \to \infty} \frac{5x^2 + 3x}{2x^2 – 7}
\]
Pane:
No ka palena o kahi hana rational i ka wā \( x \to \infty \), e hoʻohālikelike i nā kekelē kiʻekiʻe loa. ʻOiai ʻo lāua ʻelua o ke kekelē 2, ʻo ka palena ka lakio o nā coefficients kiʻekiʻe loa:
\[
\lim_{x \to \infty} \frac{5x^2 + 3x}{2x^2 – 7} = \frac{5}{2}
\]
Hopena: ʻO ka palena waiwai he 5/2.
-
Laʻana Nīnau 7: Nā Palena me ka Hoʻololi a me ka Hoʻomaʻalahi ʻana o nā Hapa
Nīnau:
E hoʻoholo:
\[
\lim_{x \to 1} \frac{x^3 – 1}{x – 1}
\]
Pane:
Hāʻawi ka hoʻololi pololei i ke ʻano \( \frac{0}{0} \). Kumu hoʻohālikelike:
\[
x^3 – 1 = (x – 1)(x^2 + x + 1)
\]
E hoʻomaʻalahi:
\[
\frac{(x – 1)(x^2 + x + 1)}{x – 1} = x^2 + x + 1
\]
Pānaʻi \( x = 1 \):
\[
1^2 + 1 + 1 = 3
\]
Hopena: ʻO ka palena palena he 3.
-
Pani
E maʻalahi loa ke aʻo ʻana i nā palena inā ʻoe e aʻo pono i nā ʻano kumu: i ka wā e hoʻohana ai i ka hoʻololi pololei, i ka wā e hoʻohālikelike ai, i ka wā e hoʻākāka ai, a i ka wā e hoʻohana ai i nā ʻano palena trigonometric. Ma ka hoʻomaʻamaʻa pinepine ʻana i nā pilikia e like me ka laʻana ma luna, e maʻa mau ʻoe i ka lawelawe ʻana i nā ʻano palena like ʻole, ʻoiai nā mea i ʻike ʻia he paʻakikī.
Inā makemake ʻoe, hiki iaʻu ke hana i kahi pūʻolo hoʻomaʻamaʻa o 20-30 mau nīnau palena me nā kūkākūkā ʻanuʻu (mai ke kumu a i ka holomua), a i ʻole e hoʻololi iā ia i ka papahana kula kiʻekiʻe/kula ʻoihana/UTBK.
