Fa'ata'ita'iga o Fesili e Talanoaina ai le Fa'aogaina o Tapula'a o Galuega Fa'atino
O le tapula'a o se galuega faatino o se manatu autū i le calculus, e masani ona fa'aaogaina e fuafua ai le amioga a se galuega faatino a'o fa'alatalata atu i se tulaga fa'apitoa. I le matematika, aemaise lava le calculus, o le malamalama i le tapula'a o se galuega faatino e taua tele mo le fa'atuina o le fa'avae mo isi manatu e pei o derivatives ma integrals. O lenei tusiga o le a aofia ai fa'ata'ita'iga o fa'afitauli ma talanoaina ai fa'aoga o tapula'a galuega faatino e tu'uina atu ai se malamalamaga loloto i lenei autu.
Fa'atomuaga i Tapula'a o Galuega
O le tapula'a o se galuega faatino e fa'amatalaina ai le tau e latalata i ai le galuega faatino a'o latalata atu le fesuia'iga i se tau patino. E lua ituaiga o tapula'a e masani ona talanoaina: tapula'a e tasi le itu (tapula'a agavale ma le tapula'a taumatau) ma tapula'a e lua itu. O le fa'ailoga lautele mo le tapula'a o se galuega faatino \( f(x) \) a'o latalata atu \( x \) \( a \) o le:
\[
\lim_{x \to a} f(x)
\]
Fa'ata'ita'iga Fesili 1: Tapula'a Fa'avae
Fesili:
Fuafua le tau o le \(\lim_{x \to 2} (3x + 1)\).
Talanoaga:
O se faʻataʻitaʻiga lea o se tapulaʻa faʻavae lea o le galuega faatino \( f(x) = 3x + 1 \) o se galuega faatino laina e faʻaauau pea i lona vaega atoa. Ona mafai lea ona tatou sui saʻo le tau o le \( x = 2 \) i totonu o le galuega faatino.
\[
\lim_{x \to 2} (3x + 1) = 3(2) + 1 = 6 + 1 = 7
\]
O lea la, \(\lim_{x \to 2} (3x + 1) = 7\).
Fa'ata'ita'iga Fesili 2: Tapula'a i le Vaevaega i le Zero
Fesili:
Fuafua le taua o le \(\lim_{x \to 3} \frac{x^2 – 9}{x – 3}\).
Talanoaga:
Afai tatou te sui sa'o le \( x = 3 \) i totonu o le galuega faatino, o le a tatou maua le fomu e le mautinoa \(\frac{0}{0}\). O le mea lea, e tatau ona tatou faafaigofieina muamua le galuega faatino.
Ia mātau o le numerator \( x^2 – 9 \) o se fa'atulagaga fa'atafafā e mafai ona fa'avasegaina:
\[
x^2 – 9 = (x – 3)(x + 3)
\]
O lea la, e mafai ona toe tusia le galuega muamua e pei ona taua i lalo:
\[
\frac{x^2 – 9}{x – 3} = \frac{(x – 3)(x + 3)}{x – 3}
\]
Mai iinei, e mafai ona tatou faafaigofieina e ala i le faaleaogaina o le \( x – 3 \) i le numerator ma le denominator, pe afai o le \( x \neq 3 \):
\[
\frac{(x – 3)(x + 3)}{x – 3} = x + 3
\]
O lea la ua mafai ona tatou fuafua sa'o le tapula'a e ala i le suiina o le \( x = 3 \):
\[
\lim_{x \to 3} (x + 3) = 3 + 3 = 6
\]
O lea la, \(\lim_{x \to 3} \frac{x^2 – 9}{x – 3} = 6\).
Faʻataʻitaʻiga 3: Tapulaʻa ma Galuega Faʻavae
Fesili:
Saili le tau o le \(\lim_{x \to 1} \frac{\sqrt{x + 3} – 2}{x – 1}\).
Talanoaga:
Afai tatou te sui sa'o le \( x = 1 \) i totonu o le galuega faatino, o le a tatou maua le foliga e le mautinoa \(\frac{0}{0}\). Mo le foia o lenei mea, e mana'omia ona tatou fa'afaigofieina le galuega faatino. O le tasi auala o le fa'amatalaina lea o le numerator.
Tatou fa'ateleina le numerator ma le denominator i le conjugate o le numerator:
\[
\frac{\sqrt{x + 3} – 2}{x – 1} \cdot \frac{\sqrt{x + 3} + 2}{\sqrt{x + 3} + 2}
\]
Ona tatou maua lea:
\[
\frac{(\sqrt{x + 3} – 2)(\sqrt{x + 3} + 2)}{(x – 1)(\sqrt{x + 3} + 2)} = \frac{(x + 3) – 4}{(x – 1)(\sqrt{x + 3} + 2)}
\]
Fa'afaigofie le numera:
\[
x + 3 – 4 = x – 1
\]
O lena la:
\[
\frac{x – 1}{(x – 1)(\sqrt{x + 3} + 2)} = \frac{1}{\sqrt{x + 3} + 2}
\]
O lea la ua mafai ona tatou fuafuaina le tapulaʻa e ala i le sui \( x = 1 \):
\[
\lim_{x \to 1} \frac{1}{\sqrt{x + 3} + 2} = \frac{1}{\sqrt{1 + 3} + 2} = \frac{1}{\sqrt{4} + 2} = \frac{1}{2 + 2} = \frac{1}{4}
\]
O lea la, \(\lim_{x \to 1} \frac{\sqrt{x + 3} – 2}{x – 1} = \frac{1}{4}\).
Fa'ata'ita'iga Fesili 4: Tapula'a ma le Trigonometry
Fesili:
Fuafua le tau o le \(\lim_{x \to 0} \frac{\sin(3x)}{x}\).
Talanoaga:
Ua tatou iloa o tapulaʻa autū o le trigonometry, o loʻo i ai tapulaʻa lauiloa nei:
\[
\lim_{x \to 0} \frac{\sin(x)}{x} = 1
\]
Mo lenei faʻafitauli, e manaʻomia ona tatou faʻafesoʻotaʻi i lena faʻatulagaga autu. Manatua o le \( 3x \) o le finauga lea o le sine. E mafai ona tatou faʻaalia le tapulaʻa e ala i le faʻaogaina e pei ona taua i lalo:
\[
\lim_{x \to 0} \frac{\sin(3x)}{x} = \lim_{x \to 0} \frac{\sin(3x)}{3x} \cdot 3
\]
Ona o le \( \lim_{u \to 0} \frac{\sin(u)}{u} = 1 \) faatasi ai ma le \( u = 3x \), o lea la:
\[
\lim_{x \to 0} \frac{\sin(3x)}{3x} = 1
\]
O lea la:
\[
\lim_{x \to 0} \frac{\sin(3x)}{x} = 1 \cdot 3 = 3
\]
O lea la, \(\lim_{x \to 0} \frac{\sin(3x)}{x} = 3\).
I'uga
Ua aofia i lenei tusiga ni faʻataʻitaʻiga o faʻafitauli ma talanoaina ai le faʻaaogaina o tapulaʻa o galuega faatino i le calculus. I faʻataʻitaʻiga taʻitasi o faʻafitauli, e amata le talanoaga i le faʻailoaina o le foliga na maua pe a suiina ni tau ona suʻesuʻe lea o auala e faʻafaigofie ai pe faʻamatala ai le galuega faatino. O le malamalama i tapulaʻa o galuega faatino ma le auala e foia ai e taua tele mo le aʻoaʻoina o manatu faʻamatematika maualuluga, e pei o derivatives ma integrals. Faatasi ai ma le faʻataʻitaʻi faifai pea, o lou malamalama i tapulaʻa o galuega faatino o le a malosi ma loloto.
