Fa'ata'ita'iga o Fesili ma Talanoaga o Ponai Sili Ona Lelei: Tau Maualalo ma Tau Maualalo
O le fuafuaina o tulaga ogaoga, o tulaga ia e oʻo atu ai se galuega faatino i lona tau maualalo poʻo le maualuga, o se manatu autū lea i le calculus ma le auʻiliʻiliga faʻamatematika. I totonu o lenei tusiga, o le a tatou suʻesuʻeina pe faʻapefea ona suʻe ma auʻiliʻili tulaga ogaoga e ala i ni faʻataʻitaʻiga o faʻafitauli e aofia ai tau maualalo ma maualuga.
Faʻamatalaga ma Aʻoaʻoga o Mataupu
A'o le'i talanoaina ni fa'ata'ita'iga o fa'afitauli, e mana'omia ona tatou malamalama i nisi o manatu ma a'oa'oga fa'avae:
1. Tulaga Taua: O le tau lea o le \( x \) lea e leai ai pe leai foi le muamua derivative \( f'(x) \) o le function \( f(x) \).
2. Tau Maualuga o le Toe Faafoi: O le tau lea o le \( f(x) \) e sili atu nai lo le tau o le \( f(x) \) i tafatafa o lena tulaga.
3. Tau Maualalo o le Toe Faafoi: O le tau lea o le \( f(x) \) e laʻititi ifo nai lo le tau o le \( f(x) \) i tafatafa o lena tulaga.
4. Teorema a Fermat: Afai o le \( f \) e iai se tau fa'alotoifale i le \( c \) ma o le derivative \( f'(c) \) e iai, ona \( f'(c) = 0 \).
Fa'ata'ita'iga Fesili 1: Galuega Fa'atino Fa'atafafā
Muamua, tatou amata i se galuega fa'afa'afa faigofie:
\[ f(x) = 2x^2 – 4x + 1 \]
Laasaga-laasaga:
1. Saili le ulua'i fua fa'atatau o le \( f'(x) \):
\[
f'(x) = \frac{d}{dx}(2x^2 – 4x + 1) = 4x – 4
\]
2. Saili ia tulaga taua e ala i le foia o le \( f'(x) = 0 \):
\[
4x – 4 = 0 \fa'aalia ai le x = 1
\]
3. Fuafua le tau o le galuega faatino i le tulaga taua:
\[
f(1) = 2(1)^2 – 4(1) + 1 = -1
\]
4. Faaaoga le fa'asologa lona lua e iloa ai le natura o le manatu:
\[
f”(x) = \frac{d}{dx}(4x – 4) = 4
\]
Talu ai o le \( f”(1) > 0 \), o le tulaga \( x = 1 \) o se tulaga maualalo i le lotoifale.
Fa'ata'ita'iga Fesili 2: Galuega Fa'atino Fa'apolinomial
Ia, seʻi o tatou taumafai nei i se galuega faatino polynomial e sili atu ona faigata:
\[ g(x) = x^3 – 3x^2 + 2 \]
Laasaga-laasaga:
1. Fuafua le ulua'i fa'aliliuga \( g'(x) \):
\[
g'(x) = \frac{d}{dx}(x^3 – 3x^2 + 2) = 3x^2 – 6x
\]
2. Saili ia tulaga taua e ala i le foia o le \( g'(x) = 0 \):
\[
3x^2 – 6x = 0 \faʻaalia 3x(x – 2) = 0 \faʻaalia x = 0 \text{ poʻo } x = 2
\]
3. Fuafua le tau o le galuega faatino i le tulaga taua:
\[
g(0) = 0^3 – 3(0)^2 + 2 = 2
\]
\[
g(2) = 2^3 – 3(2)^2 + 2 = -2
\]
4. Faaaoga le fa'asologa lona lua e iloa ai le natura o le manatu:
\[
g”(x) = \frac{d}{dx}(3x^2 – 6x) = 6x – 6
\]
\[
g”(0) = 6(0) – 6 = -6 \quad (\text{tau maualuga i le lotoifale})
\]
\[
g”(2) = 6(2) – 6 = 6 \quad (\text{tau maualalo i le lotoifale})
\]
O lea la, o le \( g(x) \) e iai lona maualuga i le lotoifale i le \( x = 0 \) ma le maualalo i le lotoifale i le \( x = 2 \).
Faʻataʻitaʻiga Fesili 3: Galuega Faʻatino Faʻa-Transcendental
Se'i o tatou tilotilo i se galuega faatino e aofia ai le fa'ateleina o le numera:
\[ h(x) = xe^{-x} \]
Laasaga-laasaga:
1. Fuafua le uluaʻi faʻaliliuga \( h'(x) \):
\[
h'(x) = \frac{d}{dx}(xe^{-x}) = e^{-x} – xe^{-x} = (1 – x)e^{-x}
\]
2. Saili le tulaga taua e ala i le foia o le \( h'(x) = 0 \):
\[
(1 – x)e^{-x} = 0 \faʻaalia 1 – x = 0 \faʻaalia x = 1
\]
3. Fuafua le tau o le galuega faatino i le tulaga taua:
\[
h(1) = 1e^{-1} = \frac{1}{e}
\]
4. Faaaoga le fa'asologa lona lua e iloa ai le natura o le manatu:
\[
h”(x) = \frac{d}{dx}((1 – x)e^{-x}) = -e^{-x} – (1 – x)e^{-x} = (x – 2)e^{-x}
\]
\[
h”(1) = (1 – 2)e^{-1} = -\frac{1}{e}
\]
Talu ai \( h”(1) < 0 \), o le tulaga \( x = 1 \) o se maualuga fa'alotoifale. Fa'ata'ita'iga Fa'afitauli 4: Galuega Fa'atatau I le iuga, matou te iloiloina le galuega fa'atatau: \[ k(x) = \frac{x^2 + 2x}{x - 1} \] La'asaga: 1. Saili le ulua'i fa'asologa e fa'aaoga ai le tulafono o le quotient: \[ k'(x) = \frac{(2x+2)(x-1) - (x^2+2x)}{(x-1)^2} = \frac{2x^2 - 2x + 2x - 2 - x^2 - 2x}{(x-1)^2} = \frac{x^2 - 2}{(x-1)^2} \] 2. Saili ia tulaga taua e ala i le foia o le \( k'(x) = 0 \): \[ \frac{x^2 - 2}{(x-1)^2} = 0 \implies x^2 - 2 = 0 \fa'ailoa mai x = \pm \sqrt{2} \] 3. Fuafua le tau o le galuega faatino i le tulaga taua: \[ k(\sqrt{2}) = \frac{(\sqrt{2})^2 + 2\sqrt{2}}{\sqrt{2} - 1} = \frac{2 + 2\sqrt{2}}{\sqrt{2} - 1} \times \frac{\sqrt{2} + 1}{\sqrt{2} + 1} = \frac{(2 + 2\sqrt{2})(\sqrt{2} + 1)}{1} = 4 + 4\sqrt{2} \] \[ k(-\sqrt{2}) = \frac{(-\sqrt{2})^2 + 2(-\sqrt{2})}{-\sqrt{2} - 1} = \frac{2 - 2\sqrt{2}}{-\sqrt{2} - 1} \times \frac{-\sqrt{2} + 1}{-\sqrt{2} + 1} = \frac{(2 - 2\sqrt{2})(-\sqrt{2} + 1)}{1} = -4 + 4\sqrt{2} \] 4. Faaaoga le derivative lona lua e siaki ai meatotino o le point: \[ k''(x) = \frac{d}{dx}\left( \frac{x^2 - 2}{(x-1)^2} \right) \] E mafai ona faia isi fuafuaga e ala i le vaevaeina \( k'(x) \) O taunuuga o le a faailoa mai ai pe o \( x = \sqrt{2} \) ma le \( x = -\sqrt{2} \) o ni point aupito maualuga po o ni point aupito maualalo i le lotoifale. Faaiuga I lenei tusiga, ua matou talanoaina ni nai faataitaiga o faafitauli e faaalia ai le auala e maua ai le extrema, o lona uiga, o tau fa'afeagai maualalo ma maualuga, o ituaiga eseese o galuega faatino. O metotia e fa'aaogaina e aofia ai le sailia o le muamua derivative e su'e ai ni vaega taua, fa'aaoga le lona lua derivative e fuafua ai le natura o vaega, ma le iloiloina o le galuega faatino i na vaega. O se fa'avae mautu lea mo le su'esu'eina loloto o le au'ili'iliga o galuega faatino i le calculus.
