Piv txwv cov lus nug thiab kev sib tham txog cov ntsiab lus siab tshaj plaws: Tus nqi rov qab tsawg kawg nkaus thiab tus nqi rov qab siab tshaj plaws
Kev txiav txim siab txog cov ntsiab lus kawg, cov ntsiab lus uas ib qho kev ua haujlwm ncav cuag nws qhov tsawg kawg nkaus lossis siab tshaj plaws, yog lub tswv yim tseem ceeb hauv kev suav lej thiab kev tshuaj xyuas lej. Hauv tsab xov xwm no, peb yuav tshawb nrhiav yuav ua li cas nrhiav thiab tshuaj xyuas cov ntsiab lus kawg los ntawm ntau qhov piv txwv teeb meem uas muaj cov nqi rov qab tsawg kawg nkaus thiab siab tshaj plaws.
Cov Lus Txhais thiab Cov Kev Xav Txog Cov Ncauj Lus
Ua ntej peb tham txog cov teeb meem piv txwv, peb yuav tsum nkag siab txog qee lub ntsiab lus yooj yim thiab cov theorem:
1. Lub Ntsiab Lus Tseem Ceeb: Puas yog tus nqi ntawm \( x \) qhov twg thawj qhov derivative \( f'(x) \) ntawm lub luag haujlwm \( f(x) \) yog xoom lossis tsis muaj nyob.
2. Tus Nqi Rov Qab Siab Tshaj Plaws: Puas yog tus nqi ntawm \( f(x) \) uas loj dua tus nqi ntawm \( f(x) \) nyob ib puag ncig qhov ntawd.
3. Tus Nqi Rov Qab Yam Tsawg Kawg Nkaus: Tus nqi ntawm \( f(x) \) puas yog me dua tus nqi ntawm \( f(x) \) nyob ib puag ncig qhov ntawd.
4. Fermat's Theorem: Yog tias \(f \) muaj tus nqi siab tshaj plaws hauv zos ntawm \(c \) thiab qhov derivative \(f'(c) \) muaj, ces \(f'(c) = 0 \).
Piv txwv lus nug 1: Quadratic Functions
Ua ntej, peb pib nrog ib qho kev ua haujlwm quadratic yooj yim:
\[ f(x) = 2x^2 – 4x + 1 \]
Cov Kauj Ruam:
1. Nrhiav thawj qhov derivative ntawm \( f'(x) \):
\[
f'(x) = \frac{d}{dx}(2x^2 – 4x + 1) = 4x – 4
\]
2. Nrhiav cov ntsiab lus tseem ceeb los ntawm kev daws \( f'(x) = 0 \):
\[
4x – 4 = 0 \ txhais tau tias x = 1
\]
3. Txheeb xyuas tus nqi ntawm lub luag haujlwm ntawm qhov tseem ceeb:
\[
f(1) = 2(1)^2 – 4(1) + 1 = -1
\]
4. Siv tus lej thib ob los txiav txim seb qhov xwm txheej ntawm lub ntsiab lus yog dab tsi:
\[
f”(x) = \frac{d}{dx}(4x – 4) = 4
\]
Vim tias \( f”(1) > 0 \), qhov point \( x = 1 \) yog qhov point tsawg kawg nkaus hauv zos.
Piv txwv lus nug 2: Polynomial Functions
Tam sim no cia peb sim nrog ib qho polynomial function uas nyuaj dua:
\[ g(x) = x^3 – 3x^2 + 2 \]
Cov Kauj Ruam:
1. Txheeb xyuas thawj qhov derivative \( g'(x) \):
\[
g'(x) = \frac{d}{dx}(x^3 – 3x^2 + 2) = 3x^2 – 6x
\]
2. Nrhiav cov ntsiab lus tseem ceeb los ntawm kev daws \( g'(x) = 0 \):
\[
3x^2 – 6x = 0 \i txhais tau tias 3x(x – 2) = 0 \i txhais tau tias x = 0 \text{ los yog } x = 2
\]
3. Txheeb xyuas tus nqi ntawm lub luag haujlwm ntawm qhov tseem ceeb:
\[
g(0) = 0^3 – 3(0)^2 + 2 = 2
\]
\[
g(2) = 2^3 – 3(2)^2 + 2 = -2
\]
4. Siv tus lej thib ob los txiav txim seb qhov xwm txheej ntawm lub ntsiab lus yog dab tsi:
\[
g”(x) = \frac{d}{dx}(3x^2 – 6x) = 6x – 6
\]
\[
g”(0) = 6(0) – 6 = -6 \quad (\text{tus nqi siab tshaj plaws hauv zos})
\]
\[
g”(2) = 6(2) – 6 = 6 \quad (\text{tus nqi tsawg kawg nkaus hauv zos})
\]
Yog li, \(g(x) \) muaj qhov siab tshaj plaws hauv zos ntawm \(x = 0 \) thiab qhov tsawg kawg nkaus hauv zos ntawm \(x = 2 \).
Piv txwv lus nug 3: Cov Haujlwm Transcendental
Cia peb saib ib qho function uas muaj exponentiation:
\[ h(x) = xe^{-x} \]
Cov Kauj Ruam:
1. Txheeb xyuas thawj qhov derivative \( h'(x) \):
\[
h'(x) = \frac{d}{dx}(xe^{-x}) = e^{-x} – xe^{-x} = (1 – x)e^{-x}
\]
2. Nrhiav qhov chaw tseem ceeb los ntawm kev daws \( h'(x) = 0 \):
\[
(1 – x)e^{-x} = 0 \ txhais tau tias 1 – x = 0 \ txhais tau tias x = 1
\]
3. Txheeb xyuas tus nqi ntawm lub luag haujlwm ntawm qhov tseem ceeb:
\[
h(1) = 1e^{-1} = \frac{1}{e}
\]
4. Siv tus lej thib ob los txiav txim seb qhov xwm txheej ntawm lub ntsiab lus yog dab tsi:
\[
h”(x) = \frac{d}{dx}((1 – x)e^{-x}) = -e^{-x} – (1 – x)e^{-x} = (x – 2)e^{-x}
\]
\[
h”(1) = (1 – 2)e^{-1} = -\frac{1}{e}
\]
Vim tias \( h”(1) < 0 \), lub ntsiab lus \( x = 1 \) yog qhov siab tshaj plaws hauv zos. Piv txwv Teeb Meem 4: Kev Ua Haujlwm Rational Thaum kawg, peb soj ntsuam qhov kev ua haujlwm rational: \[ k(x) = \frac{x^2 + 2x}{x - 1} \] Cov Kauj Ruam: 1. Nrhiav thawj qhov derivative siv txoj cai 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. Nrhiav cov ntsiab lus tseem ceeb los ntawm kev daws \( k'(x) = 0 \): \[ \frac{x^2 - 2}{(x-1)^2} = 0 \implies x^2 - 2 = 0 \implies x = \pm \sqrt{2} \] 3. Txheeb xyuas tus nqi ntawm lub luag haujlwm ntawm qhov tseem ceeb: \[ k(\sqrt{2}) = \frac{(\sqrt{2})^2 + 2\sqrt{2}}{\sqrt{2} - 1} = \frac{2 + 2\sqrt{2}}{\sqrt{2} - 1} \times \frac{\sqrt{2} + 1}{\sqrt{2} + 1} = \frac{(2 + 2\sqrt{2})(\sqrt{2} + 1)}{1} = 4 + 4\sqrt{2} \] \[ k(-\sqrt{2}) = \frac{(-\sqrt{2})^2 + 2(-\sqrt{2})}{-\sqrt{2} - 1} = \frac{2 - 2\sqrt{2}}{-\sqrt{2} - 1} \times \frac{-\sqrt{2} + 1}{-\sqrt{2} + 1} = \frac{(2 - 2\sqrt{2})(-\sqrt{2} + 1)}{1} = -4 + 4\sqrt{2} \] 4. Siv qhov thib ob derivative los xyuas cov khoom ntawm lub ntsiab lus: \[ k''(x) = \frac{d}{dx}\left( \frac{x^2 - 2}{(x-1)^2} \right) \] Cov kev xam ntxiv tuaj yeem ua tiav los ntawm kev sib txawv \( k'(x) \) Cov txiaj ntsig yuav qhia seb \( x = \sqrt{2} \) thiab \( x = -\sqrt{2} \) yog qhov siab tshaj plaws hauv zos lossis qhov tsawg kawg nkaus. Xaus Lus Hauv tsab xov xwm no, peb tau tham txog ntau qhov teeb meem piv txwv uas qhia txog yuav ua li cas nrhiav qhov extrema, uas yog, qhov tsawg kawg nkaus thiab qhov siab tshaj plaws invertible cov nqi, ntawm ntau hom kev ua haujlwm. Cov txheej txheem siv suav nrog nrhiav thawj qhov derivative los nrhiav cov ntsiab lus tseem ceeb, siv qhov thib ob derivative los txiav txim siab qhov xwm txheej ntawm cov ntsiab lus, thiab kev soj ntsuam qhov kev ua haujlwm ntawm cov ntsiab lus ntawd. Qhov no yog lub hauv paus ruaj khov rau kev nkag mus tob dua rau hauv kev tshuaj xyuas kev ua haujlwm hauv calculus.
