Mienzaniso yemibvunzo inokurukura nezveZvikamu Zvakanyanya zveKudzoka Kwemari Kushoma uye Kudzoka Kwemari Kukuru

Mibvunzo yeMienzaniso neKukurukurirana kweMapoinzi Akanyanya: Kudzoka Kushoma uye Kudzoka Kukuru

Kuziva mapoinzi akanyanya, mapoinzi apo basa rinosvika padanho rayo repamusoro kana kuti repamusoro, ipfungwa huru mukuverenga nekuongorora masvomhu. Muchinyorwa chino, tichaongorora mawaniro ekuwana nekuongorora mapoinzi akanyanya kuburikidza nematambudziko akati wandei ane chekuita nekudzoka kwemari shoma nerepamusoro.

Tsanangudzo dzeZvidzidzo neDzidziso

Tisati takurukura nezvematambudziko emuenzaniso, tinofanira kunzwisisa dzimwe pfungwa huru nedzidziso:

1. Pfungwa Inokosha: Ndiko kukosha kwe \( x \) apo derivative yekutanga \( f'(x) \) yebasa \( f(x) \) iri zero kana kuti haipo.
2. Kukosha Kwekudzoka Kwakanyanya: Kukosha kwe \( f(x) \) ndiko kwakakura kupfuura kukosha kwe \( f(x) \) padyo nenzvimbo iyoyo.
3. Kudzoserwa Kwemari Kushoma: Ndiko kukosha kwe \( f(x) \) kudiki pane kukosha kwe \( f(x) \) padyo nenzvimbo iyoyo.
4. Dzidziso yaFermat: Kana \( f \) ine kukosha kwakanyanya kwenzvimbo pa \( c \) uye derivative \( f'(c) \) iripo, saka \( f'(c) = 0 \).

Muenzaniso Mubvunzo 1: Mabasa eQuadratic

Kutanga, tinotanga nebasa riri nyore re quadratic:

\[ f(x) = 2x^2 – 4x + 1 \]

Kurongeka:

1. Tsvaga derivative yekutanga ye \( f'(x) \):
\[
f'(x) = \frac{d}{dx}(2x^2 – 4x + 1) = 4x – 4
\]

2. Tsvaga pfungwa dzakakosha nekugadzirisa \( f'(x) = 0 \):
\[
4x – 4 = 0 \zvinoreva x = 1
\]

3. Sarudza kukosha kwebasa iri panzvimbo yakakosha:
\[
f(1) = 2(1)^2 – 4(1) + 1 = -1
\]

4. Shandisa chirevo chechipiri kuti uone rudzi rwepfungwa yacho:
\[
f”(x) = \frac{d}{dx}(4x – 4) = 4
\]
Sezvo \( f”(1) > 0 \), poindi \( x = 1 \) ipoindi shoma yemuno.

Muenzaniso Mubvunzo 2: Mabasa ePolynomial

Zvino ngatiedzei nebasa repolynomial rakaoma kunzwisisa:

\[ g(x) = x^3 – 3x^2 + 2 \]

Kurongeka:

1. Sarudza derivative yekutanga \( g'(x) \):
\[
g'(x) = \frac{d}{dx}(x^3 – 3x^2 + 2) = 3x^2 – 6x
\]

2. Tsvaga pfungwa dzakakosha nekugadzirisa \( g'(x) = 0 \):
\[
3x^2 – 6x = 0 \zvinoreva 3x(x – 2) = 0 \zvinoreva x = 0 \zvinyorwa{ kana } x = 2
\]

3. Sarudza kukosha kwebasa iri panzvimbo yakakosha:
\[
g(0) = 0^3 – 3(0)^2 + 2 = 2
\]
\[
g(2) = 2^3 – 3(2)^2 + 2 = -2
\]

4. Shandisa chirevo chechipiri kuti uone rudzi rwepfungwa yacho:
\[
g”(x) = \frac{d}{dx}(3x^2 – 6x) = 6x – 6
\]
\[
g”(0) = 6(0) – 6 = -6 \quad (\text{local maximum value})
\]
\[
g”(2) = 6(2) – 6 = 6 \quad (\text{local minimum value})
\]

Saka, \( g(x) \) ine huwandu hwepamusoro hwenzvimbo pa \( x = 0 \) uye huwandu hwepamusoro hwenzvimbo pa \( x = 2 \).

Muenzaniso Mubvunzo 3: Mabasa Ekuchinjana Kwenyika

Ngatitarisei basa rinosanganisira exponentiation:

\[ h(x) = xe^{-x} \]

Kurongeka:

1. Sarudza derivative yekutanga \( h'(x) \):
\[
h'(x) = \frac{d}{dx}(xe^{-x}) = e^{-x} – xe^{-x} = (1 – x)e^{-x}
\]

2. Tsvaga chinhu chakakosha nekugadzirisa \( h'(x) = 0 \):
\[
(1 – x)e^{-x} = 0 \zvinoreva 1 – x = 0 \zvinoreva x = 1
\]

3. Sarudza kukosha kwebasa iri panzvimbo yakakosha:
\[
h(1) = 1e^{-1} = \frac{1}{e}
\]

4. Shandisa chirevo chechipiri kuti uone rudzi rwepfungwa yacho:
\[
h”(x) = \frac{d}{dx}((1 – x)e^{-x}) = -e^{-x} – (1 – x)e^{-x} = (x – 2)e^{-x}
\]
\[
h”(1) = (1 – 2)e^{-1} = -\frac{1}{e}
\]
Sezvo \( h”(1) < 0 \), poindi \( x = 1 \) iri nzvimbo yepamusoro. Muenzaniso Dambudziko 4: Mabasa Ekunzwisisa Pakupedzisira, tinoongorora basa rekunzwisisa: \[ k(x) = \frac{x^2 + 2x}{x - 1} \]

Matanho: 1. Tsvaga derivative yekutanga uchishandisa mutemo we 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. Tsvaga mapoinzi akakosha nekugadzirisa \( k'(x) = 0 \): \[ \frac{x^2 - 2}{(x-1)^2} = 0 \inoreva x^2 - 2 = 0 \inoreva x = \pm \sqrt{2} \] 3. Tsvaga kukosha kwebasa panzvimbo dzakakosha: \[ k(\sqrt{2}) = \frac{(\sqrt{2})^2 + 2\sqrt{2}}{\sqrt{2} - 1} = \frac{2 + 2\sqrt{2}}}{\sqrt{2} - 1} \times \frac{\sqrt{2} + 1}{\sqrt{2} + 1} = \frac{(2 + 2\sqrt{2})(\sqrt{2} + 1)}{1} = 4 + 4\sqrt{2} \] \[ k(-\sqrt{2}) = \frac{(-\sqrt{2})^2 + 2(-\sqrt{2})}{-\sqrt{2} - 1} = \frac{2 - 2\sqrt{2}}{-\sqrt{2} - 1} \times \frac{-\sqrt{2} + 1}{-\sqrt{2} + 1} = \frac{(2 - 2\sqrt{2})(-\sqrt{2} + 1)}{1} = -4 + 4\sqrt{2} \] 4. Shandisa derivative yechipiri kuti utarise hunhu hwepoindi: \[ k''(x) = \frac{d}{dx}\left( \frac{x^2 - 2}{(x-1)^2} \right) \] Mamwe maverengero anogona kuitwa nekusiyanisa patsva \( k'(x) \) izvo zvicharatidza kana \( x = \sqrt{2} \) uye \( x = -\sqrt{2} \) zviri local maxima kana minima. Mhedziso Muchinyorwa chino, takurukura mienzaniso yakati wandei inoratidza mawaniro ekuwana extrema, kureva minimum uye maximum inverse values, emhando dzakasiyana dzemabasa. Matekiniki anoshandiswa anosanganisira kuwana derivative yekutanga kuti uwane mapoindi akakosha, uchishandisa derivative yechipiri kuti uone hunhu hwemapoindi, uye kuongorora basa panzvimbo idzodzo. Izvi zvinopa hwaro hwakasimba hwekuongorora zvakadzama mashandiro ekuongorora mukuverenga.