Piv txwv ntawm cov lus nug sib tham txog kev siv cov derivatives
Cov derivative yog ib lub tswv yim tseem ceeb ntawm calculus uas muaj ntau yam kev siv hauv lub neej txhua hnub thiab lwm yam kev tshawb fawb, xws li physics, economics, biology, thiab engineering. Hauv tsab xov xwm no, peb yuav tham txog ntau qhov teeb meem piv txwv thiab tham txog kev siv cov derivatives, tshwj xeeb tshaj yog nyob rau hauv cov ntsiab lus ntawm kev ua kom zoo dua qub thiab kev tshuaj xyuas kev ua haujlwm.
Kev Taw Qhia Txog Cov Ntawv Thov Uas Tau Muab Los
Tus derivative ntawm ib qho kev ua haujlwm feem ntau muab cov ntaub ntawv hais txog qhov kev hloov pauv ntawm qhov kev ua haujlwm ntawd nrog rau nws cov hloov pauv ywj pheej. Piv txwv yooj yim tshaj plaws yog qhov ceev, uas yog tus derivative ntawm txoj haujlwm nrog rau lub sijhawm. Dav dua, derivatives tuaj yeem siv los nrhiav qhov siab tshaj plaws thiab qis tshaj plaws ntawm ib qho kev ua haujlwm, txiav txim siab lub sijhawm uas ib qho kev ua haujlwm nce lossis txo qis, thiab muab cov ntaub ntawv hais txog cov khoom thiab cov duab ntawm ib qho kev ua haujlwm.
Piv txwv lus nug 1: Nrhiav cov nqi siab tshaj plaws thiab qis tshaj
Lo lus nug:
Txheeb xyuas cov ntsiab lus siab tshaj plaws thiab qis tshaj plaws ntawm qhov kev ua haujlwm \( f(x) = x^3 – 3x^2 + 4 \).
Kev Sib Tham:
1. Nrhiav thawj qhov derivative:
Yuav kom nrhiav tau cov ntsiab lus tseem ceeb, peb yuav tsum nrhiav thawj qhov derivative ntawm lub function thiab muab nws sib npaug rau xoom.
\[
f'(x) = 3x^2 – 6x
\]
\[
3x^2 – 6x = 0
\]
2. Daws qhov sib npaug:
Peb muab qhov sib npaug ntawm qhov sib npaug:
\[
3x(x – 2) = 0
\]
Yog li ntawd, peb tau txais cov ntsiab lus tseem ceeb ntawm \( x = 0 \) thiab \( x = 2 \).
3. Tshuaj xyuas qhov thib ob derivative:
Yuav kom txiav txim siab seb cov ntsiab lus tseem ceeb yog maxima lossis minima, peb yuav tsum nrhiav qhov thib ob derivative ntawm lub function:
\[
f”(x) = 6x – 6
\]
Kev ntsuam xyuas ntawm cov ntsiab lus tseem ceeb:
\[
f”(0) = 6(0) – 6 = -6 \, (\text{negative, yog li\ } x = 0 \text{\ yog qhov siab tshaj plaws hauv zos})
\]
\[
f”(2) = 6(2) – 6 = 6 \, (\text{positive, yog li\ } x = 2 \text{\ yog qhov tsawg kawg nkaus hauv zos})
\]
4. Xam cov nqi siab tshaj plaws thiab qis tshaj plaws:
Hloov cov ntsiab lus tseem ceeb rau hauv qhov kev ua haujlwm qub:
\[
f(0) = 0^3 – 3 \cdot 0^2 + 4 = 4 \, (\text{maximum})
\]
\[
f(2) = 2^3 – 3 \cdot 2^2 + 4 = 8 – 12 + 4 = 0 \, (\text{yam tsawg kawg nkaus})
\]
Yog li, qhov kev ua haujlwm \( f(x) = x^3 – 3x^2 + 4 \) muaj qhov siab tshaj plaws hauv zos ntawm \( (0, 4) \) thiab qhov tsawg kawg nkaus hauv zos ntawm \( (2, 0) \).
Piv txwv lus nug 2: Kev txhim kho nrog kev txwv
Lo lus nug:
Ib tug neeg ua liaj ua teb xav ua ib lub laj kab plaub fab uas nyob ib sab ntug dej. Muab 100 meters ntawm lub laj kab, txiav txim siab seb lub laj kab loj npaum li cas kom nws thaj chaw dav tshaj plaws.
Kev Sib Tham:
1. Ua ib qho kev sib npaug:
Xav tias qhov ntev ntawm lub qhov rooj uas sib luag rau tus dej yog \(x \) meters thiab qhov dav yog \(y \) meters. Vim tias ib sab ciam teb rau tus dej, lub laj kab uas xav tau yog rau peb sab.
\[
2y + x = 100
\]
2. Nrhiav thaj chaw siab tshaj plaws:
Thaj chaw ntawm lub tawb \( A \) yog:
\[
A = x² y
\]
Los ntawm kab zauv ntawm laj kab, peb tuaj yeem qhia \( y \) hauv cov nqe lus ntawm \( x \):
\[
y = \frac{100 – x}{2}
\]
Yog li, qhov sib npaug rau thaj chaw dhau los ua:
\[
A(x) = x²⁺ (100 – x)/2 = 50x – x^2/2
\]
3. Nrhiav thawj qhov derivative:
Yuav kom nrhiav tau tus nqi siab tshaj plaws, peb nrhiav tau thawj qhov derivative ntawm \( A(x) \):
\[
A'(x) = 50 – x
\]
Sib npaug rau xoom:
\[
50 – x = 0 \ txhais tau tias x = 50
\]
4. Xam tus nqi ntawm \( y \):
Hloov \( x = 50 \) rau hauv kab zauv:
\[
y = \frac{100 – 50}{2} = 25
\]
Yog li, qhov ntev ntawm lub tawb uas muab thaj chaw siab tshaj plaws yog 50 meters rau ntev thiab 25 meters rau dav.
Piv txwv lus nug 3: Txheeb xyuas qhov ceev tshaj plaws
Lo lus nug:
Ib lub khoom me me txav mus raws txoj kab ncaj nraim nrog qhov chaw qhia ua lub luag haujlwm ntawm lub sijhawm \( s(t) = t^3 – 6t^2 + 9t + 1 \). Txheeb xyuas qhov ceev tshaj plaws ntawm lub khoom me me.
Kev Sib Tham:
1. Txheeb xyuas qhov ceev (qhov chaw derivative):
Qhov ceev ntawm ib qho khoom me me yog thawj qhov derivative ntawm txoj hauj lwm nrog rau lub sijhawm:
\[
v(t) = \frac{ds}{dt} = 3t^2 – 12t + 9
\]
2. Txheeb xyuas qhov thib ob derivative:
Yuav kom nrhiav tau cov ntsiab lus siab tshaj plaws, peb pom qhov thib ob derivative:
\[
a(t) = \frac{dv}{dt} = 6t – 12
\]
3. Nrhiav qhov tseem ceeb:
Sib npaug thawj qhov derivative ntawm qhov ceev rau xoom:
\[
3t^2 – 12t + 9 = 0
\]
Faib los ntawm 3:
\[
t^2 – 4t + 3 = 0
\]
Kev suav lej:
\[
(t – 3)(t – 1) = 0
\]
Yog li, cov ntsiab lus tseem ceeb yog \(t = 1 \) thiab \(t = 3 \).
4. Tshawb xyuas qhov kev nrawm kom nrhiav tau qhov siab tshaj plaws:
\[
a(1) = 6(1) – 12 = -6 \implies t = 1 \text{\ yog qhov siab tshaj plaws hauv zos}
\]
\[
a(3) = 6(3) – 12 = 6 \implies t = 3 \text{\ yog qhov tsawg kawg nkaus hauv zos}
\]
5. Xam qhov ceev tshaj plaws:
Hloov \(t = 1\) rau hauv kab zauv ceev:
\[
v(1) = 3(1)^2 – 12(1) + 9 = 3 – 12 + 9 = 0 \, (\text{tsis nthuav})
\]
Tshawb xyuas lwm cov kev txwv lossis cov ntsiab lus sib nrug kom paub tseeb tias qhov kev daws teeb meem zoo tshaj plaws.
Nrog cov kauj ruam no, peb tuaj yeem tsim ib qho qauv kev daws teeb meem raws li derivative rau ntau yam teeb meem thov saum toj no.
Xaus
Cov piv txwv saum toj no qhia tau hais tias yuav siv cov derivatives li cas los daws cov teeb meem hauv ntau yam xwm txheej. Kev nrhiav cov nqi siab tshaj plaws thiab qis tshaj plaws, kev ua kom zoo tshaj plaws, thiab kev tshuaj xyuas kev txav mus los tsuas yog ob peb qhov kev siv ntawm lub tswv yim ntawm derivatives. Kev paub txog cov txheej txheem thiab cov txheej txheem no yog qhov tseem ceeb rau cov neeg kawm lej siab heev thiab cov kev kawm cuam tshuam.
