Misali na tambayar tattaunawa kan amfani da abubuwan da aka samo asali
Asalin kalmar lissafi (derivative) wata babbar ma'ana ce ta lissafi wadda ke da amfani da yawa a rayuwar yau da kullum da sauran fannoni na kimiyya, kamar kimiyyar lissafi, tattalin arziki, ilmin halitta, da injiniyanci. A cikin wannan labarin, za mu tattauna misalai da dama na matsaloli kuma mu tattauna aikace-aikacen da aka samo asali, musamman a fannin ingantawa da nazarin ayyuka.
Gabatarwa ga Aikace-aikacen da aka Samu
Asalin aikin da aka samo asali daga ciki yana ba da bayanai game da saurin canjin wannan aikin dangane da canjinsa mai zaman kansa. Misali mafi sauƙi shine saurin, wanda shine asalin matsayi dangane da lokaci. Mafi faɗi, ana iya amfani da bambance-bambance don nemo matsakaicin da mafi ƙarancin ƙimar aiki, ƙayyade tazara tsakanin inda aikin ke ƙaruwa ko raguwa, da kuma samar da bayanai game da halaye da halayen zane na aikin.
Misali Tambaya ta 1: Nemo Matsakaici da Matsakaici
Tambaya:
Kayyade matsakaicin da mafi ƙarancin maki na aikin \( f(x) = x^3 – 3x^2 + 4 \).
Tattaunawa:
1. Nemo asalin farko:
Domin nemo muhimman abubuwan, muna buƙatar nemo farkon abin da aka samo daga aikin kuma mu daidaita shi da sifili.
\[
f'(x) = 3x^2 – 6x
\]
\[
3x^2 – 6x = 0
\]
2. Warware lissafin:
Mun ƙididdige lissafin:
\[
3x(x – 2) = 0
\]
Saboda haka, muna samun mahimman maki a \( x = 0 \) da \( x = 2 \).
3. Yi nazarin abin da aka samo daga na biyu:
Domin tantance ko mahimman abubuwan sune maxima ko minima, muna buƙatar nemo na biyu daga cikin aikin:
\[
f”(x) = 6x – 6
\]
Kimantawa a muhimman wurare:
\[
f”(0) = 6(0) – 6 = -6 \, (\text{negative, so\ } x = 0 \text{\ matsakaicin yanki ne})
\]
\[
f”(2) = 6(2) – 6 = 6 \, (\text{positive, so\ } x = 2 \text{\ shine mafi ƙarancin yanki})
\]
4. Lissafa matsakaicin da mafi ƙarancin ƙima:
Maye gurbin muhimman abubuwan zuwa aikin asali:
\[
f(0) = 0^3 – 3 \cdot 0^2 + 4 = 4 \, (\text{maximum})
\]
\[
f(2) = 2^3 – 3 \cdot 2^2 + 4 = 8 – 12 + 4 = 0 \, (\text{mafi ƙarancin})
\]
Don haka, aikin \( f(x) = x^3 – 3x^2 + 4 \) yana da matsakaicin yanki a \( (0, 4) \) da kuma mafi ƙarancin yanki a \( (2, 0) \).
Misali Tambaya ta 2: Ingantawa tare da Takamaiman Sharuɗɗa
Tambaya:
Manomi yana son gina katanga mai kusurwa huɗu da ke kewaye da kogi. Idan aka yi la'akari da shinge mai tsawon mita 100, a ƙayyade girman katanga don ƙara girman yankinsa.
Tattaunawa:
1. Yi lissafi:
A ce tsawon wurin da ke daidai da kogin mita ne (x) kuma faɗinsa mita ne (y) (y). Tunda gefe ɗaya yana kewaye da kogin, shingen da ake buƙata yana da ɓangarori uku.
\[
Shekaru 2 + x = 100
\]
2. Nemo matsakaicin yanki:
Yankin kejin \(A \) shine:
\[
A = x \cdot y
\]
Daga lissafin shinge, za mu iya bayyana \( y \) dangane da \( x \):
\[
y = \frac{100 – x}{2}
\]
Don haka, daidaiton yanki zai zama:
\[
A(x) = x \cdot \frac{100 – x}{2} = 50x – \frac{x^2}{2}
\]
3. Nemo asalin farko:
Domin nemo matsakaicin ƙima, mun sami farkon abin da aka samo daga \( A(x) \):
\[
A'(x) = 50 – x
\]
Daidaita zuwa sifili:
\[
50 – x = 0 \yana nufin x = 50
\]
4. Lissafa ƙimar \( y \):
Sauya \( x = 50 \) cikin lissafin:
\[
y = \frac{100 – 50}{2} = 25
\]
Saboda haka, girman kejin da ke samar da matsakaicin yanki shine mita 50 don tsayi da mita 25 don faɗi.
Misali Tambaya ta 3: Ƙayyade Matsakaicin Gudu
Tambaya:
Kwayar cuta tana tafiya a kan layi madaidaiciya tare da matsayin da aka bayyana a matsayin aikin lokaci \( s(t) = t^3 – 6t^2 + 9t + 1 \). Kayyade matsakaicin saurin ƙwayar cuta.
Tattaunawa:
1. Ƙayyade saurin (wanda aka samo daga matsayinsa):
Saurin ƙwayar cuta shine farkon abin da aka samo daga matsayi dangane da lokaci:
\[
v(t) = \frac{ds}{dt} = 3t^2 – 12t + 9
\]
2. Ƙayyade abin da aka samo asali na biyu:
Domin samun matsakaicin maki, mun sami na biyu wanda aka samo daga:
\[
a(t) = \frac{dv}{dt} = 6t – 12
\]
3. Nemo muhimmin batu:
Daidaita asalin farko na saurin zuwa sifili:
\[
3t^2 – 12t + 9 = 0
\]
Raba da 3:
\[
t^2 – 4t + 3 = 0
\]
Factoring:
\[
(t – 3)(t – 1) = 0
\]
Don haka, muhimman abubuwan sune \( t = 1 \) da \( t = 3 \).
4. Yi nazarin hanzari don nemo matsakaicin:
\[
a(1) = 6(1) – 12 = -6 \yana nufin t = 1 \text{\ matsakaicin gida ne}
\]
\[
a(3) = 6(3) – 12 = 6 \yana nufin t = 3 \text{\ shine mafi ƙarancin yanki}
\]
5. Lissafin matsakaicin gudu:
Sauya \( t = 1 \) cikin lissafin gudu:
\[
v(1) = 3(1)^2 – 12(1) + 9 = 3 – 12 + 9 = 0 \, (\rubutu{ba mai ban sha'awa ba})
\]
Duba wasu iyakoki ko wuraren tazara masu dacewa don tabbatar da mafi kyawun mafita.
Tare da waɗannan matakan, za mu iya gina tsarin mafita bisa tushen asali don matsalolin aikace-aikace daban-daban da ke sama.
Kammalawa
Misalan da ke sama sun nuna yadda za a iya amfani da abubuwan da aka samo don magance matsaloli a cikin yanayi daban-daban. Nemo mafi girma da mafi ƙarancin ƙima, ingantaccen haɓakawa mai iyaka, da nazarin motsi kaɗan ne kawai aikace-aikacen manufar abubuwan da aka samo. Kwarewar waɗannan dabaru da hanyoyin yana da mahimmanci ga waɗanda ke nazarin ilimin lissafi mai zurfi da fannoni masu alaƙa.
