Ajụjụ Ihe Nlereanya Na-ekwu Maka Ịba ụba na Nkewa nke Ọrụ
Na mgbakọ na mwepụ, ọrụ bụ mmekọrịta nke na-ejikọta ihe ọ bụla dị n'otu setịpụ na otu ihe kpọmkwem dị na setịpụ ọzọ. A na-akpọkarị ọrụ dị ka \( f(x) \), nke pụtara na \( f \) bụ ọrụ nke \( x \). Otu n'ime ọrụ enwere ike iji ọrụ rụọ bụ mmụba na nkewa. N'isiokwu a, anyị ga-enyocha ọtụtụ nsogbu atụ ma tụlee ọrụ mmụba na nkewa nke ọrụ.
Ịbawanye Ọrụ
Mmụba ọrụ bụ ọrụ ebe anyị na-amụba ọrụ abụọ, nsonaazụ ya bụ ọrụ ọhụrụ. Ka e were ya na anyị nwere ọrụ abụọ \( f(x) \) na \( g(x) \). Enwere ike ịkọwa ngwaahịa nke ọrụ abụọ a dị ka \( (f \cdot g)(x) \) ma ọ bụ \( f(x) \cdot g(x) \).
Ajụjụ Ihe atụ nke 1:
E nyere ọrụ abụọ:
– \( f(x) = 2x + 3 \)
– \( g(x) = x^2 – 4 \)
Chọta ihe si na \( f(x) \cdot g(x) \) pụta.
Azịza:
Ọrụ abụọ a bụ:
\[ (f \cdot g)(x) = f(x) \cdot g(x) \]
Ka ọ were:
\[ (f \cdot g)(x) = (2x + 3) \cdot (x^2 – 4) \]
Iji mụbaa polynomials abụọ, anyị na-eji nkesa:
\[ (2x + 3)(x^2 – 4) = 2x(x^2) + 2x(-4) + 3(x^2) + 3(-4) \]
\[ = 2x^3 – 8x + 3x^2 – 12 \]
Ya mere, ihe ikpeazụ ga-esi na ya pụta bụ:
\[ (f \cdot g)(x) = 2x^3 + 3x^2 – 8x – 12 \]
Ajụjụ Ihe atụ nke 2:
Ọrụ enyere:
– \( f(x) = \sin(x) \)
– \( g(x) = \cos(x) \)
Chọta ihe si na \( f(x) \cdot g(x) \) pụta.
Azịza:
Ọrụ abụọ a bụ:
\[ (f \cdot g)(x) = \sin(x) \cdot \cos(x) \]
Ya mere, ihe ikpeazụ ga-esi na ya pụta bụ:
\[ (f \cdot g)(x) = \sin(x) \cos(x) \]
Na trigonometry, anyị maara na:
\[ \sin(x) \cos(x) = \frac{1}{2} (\sin(2x)) \]
Ya mere, nsonaazụ nke ịba ụba ọrụ ndị a bụ:
\[ (f \cdot g)(x) = \frac{1}{2} \sin(2x) \]
Nkewa nke Ọrụ
Nkewa ọrụ bụ ọrụ nke kewaa otu ọrụ site na nke ọzọ wee nweta ọrụ ọhụrụ, ma ọ bụrụhaala na onye nkesa ahụ erughị efu. Ka e were ya na anyị nwere ọrụ abụọ \( f(x) \) na \( g(x) \). Enwere ike ịkọwa nkewa nke ọrụ abụọ a dị ka \( \left( \frac{f}{g} \right)(x) \) ma ọ bụ \( \frac{f(x)}{g(x)} \).
Ajụjụ Ihe atụ nke 3:
E nyere ọrụ abụọ:
– \( f(x) = x^2 – 1 \)
– \( g(x) = x – 1 \)
Chọta ihe si na \( \frac{f(x)}{g(x)} \) pụta.
Azịza:
Nkewa nke ọrụ abụọ a bụ:
\[ \left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} \]
Ka ọ were:
\[ \left( \frac{f}{g} \right)(x) = \frac{x^2 – 1}{x – 1} \]
Anyị nwere ike ime ka akụkụ dị mfe site na iji ọnụọgụgụ mee ihe:
\[ x^2 – 1 = (x + 1)(x – 1) \]
Ya mere:
\[ \left( \frac{f}{g} \right)(x) = \frac{(x + 1)(x – 1)}{x – 1} \]
Ebe ọ bụ na \( x \neq 1 \), anyị nwere ike ịkagbu \( (x – 1) \) na numerator na denominator:
\[ \left( \frac{f}{g} \right)(x) = x + 1 \]
Ajụjụ Ihe atụ nke 4:
E nyere ọrụ abụọ:
– \( f(x) = e^x \)
– \( g(x) = x \)
Chọta ihe si na \( \frac{f(x)}{g(x)} \) pụta.
Azịza:
Nkewa nke ọrụ abụọ a bụ:
\[ \left( \frac{f}{g} \right)(x) = \frac{e^x}{x} \]
Ya mere, ihe ikpeazụ ga-esi na ya pụta bụ:
\[ \left( \frac{f}{g} \right)(x) = \frac{e^x}{x} \]
Ajụjụ Ihe atụ nke 5:
Ọrụ enyere:
– \( f(x) = \ln(x) \)
– \( g(x) = x^2 \)
Chọta ihe si na \( \frac{f(x)}{g(x)} \) pụta.
Azịza:
Nkewa nke ọrụ abụọ a bụ:
\[ \left( \frac{f}{g} \right)(x) = \frac{\ln(x)}{x^2} \]
Ya mere, ihe ikpeazụ ga-esi na ya pụta bụ:
\[ \left( \frac{f}{g} \right)(x) = \frac{\ln(x)}{x^2} \]
Mmechi
Ịba ụba na nkewa nke ọrụ bụ echiche ndị bụ isi na mgbakọ na mwepụ ma bara uru nke ukwuu n'ọtụtụ ojiji, ma na mgbakọ na mwepụ dị ọcha ma na sayensị etinyere dịka physics na injinia. Site n'ịghọta otu esi agbakọta na kewaa ọrụ, anyị nwere ike idozi ọtụtụ nsogbu metụtara ha. Mkparịta ụka nke nsogbu ndị dị n'elu na-enye nghọta banyere otu esi arụ ọrụ ndị a na nsonaazụ e nwetara.
Nọgide na-eme ihe iji mee ka nghọta gị banyere ihe a dịkwuo omimi, ebe nghọta siri ike nke ọrụ dị oke mkpa maka ọganihu na ọmụmụ mgbakọ na mwepụ ndị ọzọ. Ọ bụrụ na ị hụta nsogbu ọ bụla, egbula oge ịjụ onye nkuzi gị ma ọ bụ chọọ ihe mmụta ndị ọzọ. Anyị nwere olileanya na isiokwu a enyerela aka n'ịghọta mmụba na nkewa nke ọrụ.
