1.1-bo'lim : Ko'rib chiqish: Funktsiyalar
8. The farq koeffitsienti \(f\left( x \right)\) funksiyasi quyidagicha aniqlanadi:
\[\frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}\]\(\displaystyle y\left( z \right) = \frac{1}{{z + 2}}\) uchun farq koeffitsientini hisoblang.
Maslahat: Funktsiya endi \(y\left( z \right)\) deb nomlanganligidan hayajonlanmang, farq koeffitsienti hali ham xuddi shunday ishlaydi, u hozirda \(f\) va \(x\) o‘rniga \(y\) va \(z\)ga ega. Shunday qilib, \(y\left( {z + h} \right)\) ni hisoblang, so'ngra suratni hisoblang va nihoyat farq koeffitsientini hisoblang.
Yechim
\[y\left( {z + h} \right) = \frac{1}{{z + h + 2}}\]
2-qadam
\[y\left( {z + h} \right) - y\left( z \right) = \frac{1}{{z + h + 2}} - \frac{1}{{z + 2}} = \frac{{z + 2 - \left( {z + h + 2} \right)}}{{\left( {z + h + 2} \right)\left( {z + 2} \right)}} = \frac{{ - h}}{{\left( {z + h + 2} \right)\left( {z + 2} \right)}}\]
E'tibor bering, farqli koeffitsientlar bilan ishlashda, keyingi bosqichga tayyorgarlik ko'rishda deyarli har doim ratsional iboralarni bitta atamaga birlashtirish tavsiya etiladi.
3-qadam
\[\frac{{y\left( {z + h} \right) - y\left( z \right)}}{h} = \frac{1}{h}\left( {y\left( {z + h} \right) - y\left( z \right)} \right) = \frac{1}{h}\left( {\frac{{ - h}}{{\left( {z + h + 2} \right)\left( {z + 2} \right)}}} \right) = \frac{{ - 1}}{{\left( {z + h + 2} \right)\left( {z + 2} \right)}}\]
Ushbu bosqichda biz surat bilan ishlashni biroz osonlashtirish uchun farq koeffitsientini biroz qayta yozdik. Biz bu erda qilayotgan barcha narsa shundan iboratki,
\[\frac{a}{b} = \left( a \right)\left( {\frac{1}{b}} \right) = \left( {\frac{1}{b}} \right)\left( a \right)\]