10.17-bo'lim : Seriya ilovalari
2. \(x = - 4\) haqidagi \(f\left( x \right) = {{\bf{e}}^{ - 6x}}\) ning Teylor seriyasi uchun \({T_2}\left( x \right)\), \({T_3}\left( x \right)\) va \({T_4}\left( x \right)\) ni yozing. \(\left[ { - 8, - 2} \right]\) oralig'i uchun bir xil grafikda Teylor ko'phadlari va \(f\left( x \right)\)ning uchtasini chizing.
Yechim
Bu erda qilishimiz kerak bo'lgan birinchi narsa - \(x = - 4\) haqida \(f\left( x \right) = {{\bf{e}}^{ - 6x}}\) uchun Teylor seriyasini olish. Yaxshiyamki, biz buni oldingi bo'limning 3-muammosida qildik. Mana biz ushbu muammoda olingan Teylor seriyasi.
\[{{\bf{e}}^{ - 6x}} = \sum\limits_{n = 0}^\infty {\frac{{{{\left( { - 6} \right)}^n}{{\bf{e}}^{24}}}}{{n!}}{{\left( {x + 4} \right)}^n}} \]2-qadam
Bu masala uchun zarur bo'lgan uchta Teylor ko'phadlari.
\[\begin{align*}{T_2}\left( x \right) & = {{\bf{e}}^{24}} - 6{{\bf{e}}^{24}}\left( {x + 4} \right) + 18{{\bf{e}}^{24}}{\left( {x + 4} \right)^2}\\ {T_3}\left( x \right) & = {{\bf{e}}^{24}} - 6{{\bf{e}}^{24}}\left( {x + 4} \right) + 18{{\bf{e}}^{24}}{\left( {x + 4} \right)^2} - 36{{\bf{e}}^{24}}{\left( {x + 4} \right)^3}\\ {T_4}\left( x \right) & = {{\bf{e}}^{24}} - 6{{\bf{e}}^{24}}\left( {x + 4} \right) + 18{{\bf{e}}^{24}}{\left( {x + 4} \right)^2} - 36{{\bf{e}}^{24}}{\left( {x + 4} \right)^3} + 54{{\bf{e}}^{24}}{\left( {x + 4} \right)^4}\end{align*}\]3-qadam
Mana bu muammo uchun grafik.
![The graphs of \(f\left( x \right) = {{\bf{e}}^{ - 6x}}\) and the Taylor polynomials \({T_2}\left( x \right)\), \({T_3}\left( x \right)\) and \({T_4}\left( x \right)\) on the interval \(\left[ { - 8, - 2} \right]\). All four graphs lie almost on top of each other near \(x = - 4\), but as you move away from that point in either direction the three polynomial graphs pull away from the exponential graph, with \({T_2}\left( x \right)\) turning down to the lower right and the others rising steeply.](/Solutions/CalcII/TaylorSeriesApps//Solutions/CalcII/TaylorSeriesApps/TaylorSeriesApps_Files/image001.webp)
Biz \(x = - 4\) “yaqinida” qolsak, ko'phadning grafiklari eksponensial funktsiya grafigiga juda yaqin ekanligini ko'rishimiz mumkin. Ammo, agar biz juda uzoqqa borsak, grafiklar haqiqatan ham eksponensial funktsiya grafigidan uzoqlasha boshlaydi.