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let-f-x-x-2n-e-3x-find-f-n-o-and-calculate-f-2021-0-

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Question Number 140637 by Mathspace last updated on 10/May/21
let f(x)=x^(2n) e^(−3x) find f^((n)) (o) and calculate f^((2021)) (0)
$${let}\:{f}\left({x}\right)={x}^{\mathrm{2}{n}} \:{e}^{−\mathrm{3}{x}} \\ $$$${find}\:\:{f}^{\left({n}\right)} \left({o}\right)\:{and} \\ $$$${calculate}\:{f}^{\left(\mathrm{2021}\right)} \left(\mathrm{0}\right) \\ $$
Answered by mathmax by abdo last updated on 12/May/21
by leibniz f^((p)) (x)=Σ_(k=0) ^p C_p ^k (x^(2n) )^((k)) (e^(−3x) )^((p−k)) (x^(2n) )^((1)) =2n x^(2n−1) ,(x^(2n) )^((2)) =2n(2n−1)x^(2n−2) .... (x^(2n) )^((k)) =2n(2n−1)....(2n−k+1)x^(2n−k) ⇒ f^((p)) (x)=Σ_(k=0) ^p C_p ^k 2n(2n−1)2n−2)....(2n−k+1)x^(2n−k) (−3)^(p−k) e^(−3x) ⇒f^((n)) (x)=Σ_(k=0) ^n C_n ^k 2n(2n−1)....(2n−k+1)x^(2n−k) (−3)^(n−k) e^(−3x) ⇒ f^((n)) (0)=0 ⇒f^((2021)) (0)=0
$$\:\mathrm{by}\:\mathrm{leibniz}\:\:\mathrm{f}^{\left(\mathrm{p}\right)} \left(\mathrm{x}\right)=\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{p}} \:\mathrm{C}_{\mathrm{p}} ^{\mathrm{k}} \:\left(\mathrm{x}^{\mathrm{2n}} \right)^{\left(\mathrm{k}\right)} \left(\mathrm{e}^{−\mathrm{3x}} \right)^{\left(\mathrm{p}−\mathrm{k}\right)} \\ $$$$\left(\mathrm{x}^{\mathrm{2n}} \right)^{\left(\mathrm{1}\right)} \:=\mathrm{2n}\:\mathrm{x}^{\mathrm{2n}−\mathrm{1}} \:,\left(\mathrm{x}^{\mathrm{2n}} \right)^{\left(\mathrm{2}\right)} \:=\mathrm{2n}\left(\mathrm{2n}−\mathrm{1}\right)\mathrm{x}^{\mathrm{2n}−\mathrm{2}} \:…. \\ $$$$\left(\mathrm{x}^{\mathrm{2n}} \right)^{\left(\mathrm{k}\right)} \:=\mathrm{2n}\left(\mathrm{2n}−\mathrm{1}\right)….\left(\mathrm{2n}−\mathrm{k}+\mathrm{1}\right)\mathrm{x}^{\mathrm{2n}−\mathrm{k}} \:\Rightarrow \\ $$$$\left.\mathrm{f}^{\left(\mathrm{p}\right)} \left(\mathrm{x}\right)=\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{p}} \:\mathrm{C}_{\mathrm{p}} ^{\mathrm{k}} \:\mathrm{2n}\left(\mathrm{2n}−\mathrm{1}\right)\mathrm{2n}−\mathrm{2}\right)….\left(\mathrm{2n}−\mathrm{k}+\mathrm{1}\right)\mathrm{x}^{\mathrm{2n}−\mathrm{k}} \:\left(−\mathrm{3}\right)^{\mathrm{p}−\mathrm{k}} \:\mathrm{e}^{−\mathrm{3x}} \\ $$$$\Rightarrow\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{x}\right)=\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \:\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} \:\mathrm{2n}\left(\mathrm{2n}−\mathrm{1}\right)….\left(\mathrm{2n}−\mathrm{k}+\mathrm{1}\right)\mathrm{x}^{\mathrm{2n}−\mathrm{k}} \left(−\mathrm{3}\right)^{\mathrm{n}−\mathrm{k}} \:\mathrm{e}^{−\mathrm{3x}} \:\Rightarrow \\ $$$$\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{0}\right)=\mathrm{0}\:\Rightarrow\mathrm{f}^{\left(\mathrm{2021}\right)} \left(\mathrm{0}\right)=\mathrm{0} \\ $$

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