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Question Number 94650 by msup by abdo last updated on 20/May/20
let f(x) =(x+1)^9 e^(−3x) calculstr f^((7)) (0) and f^((5)) (1)
$${let}\:{f}\left({x}\right)\:=\left({x}+\mathrm{1}\right)^{\mathrm{9}} \:{e}^{−\mathrm{3}{x}} \\ $$$${calculstr}\:{f}^{\left(\mathrm{7}\right)} \left(\mathrm{0}\right)\:{and}\:{f}^{\left(\mathrm{5}\right)} \left(\mathrm{1}\right) \\ $$
Commented by i jagooll last updated on 20/May/20
question what number is the integral function of the gamma sir? I do not find it
Answered by mathmax by abdo last updated on 20/May/20
we have f^((n)) (x) =Σ_(k=0) ^n C_n ^k {(x+1)^9 }^((k)) (e^(−3x) )^((n−k)) k>9 ⇒{(x+1)^9 }^((k)) =0 and for k≤9 we have (x+1)^p )^((1)) =p(x+1)^(p−1) ((x+1)^p )^((2)) =p(p−1)(x+1)^(p−1) ⇒((x+1)^p )^((k)) =p(p−1)...(p−k+1)(x+1)^(p−k) ((x+1)^9 )^((k)) =9.8....(10−k)(x+1)^(9−k) ⇒ f^((n)) (x) =(x+1)^9 (−3)^n e^(−3x) +Σ_(k=1) ^n C_n ^k 9.8...(10−k)(x+1)^(9−k) (−3)^(n−k) e^(−3x) ⇒ f^((7)) (0) =1+Σ_(k=1) ^7 C_7 ^k 9.8....(10−k)(−3)^(7−k) f^((5)) (1) =2^9 (−3)^5 e^(−3) +Σ_(k=1) ^5 C_5 ^k 9.8....(10−k)2^(5−k) e^(−3)
$$\mathrm{we}\:\mathrm{have}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{x}\right)\:=\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \:\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} \:\:\:\left\{\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{9}} \right\}^{\left(\mathrm{k}\right)} \:\left(\mathrm{e}^{−\mathrm{3x}} \right)^{\left(\mathrm{n}−\mathrm{k}\right)} \\ $$$$\mathrm{k}>\mathrm{9}\:\Rightarrow\left\{\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{9}} \right\}^{\left(\mathrm{k}\right)} \:=\mathrm{0}\:\mathrm{and}\:\:\:\:\:\mathrm{for}\:\mathrm{k}\leqslant\mathrm{9}\:\:\mathrm{we}\:\mathrm{have} \\ $$$$\left.\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{p}} \right)^{\left(\mathrm{1}\right)} \:=\mathrm{p}\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{p}−\mathrm{1}} \\ $$$$\left(\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{p}} \right)^{\left(\mathrm{2}\right)} \:=\mathrm{p}\left(\mathrm{p}−\mathrm{1}\right)\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{p}−\mathrm{1}} \:\Rightarrow\left(\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{p}} \right)^{\left(\mathrm{k}\right)} \:=\mathrm{p}\left(\mathrm{p}−\mathrm{1}\right)…\left(\mathrm{p}−\mathrm{k}+\mathrm{1}\right)\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{p}−\mathrm{k}} \\ $$$$\left(\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{9}} \right)^{\left(\mathrm{k}\right)} \:=\mathrm{9}.\mathrm{8}….\left(\mathrm{10}−\mathrm{k}\right)\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{9}−\mathrm{k}} \:\Rightarrow \\ $$$$\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{x}\right)\:=\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{9}} \left(−\mathrm{3}\right)^{\mathrm{n}} \:\mathrm{e}^{−\mathrm{3x}} +\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} \:\:\mathrm{9}.\mathrm{8}…\left(\mathrm{10}−\mathrm{k}\right)\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{9}−\mathrm{k}} \left(−\mathrm{3}\right)^{\mathrm{n}−\mathrm{k}} \:\mathrm{e}^{−\mathrm{3x}} \:\Rightarrow \\ $$$$\mathrm{f}^{\left(\mathrm{7}\right)} \left(\mathrm{0}\right)\:=\mathrm{1}+\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{7}} \:\mathrm{C}_{\mathrm{7}} ^{\mathrm{k}} \:\mathrm{9}.\mathrm{8}….\left(\mathrm{10}−\mathrm{k}\right)\left(−\mathrm{3}\right)^{\mathrm{7}−\mathrm{k}} \\ $$$$\mathrm{f}^{\left(\mathrm{5}\right)} \left(\mathrm{1}\right)\:=\mathrm{2}^{\mathrm{9}} \left(−\mathrm{3}\right)^{\mathrm{5}} \:\mathrm{e}^{−\mathrm{3}} \:+\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{5}} \:\mathrm{C}_{\mathrm{5}} ^{\mathrm{k}} \:\mathrm{9}.\mathrm{8}….\left(\mathrm{10}−\mathrm{k}\right)\mathrm{2}^{\mathrm{5}−\mathrm{k}} \:\mathrm{e}^{−\mathrm{3}} \\ $$

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