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  3. 设f(x)＝xe-x，求fn(x)．

设f(x)＝xe-x，求fn(x)．

admin2013-01-07  29
问题 设f(x)＝xe-x，求fn(x)．
选项
答案fˊ(x)＝e-x+e-x(-1)x ＝(1-x)e-x ＝-(x-1)e-x f〞(x)＝-e-x＋xe-x-e-x ＝(-1)2(x-2)e-x f’’’(x)＝(-1)3(x-3)e-x f(k)(x)＝(-1)k(x-k)e-x f(k+1)(x)＝((-1)k(x-k)e-x)ˊ ＝(-1)k[e-x+(x-k)(-e-x)] ＝(-1)k+1(x-(k+1))e-x 由数学归纳法知 f(n)(x)＝(-1)n(x-n)e-x
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