let-f-x-x-1-9-e-3x-calculstr-f-7-0-and-f-5-1-

Question Number 94650 by msup by abdo last updated on 20/May/20

l e t f (x) = {(x + 1)}^{9} e^{- 3 x}

c a l c u l s t r f^{(7)} (0) a n d f^{(5)} (1)

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) = \sum_{k = 0}^{n} C_{n}^{k} {{(x + 1)}^{9}}^{(k)} {(e^{- 3 x})}^{(n - k)}

k > 9 \Rightarrow {{(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} \Rightarrow {({(x + 1)}^{p})}^{(k)} = p (p - 1) \dots (p - k + 1) {(x + 1)}^{p - k}

{({(x + 1)}^{9})}^{(k)} = 9.8 \dots . (10 - k) {(x + 1)}^{9 - k} \Rightarrow

f^{(n)} (x) = {(x + 1)}^{9} {(- 3)}^{n} e^{- 3 x} + \sum_{k = 1}^{n} C_{n}^{k} 9.8 \dots (10 - k) {(x + 1)}^{9 - k} {(- 3)}^{n - k} e^{- 3 x} \Rightarrow

f^{(7)} (0) = 1 + \sum_{k = 1}^{7} C_{7}^{k} 9.8 \dots . (10 - k) {(- 3)}^{7 - k}

f^{(5)} (1) = 2^{9} {(- 3)}^{5} e^{- 3} + \sum_{k = 1}^{5} C_{5}^{k} 9.8 \dots . (10 - k) 2^{5 - k} e^{- 3}