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anonymous
 5 years ago
The Taylor polynomial of degree 100 for the function f about x=3 is given by
\[p(x)= (x3)^2  ((x3)^4)/2! +... + [(1)^n+1] [(x3)^n2]/n! +...  ((x3)^100)/50!]/
What is the value of f^30 (3)?
anonymous
 5 years ago
The Taylor polynomial of degree 100 for the function f about x=3 is given by \[p(x)= (x3)^2  ((x3)^4)/2! +... + [(1)^n+1] [(x3)^n2]/n! +...  ((x3)^100)/50!]/ What is the value of f^30 (3)?

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watchmath
 5 years ago
Best ResponseYou've already chosen the best response.1Plug in \(n=15\) to the expression \((1)^{n+1}/n!\)

watchmath
 5 years ago
Best ResponseYou've already chosen the best response.1Because \(f^{30}(3)\) is the coefficient of \(x^{30}\)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0So each term is\[((1)^{n+1}/n!)(x3)^{2n}\] and \[(f^{(n)}(3)/n!)(x3)^n\] so can I say \[(f^{(30)}(3)/30!)(x3)^{30}=((1)^{30+1}/30!)(x3)^{2*30}\] I don't really get where the 15 comes from

watchmath
 5 years ago
Best ResponseYou've already chosen the best response.1remember that the exponent on \((x3)\) is \(2n\). And we want this \(2n=30\). So we need to take \(n=15\). Since we are looking for the coefficient of \((x3)^{30}\)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0so \[1/15! = f^{(30)}(3)/30!\]

watchmath
 5 years ago
Best ResponseYou've already chosen the best response.1Ok, let me make this more clear. We want fo select the \(n\) so that we know the coefficient of \((x3)^{30}\). We know that the coefficient of \((x3)^{2n}\) is \((1)^{n+1}/n!\). So in order to find the coefficient of \((x3)^{30}\) we need to choolse \(n=15\). In that case the coefficient would be \((1)^{15+1}/15!=1/15!\). So \(f^{(30)}(3)=1/15!\)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Why is the coefficient of \((x3)^{n}\), \(f^{n}(3)\) and not \(f^{n}(3)/n!\) Isn't each term \((1/n!)(f^{(n)}(3))(x3)^n\)?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Thank you so much, I never would have gotten there in the first place :P
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