The Taylor Polynomial of degree 100 for the function f about x = 3 is given by P(x) = (x-3)^2 - ((x-3)^4)/2! + ((x-3)^6)/3! + ... + (-1)^(n+1) * ((x-3)^2n)/n! + .. - ((x-3)^100)/50!. What is the value of f'30(3)?
A.) -30!/15!
B.) -1/30!
C.) 1/30!
D.) 1/15!
E.) 30!/15!
Why is the answer E and not D??
A.) -30!/15!
B.) -1/30!
C.) 1/30!
D.) 1/15!
E.) 30!/15!
Why is the answer E and not D??
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By the definition of the Taylor Series, the coefficient of (x - 3)^30 is f^(30)(3) / 30!.
On the other hand, looking at the given series, it is (-1)^(15+1) / 15! (letting n = 15).
Hence, f^(30)(3) / 30! = (-1)^(15+1) / 15!
==> f^(30)(3) = 30!/15!.
I hope this helps!
On the other hand, looking at the given series, it is (-1)^(15+1) / 15! (letting n = 15).
Hence, f^(30)(3) / 30! = (-1)^(15+1) / 15!
==> f^(30)(3) = 30!/15!.
I hope this helps!