Question

Find the Taylor polynomial of degree n=4 for x near the point a=8 for the function sin4x .

Answer 1

\[\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n\]

Answer 2

thats the general form of taylor series

Answer 3

and you want a=8, so you can plug that in \[\sum_{n=0}^{\infty} \frac{f^{(n)}(8)}{n!}(x-8)^n\]

Answer 4

and plug in n as well?

Answer 5

and you need to find the nth derivative of f(x) = sin(4x)

Answer 6

\[f(x) = \sin(4x) \\ f'(x) = 4\cos(4x) \\ f ' ' (x) = -16\sin(4x) \\ f'''(x) = -64\cos(4x) \\ f^{(4)}(x)=256\sin(4x) \]

Answer 7

we get tired of writing apostrophes for f primes

Answer 8

do you plug it back into the original equation?

Answer 9

we want to expand that taylor series so that we have a fourth degree taylor polynomial

Answer 10

\[\sum_{n=0}^{4} \frac{f^{(n)}(8)}{n!}(x-8)^n = \\ \frac {f(8)}{0!}(x-8)^0 + \frac{f'(8)}{1!}(x-8)^1 + \frac{f''(8)}{2!}(x-8)^2 + \frac{f'''(8)}{3!}(x-8)^3 + \frac{f^{(4)}}{4!}(x-8)^4\]

Answer 11

thank you so much :)