Question

compute the fifth degree Taylor polynomial of the function f(x)=cos(0.3x) at a=0 and Find the largest integer k such that for all x for which |x|<1 , the Taylor polynomial T 5 (x) approximates f(x) with error less than 1/10^k .

Answer 1

i already found the fifth degree taylor poly. which is 1-.3^2/2(x^2)+(.3^4)/24*(x^4)
im just not sure how to calculate the second part

Answer 2

http://www.wolframalpha.com/input/?i=compute+the+fifth+degree+Taylor+polynomial+of+the+function+f%28x%29%3Dcos%280.3x%29+at+a%3D0+and+Find+the+largest+integer+k+such+that+for+all+x+for+which+%7Cx%7C%3C1+%2C+the+Taylor+polynomial+T+5+%28x%29+approximates+f%28x%29+wi

Answer 3

see this

Answer 4

thanks but im not sure what im suppose to be seeing in the graph? sorry i know there is a formula for the second part im just not sure how to use it in this instance

Answer 5

Taylor series is this

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

Answer 6

using this series you have to compute n derivatives. usually your teacher will tell you oho many terms

Answer 7

then after you're calculating the convergence of the series

Answer 8

Error bound: | f^6 (x) | ≤ Max
| f^6 (x) | = |- ( .3x)^6 |
= ( .3x)^6 , x ≤ 1
=> | f^6 (x) | ≤ ( .3 )^6 Max

Since Error ≤ 1/ 10^k
( .3 )^6 / 6! ≤ 1/ 10^k
-> k ≤ 5.99 ≈ 6