1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Taylor Polynomial with Remainder Question

  1. Oct 30, 2011 #1
    1. The problem statement, all variables and given/known data
    What is the minimal degree Taylor polynomial about x=0 that you need to calculate sin(1) to 3 decimal places? 6 decimal places?

    2. Relevant equations
    R_nx = f^(n+1)(c)(x-a)^(n+1)/(n+1)(factorial)

    3. The attempt at a solution
    I have attached my attempt. I am stuck on the last step, how do I solve for n? Did I even do it right up until now?
     

    Attached Files:

  2. jcsd
  3. Oct 30, 2011 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    A MUCH easier way is to start computing the terms of sin(1) one-by-one, and noting that you have an alternating series. What do you know about the "truncation" (remainder) error in an alternating series?

    RGV
     
  4. Oct 30, 2011 #3
    But don't you need a calculator for that? You would have to calculate sin1 and compare your approximations to see the difference (remainder).
     
  5. Oct 30, 2011 #4

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    No, you don't need to know the value of sin(1)---remember, sin(1) is the thing that you are trying to compute!

    RGV
     
  6. Oct 30, 2011 #5
    Wow, I totally misread the question... Ok so now I have

    Pn(x) = x - x^3/3! + x^5/5! - x^7/7! + x^9/9! - x^11/11! + ...
    Pn(1) = 1 - 1/6 + 1/120 - 1/5040 + 1/362880

    How do I know which is 3 decimal and 6 decimal places without a calculator?
     
  7. Oct 31, 2011 #6

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Are you not allowed to use a calculator to do simple addition, subtraction and division? if not, then welcome to the world of manual computation from 50 years ago: this CAN be done by hand, but it is unpleasant.

    RGV
     
  8. Oct 31, 2011 #7
    Haha, our prof said we don't need a calculator for his course. But it seems like we do for the assignments.

    Back to the question... I am still somewhat clueless. First off, when they said 3 decimals places, would that mean <10^-2? It seems like the 7th derivative at 1/5040 would be a plausible answer but the answer key says 6... What do I do :S
     
  9. Oct 31, 2011 #8

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    No. Three-decimal places of accuracy require an |error| < 0.5*10^-4 = 1/2000, so stopping at the term -1/5040 will do (but be sure to INCLUDE that term). Six decimals of accuracy need an |error| < 0.5x10^-7 = 1/20,000,000, so you can figure out where you have to stop the series.

    RGV
     
  10. Oct 31, 2011 #9

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    No. Three-decimal places of accuracy require an |error| < 0.5*10^-4 = 1/20,000, so stopping at the term -1/362,880 will do (but be sure to INCLUDE that term). Six decimals of accuracy need an |error| < 0.5x10^-7 = 1/20,000,000, so you can figure out where you have to stop the series.

    RGV
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Taylor Polynomial with Remainder Question
Loading...