Troubleshooting Newton's Method for f(x) and g(x)

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Discussion Overview

The discussion revolves around troubleshooting the application of Newton's method to find the intersection points of the functions f(x) = x and g(x) = tan(x). Participants explore the challenges faced in obtaining the correct solution and the implications of initial approximations in the method's convergence.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant describes their process of finding the intersection by defining H(x) = x - tan(x) and calculating its derivative, but expresses confusion over the discrepancy between their result and the solution manual.
  • Another participant suggests that a better first approximation is necessary for Newton's method to converge correctly.
  • A different participant questions whether the functions f(x) and g(x) are only equal in one place, noting that they intersect at multiple points.
  • It is mentioned that the value of x must be between π/2 and 3π/2 due to the asymptotes of the tangent function.
  • Further advice is given that a good first approximation is crucial, with a specific mention that starting below 4.29 may lead to convergence issues.
  • A participant expresses gratitude for the helpful advice received in the discussion.

Areas of Agreement / Disagreement

Participants generally agree on the importance of the initial approximation in Newton's method, but there is a lack of consensus on the number of intersection points between the functions and the specific range for the initial guess.

Contextual Notes

The discussion highlights the dependence on initial approximations and the potential for multiple solutions, but does not resolve the mathematical steps or assumptions regarding the behavior of the functions involved.

RadiationX
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i have the answer to this problem i just see why I'm not getting the same answer as my solution manuel. i Have two functtions f(x) = x and g(x) = tan(x)
and i have to find where these two functions are equal using Newtons method.
i subtracted the two functions to get this new function H(x) = x -tan(x) then i found the differential of this new function to be 1 - sec^2 (x). now by looking at the given graph of the first two functions i estimated the soulution to be about 4.1. then by unising Newtons method i calculated two iterations and got an answer of 8.5987, which is wrong the answer is 4.493. what could i be doing wrong? i know how to use Newtons method.
 
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You need to use a better first approximation.
 
RadiationX said:
i have the answer to this problem i just see why I'm not getting the same answer as my solution manuel. i Have two functtions f(x) = x and g(x) = tan(x)
and i have to find where these two functions are equal using Newtons method.
i subtracted the two functions to get this new function H(x) = x -tan(x) then i found the differential of this new function to be 1 - sec^2 (x). now by looking at the given graph of the first two functions i estimated the soulution to be about 4.1. then by unising Newtons method i calculated two iterations and got an answer of 8.5987, which is wrong the answer is 4.493. what could i be doing wrong? i know how to use Newtons method.

Are you sure that the two functions are only equal in one place? x and tan x intersect in *lots* of places.
 
x has to be between pi/2 and 3pi/2 these are the asmyspotes.
 
Muzza said:
You need to use a better first approximation.

You need a real good first approximation, otherwise it never converges. If you know that the tangent of 3 pi/2 is undefined and start a little lower than that, you're okay. If your guess is less than around 4.29, you won't converge.
 
thanks guys the last reply helped me out.
 

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