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Polynomial curve problem

  1. Oct 14, 2007 #1
    "let's consider the problem of connnecting a railroad track to make a smooth transition between sections of straight ttrack. Existing track along the negative x axis is to be joined smoothly to a track on the line y=1 for x>=1.

    find a polynomial P=P(x) of degree 5 such that the following function F is defined by

    .........0 if x<=1
    F(x) =P(x) if 0<x)1
    ........1 if x>= 1



    no idea how to start other than drawing a picture!!!

    Code (Text):
                    ____________ (y=1)
                   /\
    (y=0)____      ||
            /\     ||
            ||     ||
          (x=0) (x=1)

     
    **ignore all the peroids, they are for spacing as this does not put large amounts of spaces.**

    and betwwen those two lines i need an eqation for a line that looks kind of like a squished "S" on a 45 degree angle (hard to describe)

    how do i go about making that equation? where do i even start?

    edit...tried that code thing
     
    Last edited: Oct 14, 2007
  2. jcsd
  3. Oct 14, 2007 #2

    Hurkyl

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    Use [ code ] ... [ /code ] around an ASCII picture: it will retain your spaces, and all characters will have an equal width.
     
  4. Oct 14, 2007 #3
    ok thanks that worked...but im stuill stuck on the problem! :D
     
  5. Oct 14, 2007 #4

    HallsofIvy

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    So you want a polynomial P(x) of degree 5: P(x)= ax5+ bx4+ cx3+ dx2+ ex+ f such that:
    P(0)= 0
    P(1)= 1 so the tracks meet!
    P'(0)= 0
    P'(1)= 0 so there is not a "corner" where they meet.
    That much should be obvious from your picture. But's that's only 4 conditions for 6 coefficients. You need two more. In order that the transition be smooth you also need:
    P"(0)= 0
    p"(1)= 0
    That gives you 6 equations to solve for the 6 coefficients.
     
  6. Oct 14, 2007 #5
    oh that makes sense thanks

    but why does p''(0) and p''(1)mean smooth?
     
  7. Oct 14, 2007 #6
    so i get
    a=6
    b= -15
    c=10
    d=0
    e=0
    f=0

    that looks right to me?

    so my equation is 6ax^5 -15x^4+10x^3=P(x)?

    PS: how do u make the ^2 appear normal? (IE without using a ^)
     
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