What is the equation for a polynomial curve connecting two straight tracks?

[roadrunner]

"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!

                ____________ (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 equation 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

 

[Hurkyl]

Use [ code ] ... [ /code ] around an ASCII picture: it will retain your spaces, and all characters will have an equal width.

 

[roadrunner]

ok thanks that worked...but I am stuill stuck on the problem! :D

 

[HallsofIvy]

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.

 

[roadrunner]

oh that makes sense thanks

but why does p''(0) and p''(1)mean smooth?

 

[roadrunner]

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 ^)