curvature problem: Find polynomial of degree 5 such that....

[acpodgorski]

This is a very confusing problem... please help me at least get started on it!!

Let's consider the problem of designing a railroad track to make a smooth transition between sections of straight track. Existing track along the negative x-axis is to be joined smoothly to a track along the line y = 1 for x >= 1.

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

        /  0     if x <= 0
F(x) = < P(x)    if 0 < x < 1
        \  1     if x >= 1

is continuous and has continuous slope and continuous curvature

 

[stapel]

The "ends" (the part before x = 0 and the part after x = 1) are horizontal lines, and thus have slopes of zero. For the polynomial bit in the middle to join "smoothly" at the ends, the polynomial must have slope zero at the ends. In other words, you have to have max/min points of the polynomial at x = 0 and x = 1.

How does your book define "continuous curvature"?

Thank you!

Eliz.