I am designing a roller coaster using functions (ie. linear, cubic, logarithmic). I need the roller coaster to have no sudden changes in gradient - smooth transitions between say a linear function and a parabola.
If I had y = 30 (over domain {0<x<20}) & y = -((x-20)/8)^2)+30 (over domain {20<x<60}; how would I prove there is a smooth transition mathmatically? If I plot this I can tell visually but I would like to know how to prove this mathmatically if possible?
Also I would like to determine that gradient of the parabola function above (so I can work out the angle it drops at). Can someone explain a method of doing this mathmatically?
All assistance is greatly appreciated.
This is precisely the field of application of the Bezier's curve.
They are widely used in designing car bodies and similar, because they respond to the need you are telling.