The discussion revolves around taking the derivative of transformed trigonometric functions, particularly in the context of modeling a roller coaster using a combination of polynomial, trigonometric, and rational functions. Participants are exploring how to ensure the derivatives of these functions match at their connection points.
The discussion is active, with participants providing guidance on the use of piecewise-defined functions and the need for agreement in derivatives at connection points. There are multiple interpretations being explored regarding the transformation of functions and the adjustment of slopes.
Participants mention constraints related to the rate of change at connection points, specifically that it cannot vary by more than 10%. There is also a focus on the need for the y-values of the functions to agree at the connection points.
HallsofIvy said:? Without further information, all I can say is the chain rule!
O.K I have to model roller coaster by connecting three functions together the first has to be a polynomial then a trigg function which has to connect with a rational function and the point where they meet it's rate of change can not vary by more than 10%.gb7nash said:Derivatives of trigonometric functions:
http://en.wikipedia.org/wiki/Differentiation_rules#Derivatives_of_trigonometric_functions
Chain Rule:
http://en.wikipedia.org/wiki/Chain_rule
I'm not sure I understand what your question is, but does that clear anything up?
Mark44 said:Please give us some examples of the kind of functions you mean.
Mark44 said:So you have a piecewise-defined function. At the two connection points, the y values have to agree, and you want the derivatives to approximately agree. Assuming that as we go left to right, you have 1) polynomial, 2) trig function, 3) rational function, and these functions are connected at points A and B, you want the derivative of the polynomial to be about the same as the derivative of the trig functions at A, and you want the derivative of the trig function to be about the same as that of the rational function at B.
For numbers to the left of and close to A, use the polynomial function's slope. For numbers to the right of and close to A, use the trig function's slope.
For numbers to the left of and close to B, use the trig function's slope. For numbers to the right of and close to B, use the rational function's slope.
I don't see how that would work. The simplest rational function is y = f(x) = 1/x. You can stretch or compress it vertically by a scaling the y value, as in a*f(x) = a/x. You can stretch or compress it horizontally by a scaling x, as in f(cx) = 1/(cx). I think this is the way to go for your rational function.Eng_physicist said:Is there any particular way to adjust the slope of the rational function
Could I do a reverse derivative by making the slope of the rational function close to that of the trigg function in it's derivative form then turn it back into the non derivative form