If looking at a curve around a race track, how could you use optimization in calculus to find the shortest way to go around the curve?

Question

If you were to look at a curve of a formula 1 track and then proceed to make a parabola from the curve, what would looking at the derivatives and optimization give you?

Answer 1

You are definitely asking an optimization question, but it's different from the type of optimization problems that you typically encounter in Calc I, II, or III. It seems like you are asking for the best path to minimize travel time. That sounds like a simple question, but it depends on a lot more than you may have assumed at first glance.

 

Generally, we use basic optimization in calculus to find the maximum or minimum value(s) a function will assume if we allow its variables to vary within some interval of values (and very often this interval is the function's entire domain). Doing this basically amounts to finding the turning points of f(x) by means of setting its derivative to zero and solving for x. That's actually how we derive the formula for the turning point on a parabola.

 

In multivariable calculus, you extend this idea to the optimization of a function of more than one variable--f(x,y,.....,z)--except now, we set all of the function's partial derivatives to zero and obtain a system of equations that must be solved. This also introduces the concept of constrained optimization, where sometimes one variable depends on the value of other variables (e.g. if we were in a physics lab investigating an ideal gas, we couldn't just pick any values of pressure, volume, and temperature; they're constrained to satisfy the ideal gas law PV=nRT)

 

Basically, optimization amounts to "which combination of inputs gets me the biggest/smallest output". Or, "which X makes f(X) really big/small".

 

The racetrack question you're asking is different in a big way--you're trying to minimize travel time, but to do that you must pick the best path. The path tracing out where a driver moves is not a variable; it's an entire function. And, just to make things a little more complicated, travel time also depends on your velocity from moment to moment. There is a whole branch of math devoted to solving these kinds of optimization problems called calculus of variations. It's using this kind of math that we can actually prove the shortest distance between two points in space is a straight line.

 

Unfortunately there's not a simple way to answer your question (it would also need some more detailed information before any variational calculus would become useful).