How to create a model for late apex racing line and prove its efficiency? I have derived the function of geometric(normal) apex racing line. 
Car racing: How to calculate the radius of the racing line through a turn of varying lengthBut it only ensure the maximum corner speed during racing. How can I prove late apex racing line will lead to a shorter overall turning time compared to the geometric apex racing line by building a mathematical model? I'm still in high school, so don't go into too deep. I'm hoping to use differential calculus or multivariable calculus to tackle the problem.
 A: This is an interesting, though somewhat vague, claim. I find it interesting for a couple of reasons. First of all, in my motorsport experience (autocross, track, instructing both car control and autocross) I’ve never come across it. The two main benefits of a later apex that I’ve heard over and over are safety and improved exit speed, neither of which is a decrease in the overall turning time.  
Second, there’s no immediately obvious reason to believe it. Quite the opposite, I’d say. For a given lateral (centripetal) acceleration $a_{lat}$, speed and turn radius are related by $a_{lat}=v^2/r$, i.e., maximum speed is proportional to the square root of the radius. On the other hand, the distance traveled is directly proportional to the radius ($s=r\theta$). The time it takes to traverse that arc is of course $$\frac s v={r\theta\over\sqrt{ra_{lat}}}=\sqrt{r\over a_{lat}}\theta,$$ so the larger-radius turn takes more time despite allowing a greater speed. These relationships hold even when the radius, speed and lateral acceleration aren’t constant. The elapsed time along an arbitrary path $\Gamma$ is no longer a simple formula and in general is given by the line integral $\int_\Gamma ds/v(s)$. The reciprocal of speed that appears in this integral is an important quantity in these sorts of calculations. It has dimensions of time over distance, so gives the rate at which you’re using up the clock as you travel. Some authors dub this quantity “slowness.”  
Now, in what’s commonly understood to be a late-apex line, you gradually add throttle (and unwind the steering) after some point, so the path is no longer a simple circular arc. This addition of longitudinal acceleration reduces the available lateral acceleration (the “traction budget” $a_{lat}^2+a_{long}^2\le a_{max}^2$). Even assuming that the driver always manages to use all available grip, this will tend to increase the time that it takes to complete the turn instead of reducing it. On the other hand, before that point the turn radius will be smaller than that of the geometric line, which will save time. On the other other hand, the car will go deeper into the corner during that phase, which will add time...  
It’s common driving wisdom that as the corner angle increases, the apex tends to be later, so at the very least the correct apex for a turn will depend on the turn’s geometry. The car’s performance characteristics may also come into play: for high-horsepower cars, a “point and shoot” line that takes advantage of the longitudinal acceleration is often best, but that requires a lot of slowing on entry, which would be disastrous for lap times for a “momentum” car. Given all of that, it seems that this claim is at best an “it depends.” Rather than going in trying to prove it, I’d suggest that you instead explore the question of whether or not a late apex is more efficient in the sense that you defined, and if not, what its benefits are.  
Brian Beckman’s old “Physics of Racing” series, which you’ve already found, is a good starting point this investigation. In parts 17 and 18, he explores the effects on total elapsed segment time of moving the apex around and adding throttle before the turn is finished. Be sure that you read both parts, since the first is only the setup for what he does in the second. Note carefully the results that he gets in the first part: Total segment time is dominated by corner exit speed, which in his setup is attained with the earliest apex.  
The time spent turning is, not surprisingly, a monotonic function of the turn radius, exactly as predicted above. On the other hand, the total segment time up to the point that the turn is finished appears have some interesting behavior: it has a minimum at around 50° apex. This minimum moves depending on the approach speed. This behavior turns out to be an artifact of the setup. If you change things so that the apex is always at the geometric center of the turn and vary the turn radius (and change the approach line to correspond), this minimum goes away and the segment time becomes monotonic with turn radius.  
Unfortunately, once you start adding acceleration while still in the turn, the number of parameters proliferates quickly. Besides the location of the apex, there’s at least the rate at which you add throttle, the point at which you begin adding throttle, and the point and rate at which you unwind the steering. It’s going to get messy fast, which is one reason Beckmann switches from an analytic approach to an exploratory simulation. I suggest going that route yourself. You can use a spreadsheet as Beckmann did, or mock up the model in one of the excellent interactive tools that are available now, such as GeoGebra. Once you’ve gotten a feel for what’s going on, then you’ll be better able to tackle the problem analytically without following a lot of dead ends first.  
Good luck!
