| HN Mirror

On a graph, plot b/Vb, with b on the horizontal axis and b/Vb on the vertical axis. This plot shows the time you spend on the beach if you run to b on the beach.

Add to this graph a plot of w/Vw. You'll have to express w as a function of b to do this. This plot shows the time you spend in the water if you run to b on the beach and then swim to the ball.

You are trying to minimize the sum of these two plots.

You can see from the plots that as b increases, beach time goes up, and water time goes down. As b decreases, beach time goes down and water time goes up.

If b is less than the minimizing point, water time changes more than beach time as b changes, so you can lower the total by moving b in the direction that makes water time go down. That will make beach time go up, but since water time changes more than beach time, it's a net win.

When b is more than the minimizing point, beach time changes more than water time as b changes, so you can lower the total time by moving b in the direction that makes beach time go down. The will make water time go up, but since beach time changes more than water time, it's a net win.

The minimizing point is the point where the rate beach time is changing and the rate water time is changing match, so that you cannot lower the time by nudging b in either direction.

What you could then do is think about how to calculate the rate of change of a function. Say you have a function, f(x), and you want to know how fast it changes as x changes. To help with intuition, let's say f(x) is the position of a car at time x. The rate of change of position is velocity, so we are asking how you find the velocity at a given point x if you know position as a function of x. If the car was moving at a constant velocity, this would be easy. Just note the position at x, f(x), and then note the position a little bit later, say at x+t, f(x+t). Then the distance traveled is f(x+t)-f(x), in time t, so the velocity is (f(x+t)-f(x))/t.

If the car is not moving at constant velocity that just gives you the average velocity during the time t. If the car isn't accelerating too much, the smaller t the closer the average velocity over an interval t should be to the instantaneous velocity at t.

For example, suppose f(x) = x^2. Then f(x+t) = x^2 + 2xt + t^2. The average velocity is 2x + t. It's pretty obvious that as we make t smaller and smaller, this gets closer and closer to 2x, and so we can reasonably conclude that the instantaneous velocity of something whose position at time x is f(x) is 2x.

Applying those same ideas to the functions involved in the ball problem, you could figure out the rates of change, and find the b that makes the beach time and the water time rates of change match. Unlike the nice x^2 car, though, the algebra would be messy and error prone.

After you have done that, you might realize that this operation of take an average over smaller and smaller intervals and seeing if that converges to some fixed value as the interval goes to zero is quite useful, and start studying it. You'd then discover that while computing it directly can be a royal pain in the ass, it turns out that if your function is built up of other functions by adding, subtracting, multiplying, dividing, or applying functions to other functions, and you know how to do this operation on the component functions, then there are simple rules that you can follow mechanically do figure it out on the combined function.

At that point, you've pretty much invented half of calculus as it was understood back in the days of Newton and Leibniz, and are ready to tackle all kinds of minimization and maximization problems in physics, engineering, finance, and elsewhere, plus all kinds of problems involve rates of change, such as bacteria growth, velocity of water coming from a hole in the bottom of a leaky bucket, and rocket motion.