Can you find v if you know f, and vice versa, and how? If we know the velocity over the whole history of the car, we should be able to compute the total distance traveled. In other words, if the speedometer record is complete but the odometer is missing, its information could be recovered. One way to do it (without calculus) is to put in a new odometer and drive the car all over again at the right speeds. That seems like a hard way; calculus may be easier.
I like it. Don't listen to littlebit264. It's a great quote that explains the importance of math in real life. Many people believe that the things that they learned in school are all useless, but for those of us that continue on to do bigger and better things often will use things such as calculus. Even for those of us who do not use these things the real reason that we are taught Geometry and Algebra (both of which are useless to 99% of the population) is to allow us to think. So that we can learn to think in different ways and so that we can use and open our minds. It makes us smarter.