I was wondering if anyone could give me some real world problems that are best solved with calculus. I'm having trouble finding any specific real world problems that are not general examples. Such a problem should include the raw data needed to understand and solve the problem and not just be a general description of "such a problem." It should also be real-world, meaning the problem is not purely theoretical.
One rough example would be, if I drop a 5 kg ball on Earth, and there is no wind resistance, and Earth weighs 5.972E24 kg, how fast will the ball be going by the time it has fallen one meter? I don't know if that is all the information that would be needed to solve that problem or if that is even a calculus problem, but you should at least see what I'm looking for. All the data should be there and it should be best solved by calculus.
The more I learn about calculus, the more it seems like I've already been doing the equivalent of calculus in computer programs that I've written. I'm wondering if calculus is just a different way of solving the same problems or if there is something fundamentally unique about it.
scientific curiosity and the limits of understanding vs limits of description
I find the confusion over the structure of reality versus our description of reality quite interesting. I often wonder what primary goal of the average theoretical scientist is. Are they trying to uncover how the universe works, or are they simply trying to create a system that allows us to describe and predict how the universe will go. Can anyone really tell the difference?
I wonder if our understanding of the universe is limited by our ability to describe it. Forget comprehension. Let's just assume (however unrealistic) that any human mind can comprehend all of the mysteries of the universe. Then let's assume that we have no way of describing those universal workings to each other. We could all "get it" but we could never "talk about it".
Half of the greatest discoveries were not so much about getting new data, but about describing the data we already had. People had seen apples fall for millennia. Newton was not the first person to see an apple fall. In addition, I would bet that he was also not the first person to notice that the apple sped up or that it encountered wind resistance, etc. Humans are built to intuitively understand the concept of apples falling. However, we had no way of accurately describing it to each other. Newton was not highly praised for being able to predict the speed of the apple, say, after it fell 1 meter or 2.54 meters. Most people can intuitively guess that without calculus (they may not be able to give you a number, but with even a little practice, they'll know exactly when to put their hand out and catch it, or, with a little more practice, exactly when and where to fire an arrow to hit it). Newton was praised not for finding new data, but for describing the existing data, for inventing a system that allowed people to describe this process to each other with precision. And of course this same power could then be applied to planets and cannonballs.
My point is that the value was in the increased perfection of the description. So, again, that is why I wonder about what drives human curiosity. You can answer this question yourself. Would you rather know the shape of the universe in all its entirety and not be able to describe it to humans or control it as a human or would you rather have no idea about the basic structure of the universe and yet be handed human readable equations that give you full control over all matter and energy around you? Or, to put it more cryptically, what is the utility function of your scientific curiosity?
-swb November 29th '12