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I am currently studying Calculus on my own for fun. I enjoy different components of math and how they can be used to solve so many problems.

Many people, however, think I am crazy because I am studying mathematics in my spare time. Many people ask me "how will this ever be useful?" and many times I cannot think of an answer. Often, people do not understand the significance of certain things in math.

For example, I would explain to someone how the Gamma function is like an extension of the factorial. The usual response is "Okay, how is that useful?" When I attempt to explain, I then need to go into many other details about other concepts.

What are some basic uses for Calculus and its functions? I know that this is really vague, but I was hoping someone may help.

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Math doesn't have to be useful. You said it yourself: You study calculus because you just enjoy it. –  Javier Badia Apr 8 '12 at 23:48
    
But then, of course mathematics were probably initially used because it has a practical application. –  Argon Apr 9 '12 at 0:02
    
When they said "How it that useful?" and you answered "Who cares?", what did they say? –  Michael Hardy Apr 9 '12 at 1:25
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I wonder how they'd respond if you asked in return how art is useful. –  KCd Apr 9 '12 at 2:20
    
That math happens to be useful is a nice side effect. That math actually works for modeling what we see in real life is a miracle in itself. –  J. M. Apr 14 '12 at 4:49
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Here is a very general but broad class of applications: Suppose you have some quantity $q(t)$ that you want to model with respect to time, like maybe a population, or a chemical concentration, or an object's speed, or whatever. Quite often there will be a natural way to describe the quantity you're interested in by using a differential equation, i.e. an equation which relates the rate of change $\frac{dq}{dt}$ of the quantity to $q(t)$ itself.

Calculus can then be used to analyze the differential equation (which could be very complicated) and hopefully give a closed-form solution so that we can predict the quantity in the long term. If an explicit solution is found, calculus can again be used to analyze the solution to find maxima and minima, and all sorts of critical points of interest.

Differential equations aren't only useful for modelling quantities, but also positions. for example, in order to fully understand how a rocket ship blasts off into space, scientists need to take into account the fact that the burning of fuel means the mass is decreasing, and so the propulsion will cause a larger acceleration. This problem leads to solving a differential equation.

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