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Real Life Applications of Calculus?

GB_Packers_Ftw

GB_Packers_Ftw

LOOKS LIKE FAVRE GOT SHUT DOWN
Hey guys. this is my senior paper topic. the real life applications of Calculus. I backed out of 2 other topics because i just pick stupid topics and i dont have enough information on them. but i know theres ton's of cool stuff about calculus out there. im looking to narrow it down to aobut 3-5 Main big topics that pop up alot in real life. Ive considered things like sound waves...motion....diease and population control...and the one topic im sold on is conic sections. But any input as to what concepts are really big in the "real world" and especially any links would be awesome. thanks guys.
 
calculating the trajectory of an object moving through a fluid, the resistive force changes as the velocity changes so it becomes a related rates problem.
 
Are complex numbers allowed? If so, they are used for circuits dealing with impedance through a capacitor/inductor. If not, then calculus (basically, derivatives) are used to calculate rate of change for things like motion. You may also want to look into integrals as well (i.e. it's used to find the volume of non-geometric objects).
 
Cryptography is a big one. You can find tons of articles on it, although I am guessing "senior paper" refers to senior year of high school, so most of those articles will be beyond what you have studied. Here is one that utilizes a concept in number theory (the primary domain of cryptography) called Lambda Calculus:

http://www.seas.upenn.edu/~sweirich/types/archive/1999-2003/msg00480.html

Lambda Calculus, in extremely vulgar terms, is simply a method for dealing with functions that are able to return other functions instead of numbers, so you can collapse an arbitrary number of contingencies into a single-variable input. You can see why this is really useful for encryption, as well as for logic/decision theory.

Speaking of which, decision theory/game theory is another great application of calculus. Many times, we are interested in the optimal strategy not for a single move, but for an infinitely iterated number of moves. The theory of limits (which I'm sure you have covered) allows us to do that. A great example is Bayes's Theorem, which is used to calculate probabilities contingent on one another. For example, a farm has n turkeys and m chickens, all turkeys are brown, 10% of chickens are brown and 90% white, and you see me carrying a brown bird. What are the odds it is a chicken? This simple example can be solved with algebra, but the underlying proof results in a system that, through calculus, can produce optimal decisions for enormously, even arbitrarily complex probability systems.

Another example within game theory is Arrow's Theorem and the surrounding literature. It's an area that deals with the design of voting systems. Arrow's Theorem shows that no voting system can be both representative and fair. Ways of getting the closest system to actual fairness and representation (including the proof that the majority rule, which America uses, is a very good one) often include use of calculus for optimization or vector operations. Similarly, Duverger's Law (all systems with one legislator per district, one vote per person, and majority rule, such as America, will inevitably have exactly two parties) can be proved with calculus-based game theory. There are TONS of journal articles about that.
 
The punter on the street won't use calculus at all, at least not directly (they'll probably get devices or other trained people to do it for them).

In terms of practical uses, almost anything that has a physical implementation will use calculus (electronics, computers, plumbing, mechanics, etc.), as well as anything that goes on in the financial world, like stock and option pricing, etc. etc.
 
How about car safety? Speed vs. acceleration is basic derivative calculus, and with the recent Toyota debacle, I'm sure people are testing speeds on cars quite a bit.
 
99% of applications require calculus why do you think almost every scientist learns calc
 
Economics make heavy use of Calculus, considering that it's focused heavily on "marginal" costs/revenue/benefit/utility. Many of the concepts also apply things like the chain rule (ie, deriving substitution/income effects). Same for game theory -- Best Response functions to calculate Nash Equilibrium uses differentiation. Econ also gets a lot more math heavy as you go on, especially if you go on with Macroeconomics.

Statistics also make heavy use of Calculus -- anything involving continuous distributions will likely make some use of calculus. You just wont know it if you're just looking up numbers in the table though! Expected value of continuous random variables require integration, also when you transform any continuous random variables into another, you'll be using some results from multivariable calc. Anything involving Maximum Likelihood also involves differentiation, so that's a huge chunk of statistics past the basics.
 
Calculus is used all over. One of the first things that I didn't see upon a skim through of this topic was differential equations. Population models can often be modeled by differential equations. Circuits (as a matter of fact, I'm currently analyzing the RLC circuit in a crossover in a speaker for my end of the year project) are also an excellent application of differential equations.
 
thanks a bunch guys these are all very useful to me im still relatively new to calc so it helps a bunch.

and its also a little ironic that a bears fan is posting on my thread and he didnt even talk trash to me^^ :)
 
Res Ipsa Loquitur said:
Cryptography is a big one. You can find tons of articles on it, although I am guessing "senior paper" refers to senior year of high school, so most of those articles will be beyond what you have studied. Here is one that utilizes a concept in number theory (the primary domain of cryptography) called Lambda Calculus:

http://www.seas.upenn.edu/~sweirich/types/archive/1999-2003/msg00480.html

Lambda Calculus, in extremely vulgar terms, is simply a method for dealing with functions that are able to return other functions instead of numbers, so you can collapse an arbitrary number of contingencies into a single-variable input. You can see why this is really useful for encryption, as well as for logic/decision theory.
Click to expand...

This is "not even wrong."

Calculus and Lambda Calculus are not related in any way. The Lambda Calculus is a small system that can express any computable function (and is generally studied by computer scientists, not mathematicians). Calculus is (in very simple terms) the study of continuous functions.

Cryptography does make use of number theory. Because it makes use of number theory (the study of integers, discrete functions, and related friends) most of calculus is inapplicable. Furthermore, LC is not a concept of number theory...
 
Personally, I'd be tempted to discuss celestial mechanics, in the context of all these space probes that are being sent out. If you think you can cover the ground, you could pick a specific object (most likely a machine of some sort) and discuss how calculus goes into it by many aspects. The Mars rovers Spirit and Opportunity might be good - besides the journey to Mars, there's no doubt plenty of calculus used in designing and building rovers.
 
So far i have bruhsed over derivitives and have used Economics and medicine (drug dosage actually really interesting). Im moving onto integral calculus i need to fill about 6 more pages. the big ones are Volume, prolly arclength, conic setions, and the obvious motion problems. any other thoughths on integrals?
 
Integral calculus is used a lot in physics, for example, in finding total magnetic/electric fields around a charged body. And of course, there's the whole Area/Volume application.
It's used a lot to predict laws though, and things such as the equations of motion used in mechanics.
 
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