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calculus.md
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Calculus is a somewhat unpopular but immensely important area of advanced [mathematics](math.md) whose focus lies in study of **continuous change**: for example how quickly a [function](function.md) grows, how fast its growth "accelerates", in which direction a multidimensional function grows the fastest etc. This means in calculus we stop being preoccupied with actual immediate values and start focusing on their CHANGE: things like velocity, acceleration, slopes, gradients etc., in a highly generalized way. Calculus is one of the first disciplines one gets confronted with in higher math, i.e. when starting University, and for some reason it's a very feared subject among students to whom the name sounds like a curse, although the basics aren't more difficult than other areas of math (that's not to say it shouldn't be feared, just that other areas should be feared equally so). Although from high school textbooks it's easy to acquire the impression that all problems can be solved without calculus and that it will therefore be of little practical use, the opposite is in fact true: in [real world](irl.md) EVERYTHING is about change, proof of which is the fact that in [physics](physics.md) most important phenomena are described by **[differential equations](differential_equation.md)**, i.e. basically "calculus equations" -- it turns out that many things depend on rate of change of some variable rather than the variable's direct value: for example air friction depends on how fast we are moving (how quickly our position is changing), our ears hear thanks to CHANGE in air pressure, electric current gets generated by CHANGE of magnetic field etc. Calculus is very similar to (and sometimes is interchangeably used with) *mathematical analysis* (the difference is basically that analysis tries to [prove](prove.md) what calculus does, at least according to the "[Internet](internet.md)"). The word *calculus* is also sometimes used to signify any "system for making calculations", for example [lambda calculus](lambda_calculus.md).
Is this of any importance to a programmer? Fucking YES, you can't avoid it. Consider [physics engines](physics_engine.md), [machine learning](machine_learning.md), smooth [curves](curve.md) and surfaces in computer graphics, [interpolation](interpolation.md) and animation, scientific simulations, [electronics](electronics.md), [robotics](robotics.md), [signal](signal.md) processing and other kind of various shit all REQUIRE at least basics of calculus.
Is this of any importance to a programmer? Fucking YES, you can't avoid it. Consider [physics engines](physics_engine.md), [machine learning](machine_learning.md), smooth [curves](curve.md) and surfaces in computer graphics, [interpolation](interpolation.md) and animation, statistics, scientific simulations, [electronics](electronics.md), [robotics](robotics.md), [signal](signal.md) processing and other kind of various shit all REQUIRE at least basics of calculus.
In essence there are two main parts to calculus, two mathematical "operations" that work with functions and are opposite to each other: