# How would you describe calculus in simple terms?

I keep hearing about this weird type of math called calculus. I only have experience with geometry and algebra. Can you try to explain what it is to me?

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It's not a weird type of math..?? – Himadri Jul 23 '10 at 10:24

There came a time in mathematics when people encountered situations where they had to deal with really, really, really small things.

Not just small like 0.01; but small as in infinitesimally small. Think of "the smallest positive number that is still greater than zero" and you'll realize what sort of problems mathematicians began encountering.

Soon, this problem became more than just theoretical or abstract. It became very, very real.

For example, velocity. We know that average velocity is the change in position per change in time (i.e., 5 miles per hour). But what about velocity at a point in time? What does it mean to be going 5 mph at this moment?

One solution someone came up with was to say "it's the change in position divided by the change in time, where the change in time is an infinitesimally small amount of time". But how would you handle/calculate that?

Another problem came about trying to find the area under a curve. The current accepted solution was to divide the curve into rectangles, and add together the area of the rectangles. However, in order to find the exact area under the curve, you'd need to divide it into rectangles that were infinitesimally tiny, and, therefore, add up an infinite amount of tiny rectangles -- to something that was finite (area).

Calculus came about as the system of math dedicated to studying these infinitesimally small changes. In fact, I do believe some people describe calculus as "the study of continuous changes".

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This talk about "infinitesimals" would need the framework of nonstandard analysis for rigorous justification. In the epsilon-delta formulation there is no room for infinitesimals. – user1119 Aug 17 '10 at 19:34
@George I took care never to actually use the mathematical "infinitesimal" term (as a noun) in my answer, and opted for a plain-english usage of the word. I was simply trying to describe the motivations for the development of calculus in the first place, and its relation to the study of really really really "infinitesimally" small changes. – Justin L. Aug 22 '10 at 7:19
So, this means calculus is able to give answers to questions like : But what about velocity at a point in time? What does it mean to be going 5 mph at this moment? OR area under a curve ? But since the solution process deals with removal of entities which "infinitesimally" small, then how to decide, how much "perfect" the answers are ? – Vishwas G Jan 23 '13 at 20:53
@VishwasGagrani In "standard" analysis, we don't use "infinitesimals" (that is numbers which are greater than zero but less than every real number) as objects. However, we can still talk about "infinitesimal changes" via an operation called a "limit". Basically a limit tells us exactly what's happening very, very, very close to a point, without ever having to know what's actually happening AT the point. This has a rigorous definition (you'll learn about it in a class called Real Analysis) so we can be sure that we're getting the "perfect" (I assume by "perfect" you mean "correct") answer. – Bye_World Jul 13 '14 at 12:45

One of the greatest achievements of human civilization is Newton's laws of motions. The first law says that unless a force is acting then the velocity (not the position!) of objects stay constant, while the second law says that forces act by causing an acceleration (though heavy objects require more force to accellerate).

However to make sense of those laws and to apply them to real life you need to understand how to move between the following three notions:

1. Position
2. Velocity (that is the rate of change in position)
3. Acceleration (that is the rate of change of the velocity)

Moving down that list is called "taking the derivative" while moving up that list is called "taking the integral." Calculus is the study of derivatives and integerals.

In particular, if you want to figure out how objects move under some force you need to be able to integrate twice. This requires understanding a lot of calculus!

In a first semester class you usually learn about derivatives and integrals of functions of one variable, that is what you need to understand physics in 1-dimension! To understand the actual physics of the world you need to understand derivatives and integrals in 3-dimensions which requires several courses.

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Calculus is basically a way of calculating rates of changes (similar to slopes, but called derivatives in calculus), and areas, volumes, and surface areas (for starters).

It's easy to calculate these kinds of things with algebra and geometry if the shapes you're interested in are simple. For example, if you have a straight line you can calculate the slope easily. But if you want to know the slope at an arbitrary point (any random point) on the graph of some function like x-squared or some other polynomial, then you would need to use calculus. In this case, calculus gives you a way of "zooming in" on the point you're interested in to find the slope EXACTLY at that point. This is called a derivative.

If you have a cube or a sphere, you can calculate the volume and surface area easily. If you have an odd shape, you need to use calculus. You use calculus to make an infinite number of really small slices of the object you're interested in, determine the sizes of the slices, and then add all those sizes up. This process is called integration. It turns out that integration is the reverse of derivation (finding a derivative).

In summary, calculus is a tool that lets you do calculations with complicated curves, shapes, etc., that you would normally not be able to do with just algebra and geometry.

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Calculus is a field which deals with two seemingly unrelated things.

(1) the area beneath a graph and the x-axis.

(2) the slope (or gradient) of a curve at different points.

Part (1) is also called 'integration' and 'anti-differentiation', and part (2) is called 'differentiation'.

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Sure, this just about tells you what an introductory course to calculus might involve, but is far to specific to describe the subject as a whole. – Noldorin Jul 20 '10 at 21:21
I was attempting to match the degree of detail to the level of the question. If a young child asks me what a vehicle is, I might point to some cars and trucks and buses, rather than giving an exhausive definition or a complete history of how vehicles have improved in recent decades. But perhaps the community decides they prefer to have the more thorough answers, even if the questioner hasn't given the impression that they will understand them. – bryn Jul 21 '10 at 7:49
In this kind of answer I would put differentiation first and integration second, which is the way most basic calculus courses do it. Also, I think it's helpful to give brief examples from mechanics (speed and distance) as most of the other answers have done. Even so, I like the succinctness and brevity (for this level of question). – Neil Mayhew Jul 23 '10 at 5:13
Fair enough. I think I choose them in this order because I the concept of area seems more basic to understand, whereas the concept of tangents/gradients at points seems a bit more complex. But I would never teach integration before differentiation! – bryn Jul 23 '10 at 7:14

To be very brief and succinct:

Calculus is the study of how quantities change

Slightly more technically, it a subject based on infinitesimals.

It may be pointing out the obvious, but the Wikipedia article does actually provide a pretty decent beginners introduction to the subject. You'll generally want to start with differential calculus and move on quickly to integral calculus, followed by linking up the two (fundamental theorem of calculus) and moving on from there.

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Calculus is the mathematics of change. In algebra, almost nothing ever changes. Here's a comparison of some algebra vs. calc problems:

algebra: car A is driving at 50 kph. How far has it gone after 6 hours?
calc: car B starts at 10 mph and begins accelerating at the rate of 10 kph^2
(kilometers per hour per hour). How far has car B gone after 6 hours?


Note how the algebra problem nothing changes, where in the calc problem, the speed of the car is constantly changing.

calc: If a ball is rolling in a straight line at 10 fps with a diameter of 1 foot
and Q is a the point at the top of the ball when t=0, how fast is point Q moving
at time t=4 relative to the ground?


The speed of the point in relation to the ground is never the same (its zero when its at the bottom, 20fps when it's at the top. Calculus lets you figure out how fast it's going exactly at a specific moment.

There are two main branches of calculus, differential and integral. These problems pertain to differential calculus as they concern how something is changing. Integral calculus deals with how much something has changed, the opposite of differential calculus.

The number of cubic feet of oxygen circulated by Jeff lungs per hour at time t
follows the equation f(t) = t^3 + sin(t) How many cubic feet of oxygen do
Jeff's lungs cycle per day?


To find out how much something has changed when its rate of change isn't constant requires integral calculus. (the equation is purely hypothetical unless Jeff happens to be the size of a beluga whale).

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Have a look at this explanation

http://betterexplained.com/articles/a-gentle-introduction-to-learning-calculus/

I hope you love it :)

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This is the simplest explanation.

http://www.math-prof.com/Calculus_1/Calc_Ch_01.asp

ie...its simply a way of dealing with an equation where the denominator has become 0. Instead of leaving it as insoluble...you have a method of proceeding.

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