Help Needed: What is Calculus?

[Emily266]

Ok, this might be a dumb question but can anyone tell me what kind of math calculus is? I asked my current teacher but she did not give me a real answer, and anyone else I ask doesn't know.

Thanks, Emily

 

[arildno]

It's about how to calculate rates (differentiation) and accumulations (integration).
These are strongly coupled phenomena.

 

[meaw]

Emily,
Let me elaborate what arildno said.

Suppose you are tracking the distance at various time interval of a moving car from you.
The distance of the car from you varies as the following formula

X = 5t , where t is time in stopwatch.

Now if you ask what is the velocity of the car when time was 5 sec in stopwatch, it will be

v = distance travelled/ time taken

the problem here is 'distance travelled' between which two pints in time.
In ordinary maths you may take

Distance at t=5 - distance at t=6 / time taken = 1

or Distance at t=5 - distance at t=5.5 / time taken = .5

or Distance at t=5 - distance at t=5.1 / time taken = .1


right?

in calculus we reduce the time interval to a very very minuscule amount, so that we measure the velocity right at t=5.
the interval around t would reduce to such a small value that it will give the velocity at an instance t=5

That is caluculus of differential nature.

There is calculus of intergral nature too.

 

[grmnsplx]

Good question.

I'd say Calculus is a very useful branch of mathematics which lends itself very well to real physical problems. Common problems we can solve using calculus might be falling objects or the orbit of planets and moons, where our variables are time, velocity, distance, acceleration, etc.

I think it is important to note that Calculus is rigorous. By that I mean that we have some definitions and axioms to start with and then we can build all of calculus.

Before calculus, we could solve many if the same problems that we do today, except that today it's much more satisfying. Back in the day, the Greeks would use exhaustive methods.

Here is an example. Imagine a circle. Just for kicks we want to calculate the area of that circle with squares. We put one big square in the middle so that the corners touch the circle. Well, there is a lot of space not covered by the square. So let's use smaller squares. So then we use squares that are half the size and fill in the square. Still we haven't covered all of the circle. So we use smaller squares, and smaller and smaller...
But there is always some space left around the edge of the circle. Very frustrating!

And then when we are exhausted from calculating how many tiny squares will fit in the circle and have added up their areas, we give up. This is the exhaustive method. I jest.
But seriously, when we fell we are close enough we stop.

But "close enough" is just not acceptable to mathematicians. So Newton and Leibniz developed calculus. they developed a method that would allow us to continue to add up the tiny squares until their size becomes zero (we often say infinitely small or arbitrarily small).

The foundation of calculus is the limit. In the problem I stated above, we would look at the sequence of areas produced by the squares. We notice that these approximations get slightly bigger and bigger (closer to the actual area of the circle). And we take the "limit" of those approximations as those square become zero in size.

make sense?

 

[grmnsplx]

one more key point for mathematics.

the limit solves a huge problem in mathematics and that is dividing by zero.
Mathematicians avoid dividing by zero like the plague.

By taking the limit, we really don't have to worry so much about dividing by zero.